WEBVTT 00:00:00.030 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:04.000 Commons license. 00:00:04.000 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.680 continue to offer high-quality educational resources for free. 00:00:10.680 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.025 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.025 --> 00:00:17.650 at ocw.mit.edu. 00:00:27.100 --> 00:00:28.890 LORNA GIBSON: OK, so it's five after. 00:00:28.890 --> 00:00:31.820 We should probably start. 00:00:31.820 --> 00:00:33.999 So last time we were talking about honeycombs, 00:00:33.999 --> 00:00:35.790 and I just wanted to quickly kind of review 00:00:35.790 --> 00:00:37.623 what we had talked about, and then today I'm 00:00:37.623 --> 00:00:39.610 going to start deriving equations 00:00:39.610 --> 00:00:42.850 for the mechanical properties of the honeycombs, OK? 00:00:42.850 --> 00:00:45.290 So this is a slide of our honeycomb setup here. 00:00:45.290 --> 00:00:48.170 These are the hexagonal cells we're going to look at. 00:00:48.170 --> 00:00:50.000 We talked about the stress-strain behavior. 00:00:50.000 --> 00:00:53.120 The curves on the left-hand side are for compression 00:00:53.120 --> 00:00:55.752 and the ones on the right-hand side are for tension. 00:00:55.752 --> 00:00:57.460 And so what we're going to be doing today 00:00:57.460 --> 00:01:00.500 is we're going to start out by calculating a Young's modulus, 00:01:00.500 --> 00:01:01.950 this slope here. 00:01:01.950 --> 00:01:05.830 We're going to calculate the stress plateaus for failure 00:01:05.830 --> 00:01:08.850 by elastic buckling in elastomeric honeycombs, 00:01:08.850 --> 00:01:12.120 by failure from plastic yielding in, say, a metal honeycomb, 00:01:12.120 --> 00:01:14.350 and by failure by a brittle crushing 00:01:14.350 --> 00:01:15.984 in, say, a ceramic honeycomb. 00:01:15.984 --> 00:01:18.150 And if we have time, we'll get to the tension stuff. 00:01:18.150 --> 00:01:20.320 I don't know if we'll get to that today or next time. 00:01:20.320 --> 00:01:21.778 So we're going to start calculating 00:01:21.778 --> 00:01:22.900 those properties today. 00:01:22.900 --> 00:01:24.650 And these were the deformation mechanisms. 00:01:24.650 --> 00:01:27.150 Remember, we said the linear elastic behavior was related 00:01:27.150 --> 00:01:29.890 to bending of the cell walls, and then the plateau 00:01:29.890 --> 00:01:32.530 was related to buckling if it was an elastomer. 00:01:32.530 --> 00:01:34.460 And the plateau was related to yielding 00:01:34.460 --> 00:01:37.889 if it was, say, a metal that had a yield point. 00:01:37.889 --> 00:01:39.430 And then this was sort of an overview 00:01:39.430 --> 00:01:40.846 of the stress-strain curve showing 00:01:40.846 --> 00:01:42.674 those different regions, OK? 00:01:42.674 --> 00:01:44.590 So what I'm going to talk about today to start 00:01:44.590 --> 00:01:46.744 is the linear elastic behavior. 00:01:46.744 --> 00:01:49.160 And we're going to be starting with the in-plane behavior. 00:01:49.160 --> 00:01:52.730 So in-plane means in the plane of the hexagonal cells. 00:01:52.730 --> 00:01:55.860 And then next time we'll do the out-of-plane behavior, this way 00:01:55.860 --> 00:01:56.376 on. 00:01:56.376 --> 00:01:58.000 So if I had to form my little honeycomb 00:01:58.000 --> 00:02:00.130 like this, what initially happens 00:02:00.130 --> 00:02:02.530 is the inclined cell walls bend. 00:02:02.530 --> 00:02:05.050 So if you can see over here, we've kind of 00:02:05.050 --> 00:02:06.970 exaggerated it on this sketch. 00:02:06.970 --> 00:02:08.490 So this wall here is bent. 00:02:08.490 --> 00:02:11.220 This one here just kind of moves along, goes for the ride. 00:02:11.220 --> 00:02:12.410 And this guy here is bent. 00:02:12.410 --> 00:02:14.050 So this is for loading in the what 00:02:14.050 --> 00:02:16.590 we're calling the x1 direction, sigma 1. 00:02:16.590 --> 00:02:18.030 And the same kind of thing happens 00:02:18.030 --> 00:02:19.530 when we load in the other direction, 00:02:19.530 --> 00:02:22.510 in the sigma 2 direction, these guys still bend. 00:02:22.510 --> 00:02:24.590 Now the honeycomb gets wider that way. 00:02:24.590 --> 00:02:26.690 It gets shorter this way, wider that way. 00:02:26.690 --> 00:02:29.090 And we can calculate the Young's modulus 00:02:29.090 --> 00:02:33.310 if we can relate the load on the beam in the moments 00:02:33.310 --> 00:02:34.950 to this deflection here, right? 00:02:34.950 --> 00:02:38.610 So the Young's modulus is going to be related to the stiffness 00:02:38.610 --> 00:02:41.367 and the stiffness is going to be related to how much deformation 00:02:41.367 --> 00:02:43.950 you get for a certain amount of load that you put on the beam. 00:02:43.950 --> 00:02:46.730 So I'm going to calculate the modulus for the x1 direction, 00:02:46.730 --> 00:02:48.590 the thing on the left there. 00:02:48.590 --> 00:02:50.822 And you can do the same thing for the x2 direction 00:02:50.822 --> 00:02:52.280 on the right, but I won't calculate 00:02:52.280 --> 00:02:54.880 that because it's exactly the same kind of process. 00:02:54.880 --> 00:02:58.199 OK, so let me start here. 00:02:58.199 --> 00:02:58.740 Get my chalk. 00:03:01.710 --> 00:03:03.863 So I'm going to draw a one-unit cell here. 00:03:07.770 --> 00:03:12.660 So here's my unit cell there, like that. 00:03:12.660 --> 00:03:17.220 And this member here is of length h. 00:03:17.220 --> 00:03:21.360 That member there is of length l. 00:03:21.360 --> 00:03:24.270 That angle there is theta. 00:03:24.270 --> 00:03:26.560 I'm going to say all the walls have equal thickness, 00:03:26.560 --> 00:03:28.630 and I'm going to call it t. 00:03:28.630 --> 00:03:32.000 And I'm going to define an x1 and x2 axis like this. 00:03:32.000 --> 00:03:37.130 So the horizontal is x1 and the vertical axis is x2. 00:03:37.130 --> 00:03:41.580 And I'm going to say that I apply a sort of global stress 00:03:41.580 --> 00:03:44.380 to it, sigma 1. 00:03:44.380 --> 00:03:49.395 So there's a stress in the one direction there, sigma 1, OK? 00:03:49.395 --> 00:03:51.270 And I'm going to say my honeycomb has a depth 00:03:51.270 --> 00:03:58.530 b into the page, but the depth s-- 00:03:58.530 --> 00:04:00.622 is because the honeycomb's prismatic, 00:04:00.622 --> 00:04:03.080 the b's are always going to cancel out of all the equations 00:04:03.080 --> 00:04:05.010 that we're going to get, because everything's 00:04:05.010 --> 00:04:07.330 uniform in that direction. 00:04:07.330 --> 00:04:10.000 And we can think about a unit cell here, 00:04:10.000 --> 00:04:15.170 and in the x1 direction, we could 00:04:15.170 --> 00:04:24.700 say the length of our unit cell is 2l cos theta. 00:04:24.700 --> 00:04:28.820 So in the x1 direction, that's our unit cell there, 00:04:28.820 --> 00:04:34.296 and that's 2l cos of theta. 00:04:34.296 --> 00:04:37.020 And in the x2 direction, you might 00:04:37.020 --> 00:04:38.950 think that you go from this vertex 00:04:38.950 --> 00:04:41.150 up here down to that vertex there, 00:04:41.150 --> 00:04:43.950 but if you did that, then on the next layer of cells, 00:04:43.950 --> 00:04:47.440 you wouldn't have the same distance. 00:04:47.440 --> 00:04:51.880 So the unit length in the x2 direction 00:04:51.880 --> 00:04:54.074 is actually from here to here, and then you 00:04:54.074 --> 00:04:56.240 can see the next cell, you would get the same thing. 00:04:56.240 --> 00:04:58.440 You get this bit here from the inclined member, 00:04:58.440 --> 00:05:01.400 and then you would get h down here from the next member. 00:05:01.400 --> 00:05:08.240 So this bit here is equal to h plus l times sine of theta. 00:05:08.240 --> 00:05:11.142 So I can say in the x2 direction, 00:05:11.142 --> 00:05:18.870 the length of the unit cell is h plus l sine theta, OK? 00:05:18.870 --> 00:05:21.230 So that's kind of the setup. 00:05:21.230 --> 00:05:23.874 And then what we want to look at is that inclined member 00:05:23.874 --> 00:05:25.290 that bends, we want to look at how 00:05:25.290 --> 00:05:28.080 this guy bends under the load. 00:05:28.080 --> 00:05:32.290 And if we can relate the forces on it to the deflection, 00:05:32.290 --> 00:05:34.830 and we need the component of the deflection in the one 00:05:34.830 --> 00:05:37.940 direction, then we're going to be able to get the modulus. 00:05:37.940 --> 00:05:41.020 So I'm going to draw that inclined member again 00:05:41.020 --> 00:05:45.210 over here, and it's going to see some loads 00:05:45.210 --> 00:05:48.240 that I'm going to call p, and that's going 00:05:48.240 --> 00:05:51.620 to cause this thing to bend. 00:05:51.620 --> 00:05:54.840 So I've kind of exaggerated it there, but there's the bending. 00:05:54.840 --> 00:06:00.540 And there's some end deflection there, delta. 00:06:00.540 --> 00:06:05.910 And there's moments at either end of the beam, or either end 00:06:05.910 --> 00:06:07.590 of that member, as well. 00:06:07.590 --> 00:06:10.420 And this member here has a length. 00:06:10.420 --> 00:06:12.040 That length is l. 00:06:12.040 --> 00:06:15.570 OK, are we good? 00:06:15.570 --> 00:06:18.030 So it's just kind of the setup. 00:06:18.030 --> 00:06:19.900 And I'm going to draw the deflection 00:06:19.900 --> 00:06:21.730 delta bigger over here. 00:06:21.730 --> 00:06:23.190 So say that's delta. 00:06:23.190 --> 00:06:26.360 That's the same parallel as this guy here. 00:06:26.360 --> 00:06:30.170 What I'm going to want is the deflection in the x1 direction, 00:06:30.170 --> 00:06:32.900 and when I come to calculate the Poisson's ratio, 00:06:32.900 --> 00:06:35.820 I'm going to want the deflection in the x2 direction. 00:06:35.820 --> 00:06:40.360 And if this angle here is theta, between the horizontal 00:06:40.360 --> 00:06:42.820 and the inclined member, then this angle up here 00:06:42.820 --> 00:06:47.780 is also theta, and so this bit here is delta sine theta. 00:06:47.780 --> 00:06:50.350 And this bit here is delta cos theta. 00:06:55.130 --> 00:06:56.130 Ba-doop-ba-doop-ba-doop. 00:07:00.460 --> 00:07:02.230 So the Young's modulus is going to be 00:07:02.230 --> 00:07:04.210 the stress in the one direction divided 00:07:04.210 --> 00:07:06.000 by the strain in the one direction. 00:07:06.000 --> 00:07:08.050 So I need to get the stress and the strain 00:07:08.050 --> 00:07:09.890 in the one direction. 00:07:09.890 --> 00:07:14.020 So here the stress in the one direction. 00:07:14.020 --> 00:07:18.020 If I'm applying my load, like this, sigma 1, 00:07:18.020 --> 00:07:19.920 the stress in the one direction is 00:07:19.920 --> 00:07:23.130 going to be this load p-- so this load, 00:07:23.130 --> 00:07:26.530 say, p on this member here, divided by this length 00:07:26.530 --> 00:07:28.300 here, the unit cell length, and then 00:07:28.300 --> 00:07:32.830 divided by b into the board, the width end of the board. 00:07:32.830 --> 00:07:40.600 So sigma 1 is going to be p divided by h plus l sine theta 00:07:40.600 --> 00:07:41.337 times b. 00:07:44.660 --> 00:07:47.580 And epsilon 1 is going to be the strain in the one direction, 00:07:47.580 --> 00:07:50.120 is going to be the deformation in the one direction divided 00:07:50.120 --> 00:07:52.380 by the unit cell length in the one direction. 00:07:52.380 --> 00:07:59.630 So that's going to be delta sine theta divided by l cos theta. 00:07:59.630 --> 00:08:03.130 So here, even though I've said this unit cell is 2l cos theta, 00:08:03.130 --> 00:08:05.414 there'd be two of the members here 00:08:05.414 --> 00:08:07.330 that would be twice that deflection in the one 00:08:07.330 --> 00:08:09.240 direction. 00:08:09.240 --> 00:08:12.480 So it's, for one beam, it's delta sine theta 00:08:12.480 --> 00:08:14.520 over l cos theta. 00:08:14.520 --> 00:08:16.710 Are we OK so far? 00:08:16.710 --> 00:08:18.890 So far so good? 00:08:18.890 --> 00:08:20.450 So for the hexagons, because we're 00:08:20.450 --> 00:08:25.620 going to figure out the equations more or less exactly, 00:08:25.620 --> 00:08:28.221 we're going to keep track of all the geometrical factors. 00:08:28.221 --> 00:08:29.720 When we come to the foams, we're not 00:08:29.720 --> 00:08:31.997 going to keep track of all the geometrical factors. 00:08:31.997 --> 00:08:34.580 So one of the things that makes us look a little kind of hairy 00:08:34.580 --> 00:08:36.038 is just the fact that we're keeping 00:08:36.038 --> 00:08:38.630 track of all these sines and cosines and all the dimensions 00:08:38.630 --> 00:08:39.802 and things. 00:08:39.802 --> 00:08:42.696 All right, whoop. 00:08:48.160 --> 00:08:52.050 So I need to be able to relate my load p to my deformation 00:08:52.050 --> 00:08:55.820 delta to get a stiffness out of this, to get a modulus, OK? 00:08:55.820 --> 00:08:57.270 So the way I do that is, remember 00:08:57.270 --> 00:08:59.360 in 3032, we did those bending moment diagrams 00:08:59.360 --> 00:09:01.280 and we did the deflection of the beams? 00:09:01.280 --> 00:09:03.210 This is where this comes in handy. 00:09:03.210 --> 00:09:05.990 So I'm going to draw my beam a little bit differently now. 00:09:05.990 --> 00:09:08.050 I'm going to turn it on its side. 00:09:08.050 --> 00:09:11.190 So this is still my length l, but I'm 00:09:11.190 --> 00:09:13.410 going to turn on my side just so that you 00:09:13.410 --> 00:09:15.390 can see it the same kind of way we did 00:09:15.390 --> 00:09:16.580 the bending moment diagrams. 00:09:20.160 --> 00:09:24.620 So this is still my length l across here. 00:09:24.620 --> 00:09:32.100 And there's end moments, M and M here. 00:09:32.100 --> 00:09:35.060 And P sin theta is just the perpendicular component 00:09:35.060 --> 00:09:35.940 of the load p. 00:09:35.940 --> 00:09:38.260 So p sin theta is just the component perpendicular 00:09:38.260 --> 00:09:39.470 to my beam. 00:09:46.970 --> 00:09:48.760 So I could draw a shear diagram here 00:09:48.760 --> 00:09:51.070 and I could draw a bending moment diagram here. 00:09:51.070 --> 00:09:54.830 And, if you remember, the shear diagram, 00:09:54.830 --> 00:09:57.920 if I have no concentrated load along here, 00:09:57.920 --> 00:10:02.200 and I have no distributed load along here, if this is zero, 00:10:02.200 --> 00:10:06.760 down here, it's just going to go up my P sin theta, 00:10:06.760 --> 00:10:12.590 and then be horizontal, and then come down by P sin theta, OK? 00:10:12.590 --> 00:10:14.300 So that's the shear diagram. 00:10:14.300 --> 00:10:20.614 And then the bending moment diagram, 00:10:20.614 --> 00:10:21.780 I'm going to draw down here. 00:10:25.010 --> 00:10:27.850 So I've got some moment at the end here, 00:10:27.850 --> 00:10:31.912 and this would tend to bend like that. 00:10:31.912 --> 00:10:33.370 So this would be a negative moment. 00:10:33.370 --> 00:10:35.869 Remember, bending moments were negative 00:10:35.869 --> 00:10:38.160 if there was tension on the top, and they were positive 00:10:38.160 --> 00:10:39.618 if there was tension on the bottom. 00:10:39.618 --> 00:10:41.730 So over here we'd have tension on the top, 00:10:41.730 --> 00:10:44.100 so that would give us a negative bending moment. 00:10:44.100 --> 00:10:47.470 And then, if you also remember, the moment 00:10:47.470 --> 00:10:52.100 at a particular point is equal to the integral from, say, A 00:10:52.100 --> 00:10:54.760 to B-- well maybe I should write this another way. 00:10:54.760 --> 00:10:59.460 And B minus Ma is the integral of the shear 00:10:59.460 --> 00:11:02.829 diagram between the two points. 00:11:02.829 --> 00:11:03.495 A little sloppy. 00:11:06.810 --> 00:11:08.020 OK. 00:11:08.020 --> 00:11:11.610 So if I know how I have some moment here minus M, 00:11:11.610 --> 00:11:14.680 if I integrate this shear diagram up, 00:11:14.680 --> 00:11:17.630 then this is just going to be linear here, 00:11:17.630 --> 00:11:20.667 and then I'm going to be at plus M over there. 00:11:30.250 --> 00:11:33.760 So if you look at this shear and bending moment diagram, 00:11:33.760 --> 00:11:35.790 it's really just the same as the shear 00:11:35.790 --> 00:11:38.060 and bending moment diagram for two cantilevers that 00:11:38.060 --> 00:11:39.720 are attached to each other. 00:11:39.720 --> 00:11:44.730 So let me just draw over here what the cantilever looks like. 00:11:44.730 --> 00:11:45.670 Let's see. 00:11:45.670 --> 00:11:49.250 So imagine I just had a cantilever like this, 00:11:49.250 --> 00:11:52.680 and I have some force F on it like that. 00:11:52.680 --> 00:11:56.290 And I call this distance here capital L, 00:11:56.290 --> 00:11:59.940 and I'm going to call that deflection capital delta, 00:11:59.940 --> 00:12:01.500 like that. 00:12:01.500 --> 00:12:04.220 If I drew the shear diagram for that, 00:12:04.220 --> 00:12:07.710 there'd be a reaction here, F, there would be a moment here, 00:12:07.710 --> 00:12:10.030 FL. 00:12:10.030 --> 00:12:12.820 Doot, doot, yup. 00:12:12.820 --> 00:12:18.870 So this would look-- whoops-- it's a little too long. 00:12:18.870 --> 00:12:21.730 Shear diagram here would look like this. 00:12:21.730 --> 00:12:23.280 That would be zero. 00:12:23.280 --> 00:12:25.700 This would be FL. 00:12:25.700 --> 00:12:28.160 And the moment diagram would look like this. 00:12:28.160 --> 00:12:29.650 Whoops. 00:12:29.650 --> 00:12:33.070 A little too long again. 00:12:33.070 --> 00:12:35.092 And that would be minus FL. 00:12:35.092 --> 00:12:36.050 And that would be zero. 00:12:40.190 --> 00:12:44.100 So do you see how the shear and the bending moment diagram 00:12:44.100 --> 00:12:47.710 here are really just like two cantilevers, OK? 00:12:47.710 --> 00:12:54.600 So I know that the deflection for a cantilever, delta, 00:12:54.600 --> 00:13:01.560 is equal to F capital L cubed over 3EI. 00:13:01.560 --> 00:13:04.800 It's kind of a standard result. And so I 00:13:04.800 --> 00:13:07.710 can take this and apply that to this beam here. 00:13:07.710 --> 00:13:10.330 So instead of working everything out from first principles, 00:13:10.330 --> 00:13:12.820 I'm just going to say that my beam here 00:13:12.820 --> 00:13:16.840 is like two cantilevers, and instead of F, I've 00:13:16.840 --> 00:13:18.770 got P sin theta. 00:13:18.770 --> 00:13:21.840 And instead of capital L here as the length, 00:13:21.840 --> 00:13:25.889 I've got l/2 because l/2 would be the length of one 00:13:25.889 --> 00:13:26.680 of the cantilevers. 00:13:29.576 --> 00:13:38.700 OK, so for the honeycomb, I've got 00:13:38.700 --> 00:13:49.770 two cantilevers of length l/2. 00:13:49.770 --> 00:13:54.440 So delta for the inclined member on the honeycomb 00:13:54.440 --> 00:13:58.330 is going to be 2-- because I've got 00:13:58.330 --> 00:14:02.710 two cantilevers-- the force, instead of having F, 00:14:02.710 --> 00:14:04.930 I'm going to have P sin theta. 00:14:04.930 --> 00:14:07.280 And instead of having capital L, I'm 00:14:07.280 --> 00:14:11.770 going to have l/2, all cubed. 00:14:11.770 --> 00:14:17.990 So this is like F capital L cubed over 3. 00:14:17.990 --> 00:14:20.400 And here, the modulus that I want 00:14:20.400 --> 00:14:23.040 is the modulus of the solid cell wall material, 00:14:23.040 --> 00:14:29.140 so I'm going to call that ES, and over the moment of inertia. 00:14:29.140 --> 00:14:30.940 So you see how I've done it? 00:14:30.940 --> 00:14:32.370 Is that OK? 00:14:32.370 --> 00:14:35.600 So then I can just kind of simplify this thing here. 00:14:35.600 --> 00:14:40.580 I've got P sin theta l cubed. 00:14:40.580 --> 00:14:43.600 1/2 cubed is going to be 1/8. 00:14:43.600 --> 00:14:48.335 So this is going to be 2, if that's 1/8, times 3 is 24. 00:14:48.335 --> 00:14:49.785 So 2/24 is 12. 00:14:52.712 --> 00:14:56.220 So I've got delta for my honeycomb member 00:14:56.220 --> 00:15:00.620 is P sin theta l cubed over 12 EsI. 00:15:00.620 --> 00:15:08.080 And here I is the moment of inertia of that inclined 00:15:08.080 --> 00:15:12.540 member of the honeycomb. 00:15:12.540 --> 00:15:15.070 And that's BT cubed over 12, OK? 00:15:15.070 --> 00:15:19.125 So B is the depth into the board, and T is the thickness. 00:15:19.125 --> 00:15:20.990 We'll cube that and divide by 12. 00:15:20.990 --> 00:15:22.115 It's a rectangular section. 00:15:22.115 --> 00:15:22.818 Yeah? 00:15:22.818 --> 00:15:24.424 AUDIENCE: What was ES again? 00:15:24.424 --> 00:15:26.590 LORNA GIBSON: ES is the Young's modulus of the solid 00:15:26.590 --> 00:15:27.890 that it's made from. 00:15:27.890 --> 00:15:30.790 So clearly, if my honeycomb is made up 00:15:30.790 --> 00:15:33.900 of these members, whatever material 00:15:33.900 --> 00:15:36.370 the members are made of is going to affect the stiffness 00:15:36.370 --> 00:15:38.410 of the whole thing. 00:15:38.410 --> 00:15:40.410 Are we good? 00:15:40.410 --> 00:15:42.630 Because once we have this part, then we just 00:15:42.630 --> 00:15:45.260 combine these equations for the stress 00:15:45.260 --> 00:15:47.510 and the strain in the one direction. 00:15:47.510 --> 00:15:51.010 And we have this equation relating delta and P, 00:15:51.010 --> 00:15:54.500 and we're going to be able to get our Young's modulus, OK? 00:15:54.500 --> 00:15:55.910 We're happy? 00:15:55.910 --> 00:15:56.540 OK. 00:15:56.540 --> 00:15:57.390 All right. 00:16:07.206 --> 00:16:09.830 So I'm going to call the Young's modulus in the one direction E 00:16:09.830 --> 00:16:10.640 star 1. 00:16:10.640 --> 00:16:14.240 So everything with a star refers to a cellular solid property, 00:16:14.240 --> 00:16:16.550 and 1 because it's in one direction. 00:16:16.550 --> 00:16:21.030 So that's going to be sigma 1 over epsilon 1. 00:16:21.030 --> 00:16:22.870 So if I go back up there, I can say 00:16:22.870 --> 00:16:31.550 sigma 1 is equal to P divided by h plus l sin theta b. 00:16:31.550 --> 00:16:38.880 And epsilon 1 is equal to delta sine theta in the denominator 00:16:38.880 --> 00:16:40.030 over l cos theta. 00:16:45.620 --> 00:16:47.400 And now instead of having delta here, 00:16:47.400 --> 00:16:50.389 I can substitute this thing here in for delta. 00:16:50.389 --> 00:16:52.430 And then I'm going to able to cancel the P's out. 00:17:03.650 --> 00:17:14.430 So delta was equal to P sine theta l cubed over 12 Es, 00:17:14.430 --> 00:17:17.160 and there was an I, a moment of inertia, 00:17:17.160 --> 00:17:20.941 and I was equal to bt cubed over 12. 00:17:20.941 --> 00:17:23.690 And let's see here. 00:17:23.690 --> 00:17:25.329 So that's delta. 00:17:25.329 --> 00:17:26.793 And there's another sine theta here 00:17:26.793 --> 00:17:28.876 so I'm just going to square that sine theta there. 00:17:33.570 --> 00:17:36.100 So now the P's cancel out. 00:17:36.100 --> 00:17:38.195 The b's are going to cancel out. 00:17:38.195 --> 00:17:41.400 The 12s are going to cancel out. 00:17:41.400 --> 00:17:44.770 And I'm going to rearrange this a little bit. 00:17:44.770 --> 00:17:47.640 So I'm going to write Young's modulus 00:17:47.640 --> 00:17:50.530 of the solid out in the front. 00:17:50.530 --> 00:17:53.660 Then I've got a term here of t cubed 00:17:53.660 --> 00:17:56.879 and I'm going to multiply that by 1/l squared, 00:17:56.879 --> 00:17:58.670 and then everything else-- well, let's see. 00:17:58.670 --> 00:18:00.620 We can take this l cubed here. 00:18:00.620 --> 00:18:02.856 I can take that. 00:18:02.856 --> 00:18:04.230 Put it underneath that, so that's 00:18:04.230 --> 00:18:09.000 going to give me t/l cubed. 00:18:09.000 --> 00:18:11.249 And then I've got an h plus l sine theta here, 00:18:11.249 --> 00:18:12.790 and I've got an l there, so I'm going 00:18:12.790 --> 00:18:17.730 to take that to be h/l plus sine theta. 00:18:24.850 --> 00:18:25.600 Boop-boop-da-doop. 00:18:30.000 --> 00:18:32.760 So I've got this term with [? h/l's ?] in the thetas. 00:18:32.760 --> 00:18:35.570 There's a cos theta from the numerator here. 00:18:35.570 --> 00:18:38.554 This term here turns into h/l plus sine theta, 00:18:38.554 --> 00:18:40.720 and then I've got my sine squared thetas down there. 00:18:43.860 --> 00:18:46.950 And that's my result for the Young's modulus 00:18:46.950 --> 00:18:47.865 in the one direction. 00:18:51.000 --> 00:18:51.500 OK? 00:18:54.452 --> 00:18:56.330 Let's make sure that seems right. 00:18:56.330 --> 00:18:57.630 It seems good. 00:18:57.630 --> 00:18:58.700 OK. 00:18:58.700 --> 00:19:00.350 So one of the things to notice here 00:19:00.350 --> 00:19:04.260 is there's three types of parameters that are important. 00:19:04.260 --> 00:19:07.230 So one is the solid properties. 00:19:07.230 --> 00:19:11.192 So the Young's modulus of the solid comes into this. 00:19:11.192 --> 00:19:12.650 So the stiffness of the whole thing 00:19:12.650 --> 00:19:14.990 depends on the stiffness of whatever it's made from. 00:19:14.990 --> 00:19:18.040 There's this factor of t/l cubed-- 00:19:18.040 --> 00:19:21.020 that's directly related to the relative density or the volume 00:19:21.020 --> 00:19:23.330 fraction of solids. 00:19:23.330 --> 00:19:27.090 So what this is saying is the relative density goes as t/l, 00:19:27.090 --> 00:19:30.230 so the Young's modulus depends on the cube 00:19:30.230 --> 00:19:31.230 of the relative density. 00:19:31.230 --> 00:19:33.520 So it's very sensitive to the relative density. 00:19:33.520 --> 00:19:36.190 And then this factor here really is just 00:19:36.190 --> 00:19:39.069 a factor that depends on the cell geometry. 00:19:39.069 --> 00:19:41.610 Remember when we talked about the structure of the honeycomb, 00:19:41.610 --> 00:19:44.770 we said we could define the cell geometry by the ratio of h/l 00:19:44.770 --> 00:19:50.160 and theta, OK? 00:19:50.160 --> 00:19:54.260 And since we often deal with regular hexagonal honeycombs, 00:19:54.260 --> 00:19:56.030 I'm just going to write down what 00:19:56.030 --> 00:19:59.890 this works out to be for regular hexagonal honeycombs. 00:20:17.630 --> 00:20:20.640 So for a regular hexagonal honeycomb, h/l is 1. 00:20:20.640 --> 00:20:22.940 All the members have the same length. 00:20:22.940 --> 00:20:25.300 And theta's 3, and the modulus works out 00:20:25.300 --> 00:20:30.510 to 4 over root 3 times Es times t/l cubed, OK? 00:20:30.510 --> 00:20:32.080 So do you see how we do these things? 00:20:32.080 --> 00:20:35.460 So all the other properties work in a similar kind of way. 00:20:35.460 --> 00:20:38.050 You have to say something about what the sort of bulk stress 00:20:38.050 --> 00:20:39.770 is on the whole thing and relate that 00:20:39.770 --> 00:20:41.880 to the loads on the members. 00:20:41.880 --> 00:20:44.559 You have to say something about how the loads are related 00:20:44.559 --> 00:20:46.600 to deflections, or when we look at the strengths, 00:20:46.600 --> 00:20:47.860 we're going to look at moments and how 00:20:47.860 --> 00:20:49.480 the moments are related to failure 00:20:49.480 --> 00:20:51.170 moments of one sort or another. 00:20:51.170 --> 00:20:55.010 But it's all just like a little structural analysis, OK? 00:20:55.010 --> 00:20:56.046 Are we good? 00:20:56.046 --> 00:20:56.935 You good, Teddy? 00:20:56.935 --> 00:20:58.810 I thought you were going to put your hand up? 00:20:58.810 --> 00:20:59.690 No? 00:20:59.690 --> 00:21:00.210 You're OK? 00:21:00.210 --> 00:21:00.710 OK. 00:21:03.420 --> 00:21:06.850 OK, so the next property we're going to look at 00:21:06.850 --> 00:21:08.056 is Poisson's ratio. 00:21:14.179 --> 00:21:16.720 And I'm going to look at it for loading in the one direction. 00:21:36.250 --> 00:21:41.029 So Poisson's 1 2, say we load uniaxially 00:21:41.029 --> 00:21:42.570 in the one direction, we want to know 00:21:42.570 --> 00:21:44.700 what the strain is in the two direction, 00:21:44.700 --> 00:21:47.840 it's minus epsilon 2 over epsilon 1. 00:21:47.840 --> 00:21:51.610 And again, if I look at my inclined member, 00:21:51.610 --> 00:21:56.450 and I say that member's going to bend something like that, 00:21:56.450 --> 00:22:01.226 and that's my deflection delta there, and, say, 00:22:01.226 --> 00:22:04.650 got the same x1 and x2 axes. 00:22:04.650 --> 00:22:07.112 And again, if I look at delta here, 00:22:07.112 --> 00:22:08.980 it's the same little sketch I had before. 00:22:08.980 --> 00:22:12.540 That's delta sine theta. 00:22:12.540 --> 00:22:14.952 And this is delta cos theta. 00:22:14.952 --> 00:22:16.660 I'm going to need those components to get 00:22:16.660 --> 00:22:20.920 the two strains in the different directions. 00:22:20.920 --> 00:22:27.144 So epsilon 1 is going to be delta sine theta over l cos 00:22:27.144 --> 00:22:28.380 theta. 00:22:28.380 --> 00:22:32.660 And if I'm compressing it, that would get shorter. 00:22:32.660 --> 00:22:39.660 And we get-- and epsilon 2 is going 00:22:39.660 --> 00:22:49.520 to be delta cos theta divided by h plus l sine theta. 00:22:49.520 --> 00:22:50.797 And that would get longer. 00:22:58.000 --> 00:22:59.650 So these two have opposite signs, 00:22:59.650 --> 00:23:01.845 and so the minus sign is going to disappear here. 00:23:06.249 --> 00:23:07.040 Doodle-doodle-doot. 00:23:10.340 --> 00:23:11.859 So then I can get my Poisson's ratio 00:23:11.859 --> 00:23:13.650 by just taking the ratio of those two guys. 00:23:16.810 --> 00:23:18.560 So I could put a minus sign there 00:23:18.560 --> 00:23:22.390 and say that's the opposite sign to epsilon 2. 00:23:22.390 --> 00:23:29.820 Then this would be delta cos theta divided by h plus l sine 00:23:29.820 --> 00:23:30.990 theta. 00:23:30.990 --> 00:23:41.320 And epsilon 1 would be delta sine theta over l cos theta. 00:23:41.320 --> 00:23:43.550 And the thing that's convenient here 00:23:43.550 --> 00:23:45.810 is that the two deltas just cancel out. 00:23:45.810 --> 00:23:48.939 So the Poisson's ratio is the ratio of two strains. 00:23:48.939 --> 00:23:51.480 Each one of the strains is going to be proportional to delta, 00:23:51.480 --> 00:23:54.170 and so the two deltas are just going to cancel out. 00:23:54.170 --> 00:23:57.080 And so I can rewrite this thing here 00:23:57.080 --> 00:24:04.810 as cos squared theta divided by h/l plus sine 00:24:04.810 --> 00:24:10.980 theta times sine theta. 00:24:10.980 --> 00:24:12.480 And so one of the interesting things 00:24:12.480 --> 00:24:14.370 to notice that the Poisson's ratio only 00:24:14.370 --> 00:24:16.070 depends on the cell geometry. 00:24:16.070 --> 00:24:19.380 It doesn't depend on what solid the material is made from. 00:24:19.380 --> 00:24:21.550 It doesn't depend on the relative density. 00:24:21.550 --> 00:24:26.015 It only depends on the cell geometry. 00:24:49.387 --> 00:24:49.887 Oops. 00:25:25.790 --> 00:25:27.270 OK, and then we can also work out 00:25:27.270 --> 00:25:29.925 what the value is for a regular hexagonal cell. 00:25:36.610 --> 00:25:41.790 And if we plug-in h is equal to l and theta's equal to 30, 00:25:41.790 --> 00:25:43.670 you get that it's equal to 1. 00:25:43.670 --> 00:25:47.140 So one is kind of an unusual number for a Poisson's ratio. 00:25:47.140 --> 00:25:50.320 When we think of most materials, it's around 0.3, 00:25:50.320 --> 00:25:52.444 so it's kind of unusual that it's that large. 00:25:52.444 --> 00:25:53.860 The other thing that's interesting 00:25:53.860 --> 00:25:55.950 is that it can be negative. 00:26:06.950 --> 00:26:10.675 So if theta is less than 0, then you can get a negative value. 00:26:10.675 --> 00:26:12.880 If the cos squared is going to be a positive value, 00:26:12.880 --> 00:26:15.500 but you've got a sine theta down here, then that's going 00:26:15.500 --> 00:26:17.610 to give you a negative value. 00:26:17.610 --> 00:26:20.000 So you can get negative values. 00:26:20.000 --> 00:26:23.280 So let me just plug in an example. 00:26:23.280 --> 00:26:27.020 So say h/l is equal to 2, and theta 00:26:27.020 --> 00:26:36.170 is equal to minus 30 degrees, then this turns out to be 3/4. 00:26:36.170 --> 00:26:41.850 So cos of 30 is root 3/2, so square of that is 3/4. 00:26:41.850 --> 00:26:45.510 h/l is 2, sine theta is 1/2, but it's minus 1/2. 00:26:45.510 --> 00:26:49.950 So 2 minus 1/2 is 1 and 1/2. 00:26:49.950 --> 00:26:53.810 And then the sine theta is minus 1/2. 00:26:53.810 --> 00:26:55.380 And so it works out to be minus 1 00:26:55.380 --> 00:26:59.040 for that particular combination. 00:26:59.040 --> 00:27:00.800 And I brought my little honeycomb 00:27:00.800 --> 00:27:03.480 that has a negative Poisson's ratio in. 00:27:03.480 --> 00:27:05.079 So this guy here-- let's see, I don't 00:27:05.079 --> 00:27:06.370 think there's an overhead here. 00:27:06.370 --> 00:27:06.960 No overhead? 00:27:06.960 --> 00:27:07.660 Guess not. 00:27:07.660 --> 00:27:08.740 I'll just pass it around. 00:27:08.740 --> 00:27:12.240 So if you take it, put your hands on the flat side 00:27:12.240 --> 00:27:14.449 and load it like this, and don't smoosh it like that. 00:27:14.449 --> 00:27:16.740 Just load it a little bit, because you [? want to be ?] 00:27:16.740 --> 00:27:17.400 linear elastic. 00:27:17.400 --> 00:27:18.550 If you load it just a little bit, 00:27:18.550 --> 00:27:20.258 you can see that as you push it this way, 00:27:20.258 --> 00:27:22.150 it contracts in sideways that way. 00:27:22.150 --> 00:27:22.904 So don't smash it. 00:27:22.904 --> 00:27:25.070 Just load it a little bit and you can kind of see it 00:27:25.070 --> 00:27:25.980 with your hands. 00:27:25.980 --> 00:27:27.780 And if you put on a piece of lined paper, 00:27:27.780 --> 00:27:30.250 it's easier to see it. 00:27:30.250 --> 00:27:33.510 OK, so that's kind of interesting. 00:27:33.510 --> 00:27:36.216 So are we good with getting the Young's modulus in the one 00:27:36.216 --> 00:27:37.590 direction and the Poisson's ratio 00:27:37.590 --> 00:27:39.230 for loading in one direction? 00:27:39.230 --> 00:27:41.040 OK, so you can do the same sort of thing 00:27:41.040 --> 00:27:43.090 to get the Young's modulus in the two 00:27:43.090 --> 00:27:44.530 direction and the Poisson's ratio 00:27:44.530 --> 00:27:45.620 for loading in the two direction. 00:27:45.620 --> 00:27:47.286 And you get slightly different formulas, 00:27:47.286 --> 00:27:49.280 but it's the same idea. 00:27:49.280 --> 00:27:51.220 And you can also get a shear modulus this way, 00:27:51.220 --> 00:27:52.604 and in-plane shear modulus. 00:27:52.604 --> 00:27:55.020 It's a little bit-- the geometry of it's a little bit more 00:27:55.020 --> 00:27:55.950 complicated. 00:27:55.950 --> 00:27:58.590 So all of those things are derived in the book, 00:27:58.590 --> 00:27:59.830 in the cellular solids book. 00:27:59.830 --> 00:28:02.500 So if you wanted to figure those out, look at that, 00:28:02.500 --> 00:28:03.920 you could look at the book. 00:28:03.920 --> 00:28:06.149 So let me just comment on that. 00:28:36.570 --> 00:28:42.760 All right, so those are the in-plane linear elastic moduli, 00:28:42.760 --> 00:28:47.320 and remember we said that four of them 00:28:47.320 --> 00:28:51.410 describe the in-plane properties for an anisotropic honeycomb. 00:28:51.410 --> 00:28:53.830 And you can use that reciprocal relationship 00:28:53.830 --> 00:28:57.985 to relate the two Young's moduli and the two Poisson's ratios. 00:28:57.985 --> 00:29:00.110 All right, so the next thing I wanted to talk about 00:29:00.110 --> 00:29:01.580 was the compressive strength. 00:29:01.580 --> 00:29:05.000 So let me just back up here a second. 00:29:05.000 --> 00:29:06.860 So if we go back to here, remember 00:29:06.860 --> 00:29:09.780 we had for an elastomeric honeycomb, 00:29:09.780 --> 00:29:12.480 this stress plateau was related to elastic buckling. 00:29:12.480 --> 00:29:15.430 So we're going to look at that buckling stress first. 00:29:15.430 --> 00:29:17.700 And this plateau here is related to yielding. 00:29:17.700 --> 00:29:20.480 And then we'll look at the yielding stress next. 00:29:20.480 --> 00:29:22.690 And then this plateau here is related 00:29:22.690 --> 00:29:27.600 to a brittle sort of crushing, and we'll do that one third. 00:29:27.600 --> 00:29:30.400 So we're going to go through each of those next. 00:29:30.400 --> 00:29:33.612 And this is kind of a schematic for the elastic buckling. 00:29:33.612 --> 00:29:35.320 So when you look at the elastic buckling, 00:29:35.320 --> 00:29:37.370 one of the things to note is that when 00:29:37.370 --> 00:29:38.850 you load the honeycomb this way on, 00:29:38.850 --> 00:29:40.350 if you load it in the one direction, 00:29:40.350 --> 00:29:41.330 you don't get buckling. 00:29:41.330 --> 00:29:43.115 It just sort of continues to-- whoops, 00:29:43.115 --> 00:29:44.742 if I can keep it in plane. 00:29:44.742 --> 00:29:46.450 It all just kind of folds up, so you just 00:29:46.450 --> 00:29:48.240 get larger and larger bending deflections. 00:29:48.240 --> 00:29:49.510 You don't really get buckling. 00:29:49.510 --> 00:29:52.620 But when you load it this way on, these vertical members 00:29:52.620 --> 00:29:55.280 here, the ones of length h, they're going to buckle. 00:29:55.280 --> 00:29:59.320 So, see if I do that, my honeycomb looks 00:29:59.320 --> 00:30:04.170 like those cells up on the schematic there, OK? 00:30:04.170 --> 00:30:05.594 So, whoops. 00:30:13.788 --> 00:30:16.571 So we're going to look at the compressive stress or strength 00:30:16.571 --> 00:30:17.070 next. 00:30:27.040 --> 00:30:28.871 That's sometimes called the plateau stress. 00:30:43.540 --> 00:30:46.180 So we can get cell collapse by elastic buckling, 00:30:46.180 --> 00:30:49.800 if, for instance, the honeycomb is made of a polymer. 00:30:49.800 --> 00:30:52.790 And then the stress-strain curve looks something like that. 00:30:52.790 --> 00:30:54.550 And what happens is you get buckling 00:30:54.550 --> 00:31:28.270 of those vertical struts throughout the honeycomb 00:31:28.270 --> 00:31:35.130 And then you could also get a stress plateau 00:31:35.130 --> 00:31:36.290 by plastic yielding. 00:31:41.094 --> 00:31:43.010 And what happens when you get plastic yielding 00:31:43.010 --> 00:31:45.100 is you get localization of the deformation. 00:31:45.100 --> 00:31:48.390 So one band of cells will begin to yield initially, and then 00:31:48.390 --> 00:31:51.780 as the deformation proceeds, that deformation ban will 00:31:51.780 --> 00:31:53.690 propagate and get bigger and bigger, 00:31:53.690 --> 00:31:55.310 and you get a wider and wider band 00:31:55.310 --> 00:31:57.430 of cells yielding and failing. 00:32:04.100 --> 00:32:06.300 So you get localization of yield, 00:32:06.300 --> 00:32:12.870 and then as deformation progresses, 00:32:12.870 --> 00:32:16.010 the deformation band widens throughout the material. 00:32:27.310 --> 00:32:31.110 So if I go back and look-- if I look at this one 00:32:31.110 --> 00:32:35.510 here, when you look at this middle picture here, 00:32:35.510 --> 00:32:38.430 you can see how one band of cells has started to collapse 00:32:38.430 --> 00:32:39.880 and started to fail. 00:32:39.880 --> 00:32:42.820 And as you continue to compress that in the one direction, 00:32:42.820 --> 00:32:45.382 this way on, then more and more neighboring cells 00:32:45.382 --> 00:32:47.090 are going to collapse and the whole thing 00:32:47.090 --> 00:32:50.136 will get wider until the whole thing has collapsed. 00:32:50.136 --> 00:32:51.510 And that's kind of characteristic 00:32:51.510 --> 00:32:53.040 of the plastic failure. 00:32:57.040 --> 00:33:00.530 And then the third possibility is brittle crushing. 00:33:16.510 --> 00:33:19.580 And then you get these kind of serrated plateau. 00:33:19.580 --> 00:33:22.540 And the peaks and valleys correspond to fractures 00:33:22.540 --> 00:33:23.920 of individual cell walls. 00:34:31.731 --> 00:34:33.230 OK, so we're going to start off with 00:34:33.230 --> 00:34:36.120 the elastic buckling failure. 00:34:56.980 --> 00:35:00.160 And I'm going to call these plateau stresses sigma star, 00:35:00.160 --> 00:35:02.130 for the sort of compressive strength. 00:35:02.130 --> 00:35:04.640 And el means it's by elastic buckling. 00:35:04.640 --> 00:35:07.260 And as I mentioned, you don't get it in the one direction. 00:35:07.260 --> 00:35:08.270 The cells just fold up. 00:35:08.270 --> 00:35:10.311 You only get it for loading in the two direction, 00:35:10.311 --> 00:35:11.930 so it's going to be sigma star el 2. 00:35:16.820 --> 00:35:19.680 Oops, need a different piece of chalk. 00:36:35.910 --> 00:36:38.270 So you get this elastic buckling for loading 00:36:38.270 --> 00:36:42.700 in the x2 direction, and the cell walls of length h buckle. 00:36:42.700 --> 00:36:44.950 And you don't get it for loading in the one direction, 00:36:44.950 --> 00:36:46.710 the cells just fold up. 00:36:46.710 --> 00:36:50.040 So again, let me draw a little kind of unit cell here. 00:37:01.930 --> 00:37:07.370 And here is our stress sigma 2, like that. 00:37:07.370 --> 00:37:10.650 And here's our little wall of length h 00:37:10.650 --> 00:37:12.050 that's going to buckle. 00:37:12.050 --> 00:37:14.480 So if I load it up, initially it'll be linear elastic. 00:37:14.480 --> 00:37:16.120 And then eventually, at some stress, 00:37:16.120 --> 00:37:18.970 it will get large enough that this wall here will buckle. 00:37:18.970 --> 00:37:22.500 And we can relate to that plateau 00:37:22.500 --> 00:37:24.230 stress, or that compressive stress, 00:37:24.230 --> 00:37:25.750 to some Euler buckling load. 00:37:30.350 --> 00:37:33.390 So you remember, if we have a pin-ended column, so just 00:37:33.390 --> 00:37:36.030 a single column, pins on either end, 00:37:36.030 --> 00:37:37.870 the Euler buckling load says you get 00:37:37.870 --> 00:37:42.820 buckling when the critical load is equal to some end constraint 00:37:42.820 --> 00:37:44.350 factor, n squared. 00:37:44.350 --> 00:37:50.330 So n squared pi squared E, and here it's E of the solid, 00:37:50.330 --> 00:37:54.410 I over the length of the column, and in this case, 00:37:54.410 --> 00:37:57.132 the column length is h-- so h squared. 00:37:57.132 --> 00:38:00.310 OK, so that's just the Euler formula. 00:38:00.310 --> 00:38:02.480 And here, n is an end constraint factor. 00:38:10.164 --> 00:38:15.552 And if you remember for a pin column, 00:38:15.552 --> 00:38:19.020 so if our column is pinned at both ends like that, 00:38:19.020 --> 00:38:24.150 and just buckles out like that, then n is equal to 1. 00:38:24.150 --> 00:38:30.970 And if the column is fixed at both ends, 00:38:30.970 --> 00:38:35.530 something like that, then the column looks like that 00:38:35.530 --> 00:38:39.700 and then it's equal to 2, OK? 00:38:39.700 --> 00:38:43.680 So if I know what the end condition is, I know what n is 00:38:43.680 --> 00:38:46.700 and I can use my Euler formula here. 00:38:46.700 --> 00:38:50.575 So the trick to this is that it's not so obvious what n is. 00:38:50.575 --> 00:38:51.504 Yes? 00:38:51.504 --> 00:38:54.468 AUDIENCE: So, when you're loading in the x2 direction 00:38:54.468 --> 00:38:56.410 here, the first thing you're going to get 00:38:56.410 --> 00:38:58.089 is the incline members deforming? 00:38:58.089 --> 00:38:58.880 LORNA GIBSON: Yeah. 00:38:58.880 --> 00:39:00.440 AUDIENCE: And then at some point, 00:39:00.440 --> 00:39:02.682 you hit a P critical that will cause 00:39:02.682 --> 00:39:04.250 the vertical members to buckle? 00:39:04.250 --> 00:39:04.730 LORNA GIBSON: Exactly. 00:39:04.730 --> 00:39:04.995 AUDIENCE: OK. 00:39:04.995 --> 00:39:05.911 LORNA GIBSON: Exactly. 00:39:05.911 --> 00:39:08.480 That's exactly right. 00:39:08.480 --> 00:39:10.492 Hello. 00:39:10.492 --> 00:39:12.200 So the trick here is that we don't really 00:39:12.200 --> 00:39:13.940 know what this n is, initially. 00:39:13.940 --> 00:39:15.420 They're not really pinned, pinned; 00:39:15.420 --> 00:39:16.720 they're not fixed, fixed. 00:39:16.720 --> 00:39:20.220 And if you think about the setup with the honeycomb here, 00:39:20.220 --> 00:39:22.800 the constraint on that vertical member 00:39:22.800 --> 00:39:25.610 depends on how stiff the adjacent members are. 00:39:25.610 --> 00:39:27.284 So you can kind of imagine, if I'm 00:39:27.284 --> 00:39:28.950 looking at one of these vertical members 00:39:28.950 --> 00:39:32.520 here, if these two adjacent inclined members were 00:39:32.520 --> 00:39:36.240 big honking thick things, it would be more constrained. 00:39:36.240 --> 00:39:39.200 And if they were little thin, kind of teeny little membranes, 00:39:39.200 --> 00:39:40.800 it would be less constrained. 00:39:40.800 --> 00:39:43.990 And you can think of it in terms of a rotational stiffness, 00:39:43.990 --> 00:39:47.130 that when the honeycomb buckles, you kind of 00:39:47.130 --> 00:39:50.930 see the member h goes from being horizontal to sort 00:39:50.930 --> 00:39:53.190 of it buckles over like this. 00:39:53.190 --> 00:39:55.340 But that whole end joint, see the end joint 00:39:55.340 --> 00:39:57.880 at the top here or the end joint at the bottom, that 00:39:57.880 --> 00:39:59.850 whole joint rotates a little bit. 00:39:59.850 --> 00:40:03.150 And so there's some rotational stiffness of that joint. 00:40:03.150 --> 00:40:06.450 And that rotational stiffness depends on how stiff 00:40:06.450 --> 00:40:10.020 the member h is and how stiff those inclined members are. 00:40:10.020 --> 00:40:14.790 So there's a thing called the elastic line analysis that you 00:40:14.790 --> 00:40:16.510 can use to calculate what n is. 00:40:16.510 --> 00:40:18.520 And basically what that does is it matches 00:40:18.520 --> 00:40:21.490 the rotational stiffness of the column h 00:40:21.490 --> 00:40:24.000 with the rotational stiffness of those inclined members. 00:40:24.000 --> 00:40:25.240 So we're not going to get into that. 00:40:25.240 --> 00:40:26.660 I'm just going to tell you what the answer is. 00:40:26.660 --> 00:40:28.230 But if you want to go through it, 00:40:28.230 --> 00:40:29.950 it's in an appendix in the book. 00:40:29.950 --> 00:40:31.720 So you can look at it, if you want. 00:40:35.920 --> 00:40:46.260 So here I'm just going to say that the constraint 00:40:46.260 --> 00:40:51.200 n depends on the stiffness of the adjacent inclined members. 00:41:16.532 --> 00:41:18.200 And we can find that by something called 00:41:18.200 --> 00:41:19.600 the elastic line analysis. 00:41:26.700 --> 00:41:29.020 And if you have the book, you can look in the appendix 00:41:29.020 --> 00:41:32.360 and see how that works. 00:41:32.360 --> 00:41:33.930 But essentially what it does is it 00:41:33.930 --> 00:41:38.100 matches the rotational stiffness of the column h 00:41:38.100 --> 00:41:40.810 with the rotational stiffness of the inclined members. 00:42:24.130 --> 00:42:28.860 So what you find is that n depends on the ratio of h/l. 00:42:28.860 --> 00:42:32.710 And I'm just going to give you a table with a few values. 00:42:32.710 --> 00:42:39.210 So for h/l equal to 1, then n is equal to 0.686. 00:42:39.210 --> 00:42:44.890 For h equal to 1.5, it's equal to 0.76. 00:42:44.890 --> 00:42:52.810 And for h/l equal to 2, it's equal to 0.806. 00:42:52.810 --> 00:42:57.460 OK, so now if we have values for n, 00:42:57.460 --> 00:43:01.689 we can just substitute in to get the critical buckling load. 00:43:01.689 --> 00:43:03.230 And if I take that load and divide it 00:43:03.230 --> 00:43:05.260 by the area of the unit cell, I'm 00:43:05.260 --> 00:43:06.650 going to get my buckling stress. 00:43:06.650 --> 00:43:08.890 So it's pretty straightforward from this. 00:43:49.360 --> 00:43:54.180 So my buckling stress is going to be that critical load 00:43:54.180 --> 00:44:00.250 divided by my unit cell area. 00:44:00.250 --> 00:44:03.260 So it's divided by the unit cell length in the x1 direction 00:44:03.260 --> 00:44:06.477 to l cos theta times the depth b into the page. 00:44:25.610 --> 00:44:29.860 So it's equal to n squared pi squared Es times I. 00:44:29.860 --> 00:44:32.510 And I is bt cubed over 12. 00:44:32.510 --> 00:44:35.910 Divided by the length of the column, h squared, 00:44:35.910 --> 00:44:39.769 and then divided by the area of the unit cell, 2l cos theta b, 00:44:39.769 --> 00:44:40.268 OK? 00:44:43.020 --> 00:44:45.510 And I can rearrange that somewhat 00:44:45.510 --> 00:44:48.750 to put it in terms of dimensionless groups. 00:44:54.320 --> 00:44:55.990 So if I pull all the constants out, 00:44:55.990 --> 00:45:00.720 it's n squared pi squared over 24 times the modulus 00:45:00.720 --> 00:45:08.020 of the solid, t/l cubed in the numerator divided 00:45:08.020 --> 00:45:14.231 by h/l squared times cos theta in the denominator. 00:45:23.870 --> 00:45:26.080 So again, you can see that the buckling stress, 00:45:26.080 --> 00:45:29.900 the compressive sort of elastic collapse stress, 00:45:29.900 --> 00:45:31.540 depends on the solid property. 00:45:31.540 --> 00:45:34.300 So here is the modulus of the cell wall in here. 00:45:34.300 --> 00:45:38.860 Depends on the relative density through t/l cubed. 00:45:38.860 --> 00:45:43.230 And then it depends on the cell geometry through h/l cos theta, 00:45:43.230 --> 00:45:46.176 and n depends on h/l as well, OK? 00:45:48.756 --> 00:45:50.130 And then we can do the same thing 00:45:50.130 --> 00:45:52.588 where we figure out what it is for regular hexagonal cells. 00:46:07.200 --> 00:46:12.470 And it's 0.22 Es times t/l cubed. 00:46:12.470 --> 00:46:18.530 And then we can also notice that since E in the 2 direction, 00:46:18.530 --> 00:46:21.421 for a regular hexagonal cell, E is the same in the 2 00:46:21.421 --> 00:46:22.670 direction and the 1 direction. 00:46:22.670 --> 00:46:24.140 It's isotropic. 00:46:24.140 --> 00:46:31.280 So E2 is also equal to 4 over root 3 Es times t/l cubed. 00:46:31.280 --> 00:46:41.890 That's equal to-- whoops-- it's equals to 2.31 Es t/l cubed. 00:46:41.890 --> 00:46:45.869 And we can say that the strain at which that buckling happens 00:46:45.869 --> 00:46:47.035 is just equal to a constant. 00:46:55.680 --> 00:46:59.110 And for regular hexagonal honeycombs, 00:46:59.110 --> 00:47:00.650 it works out to a strain of 10%. 00:47:17.880 --> 00:47:18.785 Are we good? 00:47:18.785 --> 00:47:19.910 So we have a buckling load. 00:47:19.910 --> 00:47:21.580 We divide by the area. 00:47:21.580 --> 00:47:24.070 The only complicated thing is finding n. 00:47:24.070 --> 00:47:28.500 And you can find it by this elastic line analysis thing. 00:47:28.500 --> 00:47:30.620 So each of these calculations is like 00:47:30.620 --> 00:47:33.000 a little structural analysis, only 00:47:33.000 --> 00:47:35.875 on a little teeny weeny scale of the cells. 00:47:35.875 --> 00:47:38.000 So you see where my background in civil engineering 00:47:38.000 --> 00:47:40.320 comes in handy. 00:47:40.320 --> 00:47:41.270 Yup. 00:47:41.270 --> 00:47:41.770 OK. 00:47:47.770 --> 00:47:50.980 So the honeycombs involve the most sort 00:47:50.980 --> 00:47:53.380 of complicated equations. 00:47:53.380 --> 00:47:55.110 When we come to do the foams, we're 00:47:55.110 --> 00:47:57.230 going to use a dimensional analysis 00:47:57.230 --> 00:48:00.430 and all the equations are going to be much simpler. 00:48:00.430 --> 00:48:02.430 So this is the most kind of tedious part 00:48:02.430 --> 00:48:05.200 of the whole thing. 00:48:05.200 --> 00:48:07.100 So the next property I want to look at 00:48:07.100 --> 00:48:10.300 is the plastic collapse stress. 00:48:10.300 --> 00:48:13.430 Say we had a metal honeycomb and we 00:48:13.430 --> 00:48:18.470 wanted to calculate the stress plateau for a metal honeycomb. 00:48:18.470 --> 00:48:20.930 So we have this little schematic here, 00:48:20.930 --> 00:48:24.500 and say we load it in the one direction again. 00:48:24.500 --> 00:48:27.100 So we're loading it here. 00:48:27.100 --> 00:48:30.090 And we've got some load P, like that. 00:48:30.090 --> 00:48:33.830 And if we have our honeycomb, we load it this way on, initially, 00:48:33.830 --> 00:48:35.567 the cell walls bend. 00:48:35.567 --> 00:48:37.150 And you have linear elasticity and you 00:48:37.150 --> 00:48:38.305 have some Young's modulus. 00:48:38.305 --> 00:48:41.402 But if you have a metal, if you continue to deform it 00:48:41.402 --> 00:48:43.600 and you continue to load it more and more, 00:48:43.600 --> 00:48:46.101 eventually you're going to hit the yield stress and the cell 00:48:46.101 --> 00:48:46.600 wall. 00:48:46.600 --> 00:48:48.240 So the stresses in the cell wall are 00:48:48.240 --> 00:48:49.770 going to hit the yield stress. 00:48:49.770 --> 00:48:52.089 And initially, the stresses are just 00:48:52.089 --> 00:48:53.880 going to be-- remember, if you have a beam, 00:48:53.880 --> 00:48:56.701 the stresses are maximum at the top and the bottom of the beam. 00:48:56.701 --> 00:48:58.450 So initially you're going to hit the yield 00:48:58.450 --> 00:49:00.707 stress at the top and the bottom of the beam first. 00:49:00.707 --> 00:49:02.290 But as you continue to load it, you're 00:49:02.290 --> 00:49:04.730 going to end up yielding the cross-section 00:49:04.730 --> 00:49:06.470 through the entire section. 00:49:06.470 --> 00:49:08.770 So the entire section is going to be yielded. 00:49:08.770 --> 00:49:10.970 And once the entire section yields, 00:49:10.970 --> 00:49:13.440 it forms what's called a plastic hinge. 00:49:13.440 --> 00:49:16.830 Once the whole thing's yielded, then you can add more force 00:49:16.830 --> 00:49:18.280 and the thing just rotates. 00:49:18.280 --> 00:49:20.801 And because it rotates, it's called a plastic hinge. 00:49:20.801 --> 00:49:22.300 You know, if you take a coat hanger, 00:49:22.300 --> 00:49:24.220 and you bend it back and forth and bend it back and forth. 00:49:24.220 --> 00:49:27.320 If you bend it enough, you form a plastic hinge because it just 00:49:27.320 --> 00:49:31.640 can bend easily. 00:49:31.640 --> 00:49:36.250 So these little schematics here, if you look at the, 00:49:36.250 --> 00:49:37.900 say, one of these inclined members, 00:49:37.900 --> 00:49:39.575 the moments are maximum at the end. 00:49:39.575 --> 00:49:42.387 So Remember when we had the linear elastic deformation 00:49:42.387 --> 00:49:44.470 and I looked at the little bending moment diagram? 00:49:44.470 --> 00:49:46.590 The moments are maximum at the ends, 00:49:46.590 --> 00:49:49.180 and you're going to form those plastic hinges initially 00:49:49.180 --> 00:49:50.330 at the ends. 00:49:50.330 --> 00:49:54.100 And so these little ellipsey things here, all the ends, 00:49:54.100 --> 00:49:57.510 those kind of show where the plastic hinges are. 00:49:57.510 --> 00:49:59.010 So those plastic hinges are forming. 00:49:59.010 --> 00:50:01.250 So here's for loading in the x1 direction, 00:50:01.250 --> 00:50:05.440 and here's for loading in the x2 direction, there. 00:50:05.440 --> 00:50:07.200 So the thing we want to calculate 00:50:07.200 --> 00:50:10.860 is what stress does it take to form those plastic hinges 00:50:10.860 --> 00:50:14.510 and get this kind of plastic plateau stress? 00:50:40.090 --> 00:50:44.930 OK, so we can say we get failure by yielding in the cell walls. 00:50:56.967 --> 00:50:59.300 And I'm going to say the yield strength of the cell wall 00:50:59.300 --> 00:51:00.020 is sigma ys. 00:51:09.820 --> 00:51:12.830 So sigma y for yield and s for the solid. 00:51:12.830 --> 00:51:16.170 And the plastic hinge forms when the cross-section has fully 00:51:16.170 --> 00:51:16.670 yielded. 00:51:53.930 --> 00:51:56.610 So let's look at the stress distribution 00:51:56.610 --> 00:51:59.940 through the cross-section when its first linear elastic. 00:51:59.940 --> 00:52:05.100 So say that's the thickness t of the member. 00:52:05.100 --> 00:52:08.030 And if the beam was linear elastic, 00:52:08.030 --> 00:52:11.820 the stress would just [? vary ?] linearly, like that, right? 00:52:11.820 --> 00:52:15.190 And this would be the neutral axis, here, 00:52:15.190 --> 00:52:16.818 where there is no normal stress. 00:52:20.870 --> 00:52:23.320 So that's what happens if it's linear elastic, 00:52:23.320 --> 00:52:26.820 and I'm hoping you remember something vaguely like that. 00:52:26.820 --> 00:52:27.412 Sounds good? 00:52:30.180 --> 00:52:31.890 But as we increase the load on it, 00:52:31.890 --> 00:52:34.090 and we increase the sort of external stress, 00:52:34.090 --> 00:52:37.290 this stress in the member is going to get bigger and bigger, 00:52:37.290 --> 00:52:39.860 and eventually, that's going to reach the yield stress, OK? 00:52:39.860 --> 00:52:42.620 And once that reaches the yield stress, 00:52:42.620 --> 00:52:44.560 if we continue to load it, what happens 00:52:44.560 --> 00:52:47.220 is the yielding propagates down through the thickness 00:52:47.220 --> 00:52:48.170 of the thing here. 00:52:48.170 --> 00:52:51.270 So we get yielding through the whole cross-section. 00:52:51.270 --> 00:52:54.590 So let me scoot over here. 00:52:54.590 --> 00:52:55.570 AUDIENCE: Professor? 00:52:55.570 --> 00:52:56.320 LORNA GIBSON: Yup? 00:52:56.320 --> 00:53:00.070 AUDIENCE: When it starts to yield, does this curve change? 00:53:00.070 --> 00:53:00.820 LORNA GIBSON: Yes. 00:53:00.820 --> 00:53:01.770 I'm going to draw it for you. 00:53:01.770 --> 00:53:02.670 AUDIENCE: Oh, OK. 00:53:02.670 --> 00:53:05.040 LORNA GIBSON: That's the next step. 00:53:05.040 --> 00:53:06.367 That would be the next thing. 00:53:29.520 --> 00:53:33.190 OK, so once the stress at the outer fiber 00:53:33.190 --> 00:53:59.450 is the yield strength of the solid, 00:53:59.450 --> 00:54:03.524 then the yielding begins and it progresses through the section 00:54:03.524 --> 00:54:04.440 as the load increases. 00:54:35.900 --> 00:54:38.180 So the stress distribution starts to look something 00:54:38.180 --> 00:54:39.589 like this once it yields. 00:54:43.990 --> 00:54:47.770 OK, so that's sigma y of the solid. 00:54:47.770 --> 00:54:50.150 Actually, let me rub that out because then I 00:54:50.150 --> 00:54:53.110 can show you something else. 00:54:53.110 --> 00:54:56.680 So in 3D, this would be through the thickness of the beam. 00:54:56.680 --> 00:54:58.715 That would be the thickness of the beam there. 00:55:01.685 --> 00:55:05.090 And boop, boop. 00:55:05.090 --> 00:55:06.960 It would look something like that. 00:55:06.960 --> 00:55:08.210 OK? 00:55:08.210 --> 00:55:10.305 And then this is still our neutral axis here. 00:55:14.670 --> 00:55:17.640 And then eventually, as you load it more and more, 00:55:17.640 --> 00:55:19.626 the whole cross-section is going to yield. 00:55:19.626 --> 00:55:20.126 Whoops. 00:55:53.580 --> 00:55:56.180 And I'm assuming that the material is elastic, perfectly 00:55:56.180 --> 00:55:56.680 plastic. 00:56:12.990 --> 00:56:15.660 So the stress-strain curve from the solid I'm 00:56:15.660 --> 00:56:17.098 idealizing as-- whoops. 00:56:20.770 --> 00:56:23.150 That's not quite right. 00:56:23.150 --> 00:56:25.610 I'm idealizing as that, OK? 00:56:28.270 --> 00:56:30.050 So when you get to this point here, 00:56:30.050 --> 00:56:31.860 the entire cross-section has yielded, 00:56:31.860 --> 00:56:34.050 and that means you form the plastic hinge. 00:56:56.507 --> 00:56:58.090 The idea here is that the section then 00:56:58.090 --> 00:57:01.968 just rotates like a pin. 00:57:09.610 --> 00:57:10.110 All right. 00:57:28.990 --> 00:57:31.430 So we can figure out the plateau stress 00:57:31.430 --> 00:57:33.980 that corresponds to this by looking 00:57:33.980 --> 00:57:37.630 at the moment that's associated with the plastic hinge 00:57:37.630 --> 00:57:38.220 formation. 00:57:38.220 --> 00:57:41.040 So there's some internal moment associated with that. 00:57:41.040 --> 00:57:43.840 And then equating that to the applied moment 00:57:43.840 --> 00:57:45.770 from the applied stress. 00:57:45.770 --> 00:57:47.080 So doodle-loodle-oot. 00:57:49.830 --> 00:57:52.690 Let me see me, maybe back up here. 00:57:52.690 --> 00:57:54.940 So there's some-- if I have the stress distribution 00:57:54.940 --> 00:57:59.160 here, I could say this whole kind of stress block 00:57:59.160 --> 00:58:01.330 is equivalent to some force acting out 00:58:01.330 --> 00:58:04.350 like that and some force acting out like that. 00:58:04.350 --> 00:58:10.360 It would be sigma ys times b comes t/2 would be f. 00:58:10.360 --> 00:58:14.250 And I can say there's some plastic moment. 00:58:14.250 --> 00:58:17.240 If I think of the force here and the force there, 00:58:17.240 --> 00:58:19.865 they act as a couple and they have some moment, 00:58:19.865 --> 00:58:22.490 and that's called the plastic moment. 00:58:22.490 --> 00:58:24.020 So that's like an internal moment 00:58:24.020 --> 00:58:25.660 when the plastic hinge forms. 00:58:29.610 --> 00:58:38.079 So I'll say the internal moment at the formation 00:58:38.079 --> 00:58:38.870 of a plastic hinge. 00:58:46.580 --> 00:58:49.570 I'm going to call that Mp, for plastic moment. 00:58:53.050 --> 00:58:56.670 And we can work out Mp by looking at that stress 00:58:56.670 --> 00:58:59.600 distribution when the entire cross section has yielded. 00:58:59.600 --> 00:59:06.990 The force F is going to be sigma ys times b comes t/2. 00:59:06.990 --> 00:59:10.120 It's the stress times that area. 00:59:10.120 --> 00:59:15.440 And then the moment arm between the two forces is also t/2. 00:59:15.440 --> 00:59:20.340 And so that plastic moment is just sigma ys 00:59:20.340 --> 00:59:24.360 bt squared over 4, OK? 00:59:24.360 --> 00:59:26.740 Are we good? 00:59:26.740 --> 00:59:27.648 Sonya? 00:59:27.648 --> 00:59:30.460 AUDIENCE: What's the second [INAUDIBLE]? 00:59:30.460 --> 00:59:32.210 LORNA GIBSON: OK, so this is the force. 00:59:32.210 --> 00:59:35.760 This thing here is the force F. And I 00:59:35.760 --> 00:59:38.280 have to-- if I'm getting a moment, 00:59:38.280 --> 00:59:43.160 I'm saying that that force, if I doot-doot-doot-- 00:59:43.160 --> 00:59:47.520 the distance between those two forces there is t/2. 00:59:47.520 --> 00:59:50.769 So each force acts through the middle of the block, 00:59:50.769 --> 00:59:52.602 and so the distance between [? it is ?] t/2. 01:00:00.260 --> 01:00:04.810 And I'm going to equate that moment to the applied moment 01:00:04.810 --> 01:00:06.803 from the sort of applied stress. 01:00:15.450 --> 01:00:18.405 And then if I go back to my inclined member-- whoops, 01:00:18.405 --> 01:00:19.720 let's see. 01:00:19.720 --> 01:00:21.900 Let me get a little more inclined. 01:00:21.900 --> 01:00:24.268 That's my inclined member, there, of length l. 01:00:28.950 --> 01:00:33.800 I've got modes p that are applied 01:00:33.800 --> 01:00:35.880 at the end from sigma 1. 01:00:35.880 --> 01:00:39.170 And I've got moments that are induced at the ends. 01:00:39.170 --> 01:00:42.720 And that angle there would be theta. 01:00:42.720 --> 01:00:45.940 This length here is l, like that. 01:00:45.940 --> 01:00:47.680 And if I just use static equilibrium 01:00:47.680 --> 01:00:51.130 on that, I can say that I've got 2 times 01:00:51.130 --> 01:00:53.800 the moment, so I've got one at each end-- they're 01:00:53.800 --> 01:00:57.160 both the same sign-- minus P. And then 01:00:57.160 --> 01:00:59.590 the distance between these two P's, say I take moments 01:00:59.590 --> 01:01:03.400 about here, I've got M applied plus M applied, 01:01:03.400 --> 01:01:05.686 I've got minus P times l sine theta. 01:01:09.280 --> 01:01:10.474 That's equal to 0. 01:01:13.140 --> 01:01:18.728 So the applied moment there is just Pl sine theta over 2. 01:01:23.200 --> 01:01:25.370 So now what I'm going to do is I'm 01:01:25.370 --> 01:01:27.090 going to equate this applied moment 01:01:27.090 --> 01:01:29.190 with this plastic moment, and I'm 01:01:29.190 --> 01:01:32.780 going to relate P to my applied stress sigma 1. 01:01:32.780 --> 01:01:35.840 And then I'm going to get a strength in terms of the yield 01:01:35.840 --> 01:01:38.980 strength of the solid, there's going to be a t/l factor 01:01:38.980 --> 01:01:42.400 and there's going to be some geometrical factor. 01:01:42.400 --> 01:01:44.875 So that's just the last step. 01:01:44.875 --> 01:01:45.750 Boop-ba-doop-ba-doop. 01:02:19.000 --> 01:02:21.060 So we get plastic collapse of the honeycomb. 01:02:35.810 --> 01:02:38.902 And the stress I'm going to call sigma star plastic with a 1, 01:02:38.902 --> 01:02:40.860 because I'm going to look at the one direction. 01:02:44.450 --> 01:02:49.000 And that happens when that internal plastic moment 01:02:49.000 --> 01:02:50.363 equals the applied moment. 01:02:55.190 --> 01:02:55.920 So let's see. 01:02:55.920 --> 01:02:58.230 I've got that. 01:02:58.230 --> 01:03:01.350 Let me also write down over here, 01:03:01.350 --> 01:03:05.900 I've also got this sigma 1 is equal to P over 01:03:05.900 --> 01:03:13.670 h plus l sine theta times b. 01:03:13.670 --> 01:03:18.090 So here I can write P in terms of sigma 1, in this thing. 01:03:18.090 --> 01:03:20.960 And then write that, get the applied moment 01:03:20.960 --> 01:03:23.326 in terms of that, and then equate it to that. 01:03:48.930 --> 01:03:51.900 So this term on the left-hand side 01:03:51.900 --> 01:03:55.340 corresponds to this expression for the applied moment 01:03:55.340 --> 01:03:56.520 where I've plugged in. 01:03:56.520 --> 01:03:59.585 For P, I've plugged in sigma 1 times h plus l sine 01:03:59.585 --> 01:04:01.610 theta times b. 01:04:01.610 --> 01:04:06.210 And that's my plastic moment on the right-hand side. 01:04:06.210 --> 01:04:08.960 So if I just rearrange this, I can then 01:04:08.960 --> 01:04:11.110 solve for this plastic collapse stress. 01:04:13.930 --> 01:04:18.480 So it's equal to the yield strength of the solid times 01:04:18.480 --> 01:04:23.330 t/l squared, and then times another geometrical factor. 01:04:23.330 --> 01:04:28.850 2 times h/l plus sine theta times sine theta. 01:04:31.345 --> 01:04:31.970 Doop-doop-doop. 01:04:41.584 --> 01:04:43.000 So the same kind of thing, there's 01:04:43.000 --> 01:04:46.890 a solid property, a t/l, a relative density term, 01:04:46.890 --> 01:04:49.350 in then a cell geometry term. 01:04:49.350 --> 01:04:53.380 And we can calculate with this for regular hexagonal cells. 01:05:09.850 --> 01:05:11.930 And we can do a similar kind of calculation 01:05:11.930 --> 01:05:13.560 for loading in the other direction. 01:05:29.830 --> 01:05:34.554 And you can get a shear strength if you want to do that, too. 01:05:34.554 --> 01:05:36.595 AUDIENCE: If you're going in the other direction, 01:05:36.595 --> 01:05:40.626 only the E or the M apply changes, right? 01:05:40.626 --> 01:05:41.750 [? Or ?] like that section. 01:05:41.750 --> 01:05:43.010 LORNA GIBSON: Yeah, this thing here is the same. 01:05:43.010 --> 01:05:43.380 AUDIENCE: That stays. 01:05:43.380 --> 01:05:44.213 LORNA GIBSON: Right. 01:05:44.213 --> 01:05:46.500 And this is-- there's a different geometry to it. 01:05:46.500 --> 01:05:48.918 Because now you're loading it this way on. 01:06:11.870 --> 01:06:15.630 OK, so we've calculated an elastic buckling plateau 01:06:15.630 --> 01:06:20.070 stress and a sort of plastic collapse plateau stress. 01:06:20.070 --> 01:06:23.140 And if you have thin enough walled, say, even aluminum 01:06:23.140 --> 01:06:25.440 honeycombs, then the elastic buckling 01:06:25.440 --> 01:06:29.110 could precede the plastic collapse. 01:06:29.110 --> 01:06:31.790 And so I'm just going to work out what the criterion would 01:06:31.790 --> 01:06:32.872 be for that to happen. 01:07:12.070 --> 01:07:16.018 So the two stresses can be equated. 01:07:19.300 --> 01:07:22.193 And then that's going to give us some criterion. 01:07:33.370 --> 01:07:35.550 So the two are equal, I'm just going to write down 01:07:35.550 --> 01:07:37.400 the equations that we had. 01:07:37.400 --> 01:07:39.455 So the buckling stress was n squared 01:07:39.455 --> 01:07:46.120 pi squared over 24 times E of the solid times t/l 01:07:46.120 --> 01:07:54.350 cubed divided by h/l squared times cos of theta. 01:07:54.350 --> 01:08:00.260 And the plastic collapse stress for the 2 direction 01:08:00.260 --> 01:08:05.850 was sigma ys times t/l squared divided by 2 cos squared theta. 01:08:11.510 --> 01:08:14.140 So I can write this-- because this has a t/l cubed term, 01:08:14.140 --> 01:08:16.510 and that has a t/l squared term, I 01:08:16.510 --> 01:08:18.902 can write this in terms of a t/l critical. 01:08:22.520 --> 01:08:25.090 So if I leave it t/l here and I put everything else 01:08:25.090 --> 01:08:34.300 on the other side, I've got 12 over n squared pi squared, 01:08:34.300 --> 01:08:45.362 then h/l squared over cos theta times sigma ys over Es. 01:08:48.670 --> 01:08:50.399 So if t/l is less than that, I'm going 01:08:50.399 --> 01:08:51.659 to get elastic buckling first. 01:08:51.659 --> 01:08:52.920 And if it's more than that, I'm going 01:08:52.920 --> 01:08:54.260 to get plastic yielding first. 01:09:00.960 --> 01:09:02.710 And we can work out an exact number 01:09:02.710 --> 01:09:04.220 for regular hexagonal honeycombs, 01:09:04.220 --> 01:09:05.220 so I'm going to do that. 01:09:13.140 --> 01:09:15.010 So if I have a particular geometry, 01:09:15.010 --> 01:09:16.740 I can figure out what n is. 01:09:16.740 --> 01:09:20.930 So for regular hexagonal honeycombs, 01:09:20.930 --> 01:09:25.260 t/l critical just works out to 3 times 01:09:25.260 --> 01:09:28.710 the yield strength of the solid over the Young's modulus 01:09:28.710 --> 01:09:29.315 of the solid. 01:09:34.060 --> 01:09:36.090 So if we know that ratio of the yield 01:09:36.090 --> 01:09:38.340 strength of the modulus of the solid, 01:09:38.340 --> 01:09:43.390 we can get some idea of what that critical t/l would be. 01:09:43.390 --> 01:09:46.529 So we'll do that next. 01:09:46.529 --> 01:09:50.693 AUDIENCE: And you said if t/l is less than that critical, 01:09:50.693 --> 01:09:52.527 then you're going to get the yielding first. 01:09:52.527 --> 01:09:54.984 LORNA GIBSON: No, if it's less, you get the buckling first. 01:09:54.984 --> 01:09:57.127 If it's really skinny, it tends to buckle first. 01:10:27.510 --> 01:10:31.480 So, for example, for metals, the yield strength over the modulus 01:10:31.480 --> 01:10:35.350 is roughly 0.002, like the 0.2% yield strength. 01:10:35.350 --> 01:10:38.510 And so that means that t/l, the sort 01:10:38.510 --> 01:10:44.350 of transition or the critical value is at 0.6%. 01:10:44.350 --> 01:10:46.924 So most metal honeycombs are denser than that. 01:10:46.924 --> 01:10:48.090 That's a pretty low density. 01:11:13.730 --> 01:11:19.010 But if we look at polymers, you can get polymers 01:11:19.010 --> 01:11:22.590 with a yield strength relative to the modulus 01:11:22.590 --> 01:11:30.860 of about 3% to 5%, and then that critical t/l is 01:11:30.860 --> 01:11:32.220 equal to about 10%, 15%. 01:11:35.790 --> 01:11:38.390 So low-density polymers with yield points 01:11:38.390 --> 01:11:39.870 may buckle before they yield. 01:12:18.640 --> 01:12:22.540 So we have one more of these compressive plateau stresses, 01:12:22.540 --> 01:12:24.500 and that's for the brittle honeycomb. 01:12:24.500 --> 01:12:26.500 So I don't think I'm going to finish this today, 01:12:26.500 --> 01:12:29.139 but let me set it up and then we'll finish it next time. 01:13:23.254 --> 01:13:25.670 So the idea here is that if you have a ceramic honeycomb-- 01:13:25.670 --> 01:13:28.210 remember I showed you some of those ceramic honeycombs-- 01:13:28.210 --> 01:13:31.400 that if you compress them, they can fail by a brittle crushing 01:13:31.400 --> 01:13:32.760 mode. 01:13:32.760 --> 01:13:35.990 So ceramic honeycombs can fail in a brittle manner. 01:13:54.620 --> 01:13:57.950 And again, initially there would be some cell wall bending, 01:13:57.950 --> 01:14:00.570 but at some point, you're going to reach the bending strength 01:14:00.570 --> 01:14:02.470 of the material. 01:14:02.470 --> 01:14:04.950 And bending strengths are called modulus of rupture. 01:14:04.950 --> 01:14:07.940 So you reach the modulus of rupture of the cell wall. 01:14:47.757 --> 01:14:49.590 So I'm not going to write the equations down 01:14:49.590 --> 01:14:51.506 today because we're not going to get very far, 01:14:51.506 --> 01:14:52.782 so I'll do that next time. 01:14:52.782 --> 01:14:54.740 But we're going to set this up exactly the same 01:14:54.740 --> 01:14:57.260 as we did for the last one, for the plastic yielding. 01:14:57.260 --> 01:14:59.540 But instead of getting that sort of blocky, 01:14:59.540 --> 01:15:03.007 fully yielded cross-section stress distribution, 01:15:03.007 --> 01:15:05.090 we're just going to have the linear elastic stress 01:15:05.090 --> 01:15:07.150 distribution, and when the maximum stress 01:15:07.150 --> 01:15:09.400 reaches that modulus [? rupture, ?] the thing's 01:15:09.400 --> 01:15:10.360 going to fail. 01:15:10.360 --> 01:15:12.020 So the form of the equations is going 01:15:12.020 --> 01:15:15.060 to be very similar to what we had for the plastic collapse 01:15:15.060 --> 01:15:17.800 stress, but there's a slightly different geometrical factor-- 01:15:17.800 --> 01:15:18.616 that's all. 01:15:18.616 --> 01:15:19.740 So we'll do that next time. 01:15:19.740 --> 01:15:21.198 And then next time we're also going 01:15:21.198 --> 01:15:23.450 to talk about the tensile behavior 01:15:23.450 --> 01:15:24.860 of honeycombs in-plane. 01:15:24.860 --> 01:15:26.516 We'll work out a fracture toughness 01:15:26.516 --> 01:15:28.640 and then we'll start talking about the out-of-plane 01:15:28.640 --> 01:15:29.960 properties, as well. 01:15:29.960 --> 01:15:34.200 So on Wednesday, we'll do the out-of-plane properties, OK? 01:15:34.200 --> 01:15:36.530 So hopefully we'll finish the out-of-plane properties 01:15:36.530 --> 01:15:37.465 Wednesday. 01:15:37.465 --> 01:15:39.090 And then next week, I was going to talk 01:15:39.090 --> 01:15:41.130 about some natural materials that 01:15:41.130 --> 01:15:44.690 have honeycomb-like structures, so things like wood and cork, 01:15:44.690 --> 01:15:45.860 OK? 01:15:45.860 --> 01:15:48.640 All right, so this is the kind of most equationy lecture 01:15:48.640 --> 01:15:50.540 in the whole course.