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. 00:00:26.570 --> 00:00:28.070 LORNA GIBSON: I think, last time, we 00:00:28.070 --> 00:00:31.210 got as far as talking about processing of foams 00:00:31.210 --> 00:00:33.960 and we talked a bit about processing of polymer foams. 00:00:33.960 --> 00:00:36.420 And today, I want to pick up where we left off. 00:00:36.420 --> 00:00:38.850 And I was going to talk about processing metal foams. 00:00:38.850 --> 00:00:42.040 We'll talk a little bit about carbon foams, ceramic foams, 00:00:42.040 --> 00:00:43.530 and glass foams. 00:00:43.530 --> 00:00:45.630 We'll finish this section on processing today 00:00:45.630 --> 00:00:47.630 and then we'll start talking about the structure 00:00:47.630 --> 00:00:48.870 of cellular materials. 00:00:48.870 --> 00:00:50.870 And hopefully-- we won't finish that today, 00:00:50.870 --> 00:00:53.866 but we'll finish it the start of tomorrow's lecture. 00:00:53.866 --> 00:00:55.990 And then, we'll start doing mechanics of honeycombs 00:00:55.990 --> 00:00:56.590 tomorrow. 00:00:56.590 --> 00:00:57.720 OK? 00:00:57.720 --> 00:00:59.571 So that's the scheme. 00:00:59.571 --> 00:01:02.070 I thought what I'd do-- I have a whole series of slides that 00:01:02.070 --> 00:01:05.519 show, schematically, a variety of different processes 00:01:05.519 --> 00:01:07.220 for making metal foams. 00:01:07.220 --> 00:01:09.440 So I thought I would just go through the slides- 00:01:09.440 --> 00:01:11.360 and I think I did this really quickly last time-- but I would 00:01:11.360 --> 00:01:13.900 write down a little bit of notes on each of the slides 00:01:13.900 --> 00:01:16.740 so that you've got some notes on it, too. 00:01:16.740 --> 00:01:18.710 This is the first method here. 00:01:18.710 --> 00:01:21.950 And many of these methods were developed for aluminum foams. 00:01:21.950 --> 00:01:24.920 But you could, in principle, use them for other types of foams, 00:01:24.920 --> 00:01:26.020 as well. 00:01:26.020 --> 00:01:27.800 This first method here-- let me see 00:01:27.800 --> 00:01:29.540 if I can get my little pointer. 00:01:29.540 --> 00:01:32.850 [INAUDIBLE] The idea here is that you take molten aluminum. 00:01:32.850 --> 00:01:35.620 Down in this bath here, you've got molten aluminum. 00:01:35.620 --> 00:01:38.860 And they put, into the aluminum, silicon carbide particles. 00:01:38.860 --> 00:01:40.700 And the silicon carbide particles 00:01:40.700 --> 00:01:42.750 adjust the viscosity of the melt. 00:01:42.750 --> 00:01:44.260 They make it more viscous. 00:01:44.260 --> 00:01:47.500 And then they just have a-- I've got this thing here-- 00:01:47.500 --> 00:01:50.934 they've got a tube here that they blow gas in with. 00:01:50.934 --> 00:01:52.600 And if you go to the bottom of the tube, 00:01:52.600 --> 00:01:56.390 there's an impeller, or a little paddle, that stirs the gas up. 00:01:56.390 --> 00:01:59.764 And so the gas just forms in the molten aluminum here. 00:01:59.764 --> 00:02:01.180 And then they have conveyors which 00:02:01.180 --> 00:02:07.110 pull off the molt-- or the metal foam, and then it cools. 00:02:07.110 --> 00:02:09.060 The idea is that if you just had the aluminum, 00:02:09.060 --> 00:02:11.268 you couldn't really do this because the bubbles would 00:02:11.268 --> 00:02:14.260 collapse before they cooled down and became a solid. 00:02:14.260 --> 00:02:17.520 But adding the silicon carbide particles increases 00:02:17.520 --> 00:02:19.400 the viscosity of the melt. It helps 00:02:19.400 --> 00:02:21.040 prevent drainage of the foam. 00:02:21.040 --> 00:02:23.352 Normally, if you have a liquid foam, just from gravity, 00:02:23.352 --> 00:02:24.810 you're going to have some drainage. 00:02:24.810 --> 00:02:27.510 It's going to tend to-- some of the liquid is going to tend 00:02:27.510 --> 00:02:30.060 to sink, just from gravity. 00:02:30.060 --> 00:02:32.330 By putting the silicon carbide particles in, 00:02:32.330 --> 00:02:33.690 you increase the viscosity. 00:02:33.690 --> 00:02:35.430 It helps prevent the drainage. 00:02:35.430 --> 00:02:38.890 And you can get the foam bubbles to be stable. 00:02:38.890 --> 00:02:41.810 Let me just write one little note here. 00:02:41.810 --> 00:02:47.630 The first method here involves just bubbling gas 00:02:47.630 --> 00:02:49.030 into the molten aluminum. 00:02:55.350 --> 00:02:59.730 And that molten aluminum is stabilized 00:02:59.730 --> 00:03:01.640 by silicon carbide particles. 00:03:01.640 --> 00:03:12.190 Sometimes people use aluminum particles and the particles 00:03:12.190 --> 00:03:29.240 increase the viscosity of the melt. 00:03:29.240 --> 00:03:42.924 And that reduces the drainage and it stabilizes the foam. 00:03:59.810 --> 00:04:03.340 That process was developed at Alcan in Canada 00:04:03.340 --> 00:04:06.930 and at Norsk Hydro in Norway. 00:04:06.930 --> 00:04:10.780 And this foam here is an example of the foam that they've made. 00:04:10.780 --> 00:04:12.880 And you can kind of see-- there's a density 00:04:12.880 --> 00:04:13.967 gradient in this foam. 00:04:13.967 --> 00:04:14.800 I'll pass it around. 00:04:14.800 --> 00:04:15.530 You can see it. 00:04:15.530 --> 00:04:18.360 But the bubbles are smaller down here and there's fewer of them 00:04:18.360 --> 00:04:19.630 than there are up here. 00:04:19.630 --> 00:04:21.630 And that's partly from the drainage. 00:04:21.630 --> 00:04:23.290 The molten aluminum is draining down 00:04:23.290 --> 00:04:24.860 and you get more liquid at the bottom 00:04:24.860 --> 00:04:26.609 and then you get more solid at the bottom. 00:04:26.609 --> 00:04:28.190 So that's the Alcan foam. 00:04:31.080 --> 00:04:31.580 OK. 00:04:31.580 --> 00:04:33.621 Another-- there's half a dozen of these processes 00:04:33.621 --> 00:04:34.810 for metal foams. 00:04:34.810 --> 00:04:39.350 Another version of the process involves taking metal powder 00:04:39.350 --> 00:04:42.317 and combining it with titanium hydride powder. 00:04:42.317 --> 00:04:44.150 And then you consolidate it and you heat it. 00:04:44.150 --> 00:04:48.149 So if I can show through the schematics here. 00:04:48.149 --> 00:04:50.690 In the first step, you take the two powders, you mix them up. 00:04:50.690 --> 00:04:53.880 So the second thing down here is mixing the powders up. 00:04:53.880 --> 00:04:55.520 And then you press them together. 00:04:55.520 --> 00:04:57.530 There's a dye here that you press it, 00:04:57.530 --> 00:04:58.950 and then you get pieces. 00:04:58.950 --> 00:05:01.230 And then you can heat that up. 00:05:01.230 --> 00:05:03.540 And the way that this works is that the titanium 00:05:03.540 --> 00:05:06.490 hydride decomposes at a temperature of about 00:05:06.490 --> 00:05:09.530 300 degrees C. And if the other powder is something 00:05:09.530 --> 00:05:12.800 like aluminum-- aluminum melts at something like 660 degrees 00:05:12.800 --> 00:05:16.670 C-- the aluminum has become soft at 300 degrees C, 00:05:16.670 --> 00:05:18.710 but it's not molten. 00:05:18.710 --> 00:05:20.920 And the titanium hydride-- when it decomposes-- 00:05:20.920 --> 00:05:22.320 forms hydrogen gas. 00:05:22.320 --> 00:05:24.000 So the hydrogen gas forms the bubbles 00:05:24.000 --> 00:05:25.940 that you need to make the foam. 00:05:25.940 --> 00:05:27.550 Are you riding your bicycle? 00:05:27.550 --> 00:05:31.300 You are tough. 00:05:31.300 --> 00:05:32.369 Very tough. 00:05:32.369 --> 00:05:34.410 I ride my bike, but I gave up a couple weeks ago. 00:05:34.410 --> 00:05:35.701 Well, three weeks ago, I guess. 00:05:35.701 --> 00:05:38.150 When it started snowing, I gave up. 00:05:38.150 --> 00:05:42.430 The idea with this process is you can use titanium hydride 00:05:42.430 --> 00:05:45.070 powder with aluminum and the aluminum becomes 00:05:45.070 --> 00:05:48.690 soft at the temperature that the titanium hydro decomposes. 00:05:48.690 --> 00:05:50.310 And when it forms the hydrogen gas, 00:05:50.310 --> 00:05:52.290 that gives you the bubbles, and so you get 00:05:52.290 --> 00:05:55.010 the foam made by that process. 00:05:55.010 --> 00:05:58.560 And that was developed by a place called Fraunhofer. 00:05:58.560 --> 00:06:00.330 And I have one of their little foams here. 00:06:00.330 --> 00:06:04.390 This is an example of their foam made by that powder metallurgy 00:06:04.390 --> 00:06:05.434 process there. 00:06:08.800 --> 00:06:09.340 OK. 00:06:09.340 --> 00:06:11.190 Let me just write a little note about that. 00:06:16.630 --> 00:06:21.460 So you can mix up titanium hydride powder with a metal 00:06:21.460 --> 00:06:27.286 powder and then heat that up. 00:07:04.950 --> 00:07:08.010 When you do this, you need to have a metal powder that's 00:07:08.010 --> 00:07:12.360 going to be deforming by, say, high temperature creep 00:07:12.360 --> 00:07:15.100 at the temperature that the decomposition of the titanium 00:07:15.100 --> 00:07:15.920 hydride happens. 00:07:15.920 --> 00:07:17.650 And for aluminum, that works. 00:07:17.650 --> 00:07:20.780 So you need to have the metal material be 00:07:20.780 --> 00:07:22.370 able to deform fairly easily, in order 00:07:22.370 --> 00:07:25.029 to get these bubbles to form a nice foam. 00:07:25.029 --> 00:07:26.570 But that's another way you can do it, 00:07:26.570 --> 00:07:29.410 is by consolidating these two powders. 00:07:29.410 --> 00:07:31.800 Here's another way to do it. 00:07:31.800 --> 00:07:34.800 You can also make use of this property of the titanium 00:07:34.800 --> 00:07:37.060 hydride-- that it'll decompose and form 00:07:37.060 --> 00:07:39.930 hydrogen gas-- by just putting it into molten aluminum. 00:07:39.930 --> 00:07:44.260 And in this example here, you've got an aluminum melt in here. 00:07:44.260 --> 00:07:47.730 And this time, they've added 2% calcium to it. 00:07:47.730 --> 00:07:50.230 Again, to adjust the viscosity. 00:07:50.230 --> 00:07:52.882 And then they add the titanium hydride in this step here 00:07:52.882 --> 00:07:54.590 and they've got a little mixer thing here 00:07:54.590 --> 00:07:56.440 that's spinning around and will mix it. 00:07:56.440 --> 00:08:00.100 And then they'll put a lid on it to control the pressure 00:08:00.100 --> 00:08:02.880 and the titanium hydride will decompose, 00:08:02.880 --> 00:08:04.690 the hydrogen gas will evolve, and you'll 00:08:04.690 --> 00:08:07.550 get the foam made by that method, too. 00:08:07.550 --> 00:08:10.890 So you can stir titanium hide right into a molten metal, 00:08:10.890 --> 00:08:11.496 as well. 00:08:45.540 --> 00:08:46.040 OK. 00:08:46.040 --> 00:08:47.710 So that's another method. 00:08:47.710 --> 00:08:52.240 And I think that was made by something called the Alporas 00:08:52.240 --> 00:08:52.890 process. 00:08:52.890 --> 00:08:54.765 And this is an example of one of their foams. 00:08:54.765 --> 00:08:58.150 I'm pretty sure that's an Alporas foam there. 00:08:58.150 --> 00:08:59.002 Yep. 00:08:59.002 --> 00:08:59.877 AUDIENCE: [INAUDIBLE] 00:09:03.770 --> 00:09:06.000 LORNA GIBSON: They can't really control it perfectly. 00:09:06.000 --> 00:09:08.130 In that first example that's going around-- maybe 00:09:08.130 --> 00:09:10.390 you might have missed it-- there's this drainage. 00:09:10.390 --> 00:09:13.630 The first one's made by a molten process and you get drainage. 00:09:13.630 --> 00:09:15.600 And you get different sized bubbles. 00:09:15.600 --> 00:09:17.500 And you get different-- you get a density 00:09:17.500 --> 00:09:19.280 gradient in the thing, as well. 00:09:19.280 --> 00:09:21.710 So you can't control these things perfectly. 00:09:24.420 --> 00:09:24.920 OK. 00:09:24.920 --> 00:09:28.130 Here's another method for the metal foams. 00:09:28.130 --> 00:09:30.820 This one here involves replication. 00:09:30.820 --> 00:09:33.010 In this method here, what you do is you start out 00:09:33.010 --> 00:09:35.700 with an open-cell polymer foam. 00:09:35.700 --> 00:09:39.340 In this step up here, there's an open-cell polymer foam. 00:09:39.340 --> 00:09:41.180 You fill that with sand. 00:09:41.180 --> 00:09:44.960 So you fill up all the open parts of the cells with sand. 00:09:44.960 --> 00:09:47.560 Then you burn off the polymer. 00:09:47.560 --> 00:09:49.520 And so you've got a little channels 00:09:49.520 --> 00:09:51.180 where the polymer used to be. 00:09:51.180 --> 00:09:55.790 And then you infiltrate those channels with the molten metal 00:09:55.790 --> 00:09:56.820 that you want to use. 00:09:56.820 --> 00:09:59.870 So you replicate the polymer foam structure. 00:09:59.870 --> 00:10:02.350 This is the infiltration process here. 00:10:02.350 --> 00:10:04.012 This little thing here is your furnace. 00:10:04.012 --> 00:10:05.720 And then you get rid of the sand and then 00:10:05.720 --> 00:10:12.380 you're left with a metal replica of the-- a replica 00:10:12.380 --> 00:10:13.960 of the original polymer foam. 00:10:13.960 --> 00:10:16.450 And this example here is one of these things that's made 00:10:16.450 --> 00:10:18.200 by this replication process. 00:10:18.200 --> 00:10:19.844 So that's an open-celled aluminum. 00:10:22.165 --> 00:10:24.540 I think the-- if you look at the density of these things, 00:10:24.540 --> 00:10:25.480 they're fairly low. 00:10:25.480 --> 00:10:28.100 And so there's quite a large volume of pores 00:10:28.100 --> 00:10:29.480 and they're all interconnected. 00:10:29.480 --> 00:10:31.420 So it's not that hard to get the sand out. 00:10:52.410 --> 00:10:54.460 This method would involve replication 00:10:54.460 --> 00:10:57.930 of the open-cell foam-- the polymer foam-- by casting. 00:10:57.930 --> 00:11:07.805 So you fill the open-cell polymer foam with sand. 00:11:12.030 --> 00:11:13.095 You burn off the polymer. 00:11:19.710 --> 00:11:22.000 And then you infiltrate the metal into that. 00:11:36.860 --> 00:11:39.630 And then you remove the sand. 00:11:39.630 --> 00:11:41.500 OK? 00:11:41.500 --> 00:11:45.640 Then, another process involves just using vapor depositions. 00:11:45.640 --> 00:11:48.420 So you take an open-cell polymer foam again. 00:11:48.420 --> 00:11:50.100 Here-- let me use my little arrow. 00:11:50.100 --> 00:11:52.690 Here's the open-cell polymer foam up here. 00:11:52.690 --> 00:11:57.490 And you have a furnace here with a vapor deposition system. 00:11:57.490 --> 00:12:01.730 And they use a nickel CO4 system to do this. 00:12:01.730 --> 00:12:04.680 You then burn out the polymer. 00:12:04.680 --> 00:12:07.480 You're left with a metal foam with hollow cell walls, 00:12:07.480 --> 00:12:08.739 where the polymer used to be. 00:12:08.739 --> 00:12:10.780 And then usually, what they do is they center it. 00:12:10.780 --> 00:12:14.070 They heat it up again to try to densify the walls. 00:12:14.070 --> 00:12:16.720 The only teeny weeny problem with this process 00:12:16.720 --> 00:12:19.780 is that the gas they use this incredibly toxic. 00:12:19.780 --> 00:12:24.099 And so it's not cheap and it's got health hazards, as well, 00:12:24.099 --> 00:12:24.890 associated with it. 00:12:24.890 --> 00:12:26.023 But you can do it. 00:12:28.641 --> 00:12:31.140 That gives you a nickel foam when you're finished with that. 00:12:37.820 --> 00:12:40.755 And you could also use an electrodeposition technique 00:12:40.755 --> 00:12:41.380 that's similar. 00:13:13.180 --> 00:13:15.690 OK. 00:13:15.690 --> 00:13:18.440 That's another method. 00:13:18.440 --> 00:13:22.040 Another method is shown here. 00:13:22.040 --> 00:13:25.000 This is the entrapped gas expansion method. 00:13:25.000 --> 00:13:28.600 And what they do in this method is they have a can 00:13:28.600 --> 00:13:30.450 and the can has a metal powder in it. 00:13:30.450 --> 00:13:32.810 It's whatever metal you want to make the foam out of. 00:13:32.810 --> 00:13:35.060 In this example here-- it's probably too small for you 00:13:35.060 --> 00:13:38.360 to read in the seats-- but it's titanium. 00:13:38.360 --> 00:13:40.690 They've taken a titanium alloy. 00:13:40.690 --> 00:13:42.690 They've got a powder of titanium alloy. 00:13:42.690 --> 00:13:44.127 They then evacuate the can. 00:13:44.127 --> 00:13:45.460 They take all the air out of it. 00:13:45.460 --> 00:13:47.880 And then they back fill it with argon gas. 00:13:47.880 --> 00:13:49.990 They put in an inert gas in there. 00:13:49.990 --> 00:13:52.370 And then what they do is they pressurize and heat 00:13:52.370 --> 00:13:55.820 the thing up and so the gas is internally 00:13:55.820 --> 00:13:58.110 pressurized by doing that. 00:13:58.110 --> 00:13:59.780 And then, sometimes when they do this, 00:13:59.780 --> 00:14:03.210 they want to have a skin on the two faces. 00:14:03.210 --> 00:14:05.110 So in this next little image here, 00:14:05.110 --> 00:14:08.500 it's done where they roll the can 00:14:08.500 --> 00:14:12.350 and produce a panel that's got solid faces. 00:14:12.350 --> 00:14:14.880 And then when you heat it up, the gas expands 00:14:14.880 --> 00:14:17.220 and you end up with a sandwich panel by doing this. 00:14:17.220 --> 00:14:19.446 So this bottom figure down here-- 00:14:19.446 --> 00:14:21.430 I'm not having much luck with the pointer. 00:14:21.430 --> 00:14:24.240 This bottom figure down here, they've heated it up. 00:14:24.240 --> 00:14:26.860 The gas expands and you've got this solid skin 00:14:26.860 --> 00:14:30.130 on the thing, which is from where the can use to be. 00:14:30.130 --> 00:14:32.815 So that's trapping of a gas. 00:15:37.420 --> 00:15:38.480 OK? 00:15:38.480 --> 00:15:41.910 There's a couple more methods for the metal foams. 00:15:41.910 --> 00:15:45.310 One involves centering hollow metal spheres together. 00:15:45.310 --> 00:15:48.150 And the trick there is to make the hollow spheres. 00:15:48.150 --> 00:15:52.670 And the way that can be done is by taking titanium hydride 00:15:52.670 --> 00:15:55.170 again, if you want to make titanium spheres. 00:15:55.170 --> 00:15:57.440 You put it in an organic binder-- in a solvent-- 00:15:57.440 --> 00:15:59.530 so you've got a slurry here. 00:15:59.530 --> 00:16:02.240 And you've got a tube that you blow gas through. 00:16:02.240 --> 00:16:05.390 And as you do that, you get hollow titanium hydride 00:16:05.390 --> 00:16:06.004 bubbles. 00:16:06.004 --> 00:16:08.420 And then you can do the same thing where you heat that up. 00:16:08.420 --> 00:16:11.510 The hydrogen gas evolves off, and you're left with titanium. 00:16:11.510 --> 00:16:13.020 You're left with titanium spheres 00:16:13.020 --> 00:16:14.210 down at the bottom here. 00:16:14.210 --> 00:16:15.840 And then you can pack those together 00:16:15.840 --> 00:16:19.810 and press those and form a cellular material that way. 00:16:19.810 --> 00:16:22.865 You can also center hollow metal spheres. 00:16:34.540 --> 00:16:38.139 And the last method I've got for the metal foams 00:16:38.139 --> 00:16:39.680 is that you can use a fugitive phase. 00:16:48.510 --> 00:16:51.620 With the fugitive phase, you would take some material 00:16:51.620 --> 00:16:54.930 that you could get rid of at the end of the process. 00:16:54.930 --> 00:16:57.500 Say, something like salt, that you could leach out. 00:16:57.500 --> 00:17:00.330 Here, we have our bed filled with salt. 00:17:00.330 --> 00:17:02.660 And then you would infiltrate that with a liquid metal, 00:17:02.660 --> 00:17:04.270 typically under pressure. 00:17:04.270 --> 00:17:07.180 And then, after the metals cooled, you leech out the salt. 00:17:07.180 --> 00:17:08.560 You get rid of that. 00:17:08.560 --> 00:17:14.349 You can pressure infiltrate a leachable bed of particles, 00:17:14.349 --> 00:17:16.863 and then leach the particles out. 00:17:56.220 --> 00:17:57.700 OK. 00:17:57.700 --> 00:17:59.430 We have a whole variety of methods 00:17:59.430 --> 00:18:01.840 that have been developed to make metal foams. 00:18:01.840 --> 00:18:04.230 And most of these have been developed, probably, 00:18:04.230 --> 00:18:05.210 in the last 20 years. 00:18:05.210 --> 00:18:05.810 Something like that. 00:18:05.810 --> 00:18:07.726 Some of them are a little bit older than that. 00:18:07.726 --> 00:18:10.730 But there's been a lot of interest in this recently. 00:18:10.730 --> 00:18:12.754 Those are all methods to make metal foams. 00:18:12.754 --> 00:18:15.170 I wanted to talk just a little bit about a few other types 00:18:15.170 --> 00:18:16.250 of foams. 00:18:16.250 --> 00:18:17.760 People make carbon foams. 00:18:17.760 --> 00:18:19.260 And they use the same kind of method 00:18:19.260 --> 00:18:21.990 as they do to make those bio carbon templates 00:18:21.990 --> 00:18:23.055 I told you about. 00:18:23.055 --> 00:18:25.555 When you take wood and you heat it up in an inert atmosphere 00:18:25.555 --> 00:18:27.660 and it turns into a carbon template, 00:18:27.660 --> 00:18:30.180 you can do the same thing where you take a polymer foam, 00:18:30.180 --> 00:18:32.440 you heat it up in an inert atmosphere, 00:18:32.440 --> 00:18:34.482 and everything except the carbon is driven off. 00:18:34.482 --> 00:18:35.940 And you're left with a carbon foam. 00:18:35.940 --> 00:18:39.080 It's the same process they used to make carbon fibers. 00:18:39.080 --> 00:18:40.200 There's carbon foams. 00:19:36.050 --> 00:19:38.940 There's also ceramic foams that you can make. 00:19:38.940 --> 00:19:41.170 I brought the little sample of ceramic foaming again. 00:19:41.170 --> 00:19:42.530 You can pass that around. 00:19:42.530 --> 00:19:45.280 And those are typically made by taking an open-cell polymer 00:19:45.280 --> 00:19:49.420 foam and passing a ceramic slurry through the polymer foam 00:19:49.420 --> 00:19:51.280 so that you coat the cell walls. 00:19:51.280 --> 00:19:55.210 And then you fire it so that you bond the ceramic together 00:19:55.210 --> 00:19:56.730 and you burn the polymer off. 00:19:56.730 --> 00:19:59.560 And you're left with a foam that's got hollow cell walls. 00:19:59.560 --> 00:20:02.410 You can also make ceramic foams by doing a CVD process 00:20:02.410 --> 00:20:04.090 on the carbon foam that you could 00:20:04.090 --> 00:20:05.563 make by the previous process. 00:21:07.305 --> 00:21:10.670 And people also make glass foams. 00:21:10.670 --> 00:21:12.330 And to make glass foams, they use 00:21:12.330 --> 00:21:14.180 some of the same kinds of processes 00:21:14.180 --> 00:21:16.720 as people use for polymer foams. 00:21:16.720 --> 00:21:21.285 I'll just say similar processes to polymer foams. 00:21:29.600 --> 00:21:31.280 OK. 00:21:31.280 --> 00:21:33.640 That covers making the foams, and I 00:21:33.640 --> 00:21:35.950 think we talked about the honeycombs last time. 00:21:35.950 --> 00:21:38.390 I wanted to talk a little bit about making 00:21:38.390 --> 00:21:40.880 what are called 3D lattice materials, or 3D 00:21:40.880 --> 00:21:43.170 truss materials, as well. 00:21:43.170 --> 00:21:45.533 Let me [? strip ?] that up there. 00:21:59.660 --> 00:22:00.376 Yeah? 00:22:00.376 --> 00:22:01.250 AUDIENCE: [INAUDIBLE] 00:22:01.250 --> 00:22:02.916 LORNA GIBSON: Chemical vapor deposition. 00:22:09.189 --> 00:22:11.997 AUDIENCE: [INAUDIBLE] use this for metal foams? 00:22:11.997 --> 00:22:14.580 LORNA GIBSON: Well, people were quite interested in using them 00:22:14.580 --> 00:22:17.490 for sandwich panels-- the cores of sandwich panels-- 00:22:17.490 --> 00:22:18.840 lightweight panels. 00:22:18.840 --> 00:22:21.620 There was interest in using them for energy absorption, 00:22:21.620 --> 00:22:22.932 say, car bumpers. 00:22:22.932 --> 00:22:25.140 The automotive industry was quite interested in this, 00:22:25.140 --> 00:22:28.110 in terms of trying to make components with sandwich 00:22:28.110 --> 00:22:29.960 structures that would be lighter weight, 00:22:29.960 --> 00:22:31.930 or energy absorption for bumpers. 00:22:31.930 --> 00:22:35.120 Or filling up-- if you take-- say you take a metal tube. 00:22:35.120 --> 00:22:38.740 If you think of a car chassis and it's made of tubes, 00:22:38.740 --> 00:22:41.352 if you fill those tubes with a foam-- 00:22:41.352 --> 00:22:43.310 especially if you fill them with a metal foam-- 00:22:43.310 --> 00:22:45.900 you can increase the energy absorption quite substantially. 00:22:45.900 --> 00:22:49.040 So when you have a tube in a chassis, 00:22:49.040 --> 00:22:51.480 if it's loaded axially, it will fold up 00:22:51.480 --> 00:22:54.170 and you get all these wavelengths of buckling. 00:22:54.170 --> 00:22:55.770 And if you've put a foam in there, 00:22:55.770 --> 00:22:57.380 it changes the buckling wavelength 00:22:57.380 --> 00:23:00.410 and it increases the amount of energy you can absorb. 00:23:00.410 --> 00:23:02.740 So not only is the energy absorbed by the foam 00:23:02.740 --> 00:23:06.460 itself, it actually changes the buckling of the tube 00:23:06.460 --> 00:23:08.130 so you can absorb more energy. 00:23:08.130 --> 00:23:09.842 So there was a lot of interest in that. 00:23:09.842 --> 00:23:11.300 There was an interest in using them 00:23:11.300 --> 00:23:17.490 for cooling devices for, say, electronic components. 00:23:17.490 --> 00:23:19.100 The idea was you would take, say, 00:23:19.100 --> 00:23:21.430 an aluminium open-cell foam and you 00:23:21.430 --> 00:23:23.190 would flow air through that. 00:23:23.190 --> 00:23:26.180 And say you have your device that's generating heat, 00:23:26.180 --> 00:23:27.880 you'd have a foam underneath it. 00:23:27.880 --> 00:23:31.020 And the aluminum conducts the heat fairly well. 00:23:31.020 --> 00:23:32.644 And then you would blow air through it 00:23:32.644 --> 00:23:33.560 to try to cool it off. 00:23:33.560 --> 00:23:35.330 So there were a bunch of different applications 00:23:35.330 --> 00:23:36.704 people have had in mind for them. 00:23:36.704 --> 00:23:38.052 AUDIENCE: What about glass? 00:23:38.052 --> 00:23:39.510 LORNA GIBSON: Glass foams, I think, 00:23:39.510 --> 00:23:41.610 are largely used for insulation in buildings, 00:23:41.610 --> 00:23:42.360 believe it or not. 00:23:42.360 --> 00:23:45.220 I think, actually, one of the dorms at MIT-- maybe 00:23:45.220 --> 00:23:48.995 the Simmons dorm-- has a glass foam insulation. 00:23:48.995 --> 00:23:51.296 AUDIENCE: [INAUDIBLE] 00:23:53.644 --> 00:23:55.560 LORNA GIBSON: Well, I think because the foam-- 00:23:55.560 --> 00:23:59.510 because the cells are closed, the gas is trapped in the cell. 00:23:59.510 --> 00:24:03.240 Whereas with a fiberglass, gas could move through the fibers 00:24:03.240 --> 00:24:04.560 more easily. 00:24:04.560 --> 00:24:06.600 So I think that's partly how it works. 00:24:06.600 --> 00:24:07.100 OK. 00:24:07.100 --> 00:24:09.558 Well, let me talk a little bit about the lattice materials, 00:24:09.558 --> 00:24:11.260 too, and how we make those. 00:24:11.260 --> 00:24:13.550 We're going to start talking about the modeling 00:24:13.550 --> 00:24:15.290 of honeycombs and foams. 00:24:15.290 --> 00:24:17.850 And when we do that, we're going to see that if we have 00:24:17.850 --> 00:24:21.120 a structure that deforms by bending, 00:24:21.120 --> 00:24:24.010 the properties vary with the amount of material, 00:24:24.010 --> 00:24:25.180 in a certain way. 00:24:25.180 --> 00:24:29.400 But if we have materials where the deformation is controlled 00:24:29.400 --> 00:24:32.360 by axial deformation, the stiffness and the strength 00:24:32.360 --> 00:24:35.110 are going to be higher at the same density. 00:24:35.110 --> 00:24:37.832 People made these lattice-type materials 00:24:37.832 --> 00:24:39.290 to try to get something with a more 00:24:39.290 --> 00:24:40.950 regular structure, and especially 00:24:40.950 --> 00:24:42.210 a triangulated structure. 00:24:42.210 --> 00:24:44.520 You see how these things are like little trusses? 00:24:44.520 --> 00:24:45.950 Triangulated? 00:24:45.950 --> 00:24:47.570 Triangulated structures, when you 00:24:47.570 --> 00:24:49.570 load them-- say I load this like this-- 00:24:49.570 --> 00:24:52.350 there's just axial components-- axial forces 00:24:52.350 --> 00:24:53.365 in each of the members. 00:24:53.365 --> 00:24:54.740 And so, theoretically, this would 00:24:54.740 --> 00:24:58.030 be higher stiffness and strength for a given weight 00:24:58.030 --> 00:24:59.464 than, say, a foam would be. 00:24:59.464 --> 00:25:01.630 So people were interested in these lattice material. 00:25:01.630 --> 00:25:03.269 This one here is made of aluminum. 00:25:03.269 --> 00:25:05.060 And I wanted to talk a little bit about how 00:25:05.060 --> 00:25:06.970 you can make these things. 00:25:06.970 --> 00:25:10.300 One way you can do it is by injection molding. 00:25:10.300 --> 00:25:14.450 And this here is just the centerpiece of something 00:25:14.450 --> 00:25:15.860 that would look like this. 00:25:15.860 --> 00:25:17.840 So there would be a-- I didn't bring it, 00:25:17.840 --> 00:25:20.790 but there's a top face and a bottom face that fit onto this. 00:25:20.790 --> 00:25:23.140 And they would be injection molded 00:25:23.140 --> 00:25:25.990 as three different pieces, and then assembled together. 00:25:25.990 --> 00:25:30.450 So you can make a mold in this complicated geometry, 00:25:30.450 --> 00:25:35.190 and you can make a lattice material by injection molding. 00:25:35.190 --> 00:25:40.363 We'll start with polymer lattices first. 00:25:45.280 --> 00:25:46.856 One way is injection molding. 00:25:50.330 --> 00:25:54.820 Another way to do it is by 3-D printing. 00:25:54.820 --> 00:25:58.170 You can generate a structure like that by 3-D printing. 00:25:58.170 --> 00:26:03.600 You can also make trusses in 2D and you 00:26:03.600 --> 00:26:07.630 can make them so that you can snap fit those together. 00:26:07.630 --> 00:26:12.299 So you can make little 2D trusses. 00:26:12.299 --> 00:26:13.840 Here's a little truss here and here's 00:26:13.840 --> 00:26:16.040 a little piece of a truss here. 00:26:16.040 --> 00:26:19.750 And you can make a little snap fit joints. 00:26:19.750 --> 00:26:23.710 Do you see how these ones have little divots in them? 00:26:23.710 --> 00:26:26.170 And you can make it so that these things will fit together. 00:26:26.170 --> 00:26:29.640 I think these guys-- can I do it? 00:26:29.640 --> 00:26:30.457 No. 00:26:30.457 --> 00:26:31.540 You'd have to take-- oops. 00:26:31.540 --> 00:26:32.123 Wait a minute. 00:26:32.123 --> 00:26:34.740 No, it's not that way. 00:26:34.740 --> 00:26:36.280 There we go. 00:26:36.280 --> 00:26:39.600 So you can snap them together like that. 00:26:39.600 --> 00:26:40.100 OK. 00:26:40.100 --> 00:26:43.229 I can't get it to-- there we go. 00:26:43.229 --> 00:26:44.020 So you can do that. 00:26:44.020 --> 00:26:45.600 And you do that over and over again. 00:26:45.600 --> 00:26:46.640 And if you do it over and over again, 00:26:46.640 --> 00:26:48.400 you get something that looks like that. 00:26:48.400 --> 00:26:49.450 OK? 00:26:49.450 --> 00:26:51.210 You can make a snap fit thing. 00:26:51.210 --> 00:26:52.790 Let me pass all these guys around 00:26:52.790 --> 00:26:54.105 and you can play with those. 00:26:56.805 --> 00:26:58.180 That's the injection molding one. 00:26:58.180 --> 00:27:01.770 This is the snap-fit one. 00:27:01.770 --> 00:27:02.270 Got that? 00:27:06.270 --> 00:27:06.940 Let's see. 00:27:06.940 --> 00:27:08.481 I think I have a little picture here. 00:27:08.481 --> 00:27:11.430 This is an example of the snap fit truss here. 00:27:11.430 --> 00:27:14.520 It's the thing that's getting passed around. 00:27:14.520 --> 00:27:17.180 And another clever way that was developed 00:27:17.180 --> 00:27:21.140 was by taking a monomer that's sensitive to light. 00:27:21.140 --> 00:27:23.490 You take a photo sensitive monomer. 00:27:23.490 --> 00:27:27.410 And you put a mask on top of it and the mask has holes in it. 00:27:27.410 --> 00:27:29.710 And then you shine collimated light on it. 00:27:29.710 --> 00:27:31.550 You shine, say, a laser on it. 00:27:31.550 --> 00:27:35.030 And the light goes through the holes in the mask. 00:27:35.030 --> 00:27:37.660 And it starts to polymerize the polymer 00:27:37.660 --> 00:27:39.330 because it's photosensitive. 00:27:39.330 --> 00:27:43.970 And then, as the polymer-- as it polymerizes and becomes solid, 00:27:43.970 --> 00:27:47.120 it then acts as a waveguide and draws the light down 00:27:47.120 --> 00:27:49.200 deeper into the monomer. 00:27:49.200 --> 00:27:51.720 And so the polymer acts as a wave guide. 00:27:51.720 --> 00:27:54.080 It brings the light down. 00:27:54.080 --> 00:27:58.280 And this is a schematic over here. 00:27:58.280 --> 00:28:00.340 This is a schematic showing the set up. 00:28:00.340 --> 00:28:02.960 And these are some examples of some 3D trusses 00:28:02.960 --> 00:28:05.500 that they've made using this technique. 00:28:05.500 --> 00:28:07.050 And one of the nice things about this 00:28:07.050 --> 00:28:09.160 is you can get a very small size cell size. 00:28:09.160 --> 00:28:13.141 So this is-- I think that bar-- it says 1,500 microns. 00:28:13.141 --> 00:28:13.640 That's what? 00:28:13.640 --> 00:28:14.765 One and a half millimeters. 00:28:14.765 --> 00:28:18.480 So you can get a nice, small cell size if you want that. 00:28:18.480 --> 00:28:19.753 Let me write that down. 00:28:46.700 --> 00:28:52.900 You take a photosensitive monomer 00:28:52.900 --> 00:28:55.200 and then you have it in a mold beneath a mask. 00:29:00.020 --> 00:29:07.130 And then you shine collimated light on it. 00:29:15.910 --> 00:29:19.375 And as the light shines on it, it polymerizes the monomer. 00:29:31.620 --> 00:29:34.770 So then it solidifies and then it guides the light deeper 00:29:34.770 --> 00:29:35.890 into the monomer. 00:30:02.564 --> 00:30:04.080 OK. 00:30:04.080 --> 00:30:05.300 That's that. 00:30:05.300 --> 00:30:08.360 And then finally, there's metal lattices, as well. 00:30:15.030 --> 00:30:17.670 And so this is, obviously, a metal lattice here. 00:30:17.670 --> 00:30:19.140 It's an aluminum alloy. 00:30:19.140 --> 00:30:21.280 And the metal lattice is made by taking 00:30:21.280 --> 00:30:25.310 that polymer lattice that was made by the injection molding 00:30:25.310 --> 00:30:26.300 technique. 00:30:26.300 --> 00:30:28.620 You coat that with a ceramic slurry. 00:30:28.620 --> 00:30:30.130 You burn off the polymer, and then 00:30:30.130 --> 00:30:34.110 you infiltrate the metal where the polymer used to be. 00:31:16.030 --> 00:31:17.250 OK. 00:31:17.250 --> 00:31:20.150 That's the section on processing of the honeycombs 00:31:20.150 --> 00:31:21.872 and the foams and the lattices. 00:31:21.872 --> 00:31:23.705 So there's a variety of different techniques 00:31:23.705 --> 00:31:25.730 that people have developed for making 00:31:25.730 --> 00:31:27.590 these kinds of materials. 00:31:27.590 --> 00:31:30.990 And I thought it'd be useful to just describe 00:31:30.990 --> 00:31:32.140 some of the techniques. 00:31:32.140 --> 00:31:34.455 As I think I said last time, this isn't comprehensive. 00:31:34.455 --> 00:31:36.060 This doesn't cover every technique. 00:31:36.060 --> 00:31:38.039 But it gives you a flavor of what 00:31:38.039 --> 00:31:39.830 techniques people have developed for making 00:31:39.830 --> 00:31:41.990 these kinds of materials. 00:31:41.990 --> 00:31:42.940 OK? 00:31:42.940 --> 00:31:44.100 Are we good? 00:31:44.100 --> 00:31:44.900 We're good. 00:31:44.900 --> 00:31:45.740 OK. 00:31:45.740 --> 00:31:48.220 The next part, I want to do on the structure 00:31:48.220 --> 00:31:50.360 of cellular materials. 00:31:50.360 --> 00:31:53.877 And I have a little overview. 00:31:53.877 --> 00:31:55.460 I don't think we'll finish this today, 00:31:55.460 --> 00:31:57.090 but we'll finish it tomorrow. 00:32:50.480 --> 00:32:50.990 Yeah? 00:32:50.990 --> 00:32:53.811 AUDIENCE: [INAUDIBLE] 00:32:53.811 --> 00:32:54.810 LORNA GIBSON: Down here? 00:32:54.810 --> 00:32:57.010 AUDIENCE: What happens to the ceramic? 00:32:57.010 --> 00:32:58.760 LORNA GIBSON: They get rid of the ceramic. 00:32:58.760 --> 00:33:01.320 Typically, the ceramic is not very strong 00:33:01.320 --> 00:33:04.075 and it's just fired enough so they can infiltrate it 00:33:04.075 --> 00:33:04.700 with the metal. 00:33:04.700 --> 00:33:08.300 And then they-- yeah, I think with mechanical smushing 00:33:08.300 --> 00:33:11.280 around, you can get rid of the ceramic. 00:33:11.280 --> 00:33:13.880 And the ceramic's brittle, so if you break the ceramic, 00:33:13.880 --> 00:33:15.530 you're not going to break the metal. 00:33:15.530 --> 00:33:17.030 AUDIENCE: I'm wondering if you could 00:33:17.030 --> 00:33:19.678 make a type of metal lattice [INAUDIBLE] 00:33:19.678 --> 00:33:21.625 with reducing [? the ?] oxides? 00:33:21.625 --> 00:33:24.000 LORNA GIBSON: I guess you could, if you could-- but you'd 00:33:24.000 --> 00:33:25.958 have to then make the oxide in that shape, too. 00:33:25.958 --> 00:33:28.869 You've always got to make something in that shape. 00:33:28.869 --> 00:33:29.744 AUDIENCE: [INAUDIBLE] 00:33:35.640 --> 00:33:37.640 LORNA GIBSON: Yeah, maybe you could make a foam. 00:33:37.640 --> 00:33:40.210 But to make these lattices, you need this really regular kind 00:33:40.210 --> 00:33:43.290 of structure and be able to control the structure. 00:33:43.290 --> 00:33:44.680 OK. 00:33:44.680 --> 00:33:48.820 Let me scoot out of this set of slides and get the next set up. 00:33:51.800 --> 00:33:54.370 OK. 00:33:54.370 --> 00:33:57.270 We want to talk about the structure of cellular solids. 00:33:57.270 --> 00:34:01.410 And we classify cellular materials into two main groups. 00:34:01.410 --> 00:34:03.460 One's called honeycombs. 00:34:03.460 --> 00:34:05.660 This thing down here is a honeycomb. 00:34:05.660 --> 00:34:09.389 And honeycombs have polygonal cells that fill a plane 00:34:09.389 --> 00:34:11.730 and then they're prismatic in the third direction. 00:34:11.730 --> 00:34:15.120 So you can think of them as just being a prismatic-- 00:34:15.120 --> 00:34:17.239 and they can be hexagons, they can be squares, 00:34:17.239 --> 00:34:19.365 they can be triangles-- but you can think 00:34:19.365 --> 00:34:21.100 of them as prismatic cells. 00:34:21.100 --> 00:34:24.020 And the cells are just in a 2D plane. 00:34:24.020 --> 00:34:25.389 And then we also have foams. 00:34:25.389 --> 00:34:29.060 All of these ones over here are foamed materials. 00:34:29.060 --> 00:34:31.120 And they're made up of polyhedral cells. 00:34:31.120 --> 00:34:34.469 The cells themselves are three-dimensional polyhedra. 00:34:34.469 --> 00:34:36.659 And this slide here shows a number 00:34:36.659 --> 00:34:37.889 of different types of foams. 00:34:37.889 --> 00:34:39.790 These ones are polymers up here. 00:34:39.790 --> 00:34:41.159 These are two metals. 00:34:41.159 --> 00:34:42.550 These are two ceramics. 00:34:42.550 --> 00:34:44.699 This is a glass foam down here. 00:34:44.699 --> 00:34:47.080 And this is another polymer foam down here. 00:34:47.080 --> 00:34:48.376 OK? 00:34:48.376 --> 00:34:49.810 AUDIENCE: [INAUDIBLE] 00:34:49.810 --> 00:34:50.600 LORNA GIBSON: No. 00:34:50.600 --> 00:34:51.520 I just know that. 00:34:51.520 --> 00:34:52.300 AUDIENCE: OK. 00:34:52.300 --> 00:34:54.567 LORNA GIBSON: I took those pictures so I know that. 00:34:54.567 --> 00:34:56.150 No, you can't tell by looking at them. 00:34:56.150 --> 00:34:58.250 In fact, that's one of the things about how we 00:34:58.250 --> 00:35:00.710 model the cellular materials. 00:35:00.710 --> 00:35:03.000 The fact that their structure is so similar 00:35:03.000 --> 00:35:04.810 is what gives them similar properties. 00:35:04.810 --> 00:35:06.768 And they behave in similar ways because they've 00:35:06.768 --> 00:35:09.751 got similar structures. 00:35:09.751 --> 00:35:10.251 OK. 00:35:25.110 --> 00:35:34.310 We've got 2D honeycombs, where we have polygonal cells 00:35:34.310 --> 00:35:35.700 that pack to fill a plane. 00:35:41.650 --> 00:35:44.110 And then they're prismatic in the third direction. 00:35:56.080 --> 00:36:00.290 And then we have what we call 3D cellular materials, which 00:36:00.290 --> 00:36:02.805 are foams, which have polyhedral cells. 00:36:07.560 --> 00:36:08.950 And then they pack to fill space. 00:36:16.350 --> 00:36:19.200 The properties of all of these materials 00:36:19.200 --> 00:36:21.062 depend, essentially, on three things. 00:36:24.922 --> 00:36:27.380 They're going to depend on the solid that you make it from. 00:36:27.380 --> 00:36:30.182 If you make the material from a rubber or from an aluminum, 00:36:30.182 --> 00:36:31.890 you're going to get different properties. 00:36:31.890 --> 00:36:33.806 So they depend on the properties of the solid. 00:36:45.300 --> 00:36:46.860 And some of the properties that we're 00:36:46.860 --> 00:36:52.137 going to use that are important for this type of modeling 00:36:52.137 --> 00:36:54.470 are a density of the solid-- which I'm going to call rho 00:36:54.470 --> 00:36:56.990 s-- a Young's modulus of the solid-- which I'm going to call 00:36:56.990 --> 00:36:59.580 es-- and some sort of strength of the solid-- 00:36:59.580 --> 00:37:02.390 which I'm going to call sigma ys for now. 00:37:02.390 --> 00:37:03.890 And you could think of other things. 00:37:03.890 --> 00:37:05.973 There could be a fractured toughness of the solid. 00:37:05.973 --> 00:37:07.920 There could be other kinds of things. 00:37:07.920 --> 00:37:10.800 One thing that the properties of the cellular material depend on 00:37:10.800 --> 00:37:12.560 is the properties of the solid. 00:37:12.560 --> 00:37:18.840 Another is the relative density of the cellular material. 00:37:18.840 --> 00:37:21.110 And that's the density of the cellular thing divided 00:37:21.110 --> 00:37:22.810 by the density of the solid. 00:37:22.810 --> 00:37:26.295 And that's equivalent to the volume fraction of solids. 00:37:36.109 --> 00:37:38.150 So it makes sense that the more solid you've got, 00:37:38.150 --> 00:37:40.370 the stiffer and stronger the material's going to be. 00:37:40.370 --> 00:37:43.180 So it's going to depend on how much material you've got. 00:37:43.180 --> 00:37:45.570 And it also depends-- the properties also 00:37:45.570 --> 00:37:46.972 depend on the cell geometry. 00:38:06.280 --> 00:38:09.370 The cell shape can control things like whether or not 00:38:09.370 --> 00:38:12.460 the honeycomb or the foam is isotropic or anisotropic. 00:38:12.460 --> 00:38:14.520 You can imagine, if you have a foam, 00:38:14.520 --> 00:38:16.320 and you've got equiaxed cells, you 00:38:16.320 --> 00:38:18.820 might expect to have the same properties in all directions. 00:38:18.820 --> 00:38:21.300 But if you had cells that were elongated in some way, 00:38:21.300 --> 00:38:23.300 you might expect you'd have different properties 00:38:23.300 --> 00:38:25.300 in the direction that they're elongated relative 00:38:25.300 --> 00:38:26.640 to the other plane. 00:38:26.640 --> 00:38:28.280 So cell shape can lead to anisotropy. 00:38:33.530 --> 00:38:36.560 For the foams, you can also have what 00:38:36.560 --> 00:38:39.375 we call open-cell and closed-cell foams. 00:38:48.570 --> 00:38:51.620 If you look at this slide here, and we 00:38:51.620 --> 00:38:55.840 look at this top right images-- these two up here-- 00:38:55.840 --> 00:38:59.690 the one on the left in the top is an open-celled foam. 00:38:59.690 --> 00:39:01.440 There's just material in the edges. 00:39:01.440 --> 00:39:02.330 There's no faces. 00:39:02.330 --> 00:39:05.515 And so a gas can flow between one cell and another. 00:39:05.515 --> 00:39:07.390 And then if you look at the one on the right, 00:39:07.390 --> 00:39:08.750 this is a closed-celled foam. 00:39:08.750 --> 00:39:09.500 There's faces. 00:39:09.500 --> 00:39:11.041 If you think of the polyhedra, you've 00:39:11.041 --> 00:39:14.412 got solid faces covering the faces of the polyhedra. 00:39:17.510 --> 00:39:21.430 For an open-cell foam, you've only got solid 00:39:21.430 --> 00:39:23.065 in the edges of the polyhedra. 00:39:27.840 --> 00:39:31.190 And the voids are continuous, so they're connected together. 00:39:35.230 --> 00:39:36.930 And for a closed-cell foam, you've 00:39:36.930 --> 00:39:38.765 got solid in the edges and the faces. 00:39:43.210 --> 00:39:46.270 And then the voids are separated off from each other. 00:39:46.270 --> 00:39:49.830 So we'll say, the cells are closed off from one another. 00:39:58.210 --> 00:40:01.840 Another feature of the cell geometry is the cell size. 00:40:05.470 --> 00:40:07.690 And the cell size can be important for things 00:40:07.690 --> 00:40:09.790 like the thermal properties of foams. 00:40:09.790 --> 00:40:11.540 It's important for things like the surface 00:40:11.540 --> 00:40:12.776 area per unit volume. 00:40:12.776 --> 00:40:14.650 But typically, for the mechanical properties, 00:40:14.650 --> 00:40:15.650 it's not that important. 00:40:23.867 --> 00:40:25.950 And we'll see why that is when we do the modeling. 00:40:59.135 --> 00:40:59.635 OK. 00:41:06.100 --> 00:41:08.300 Yes? 00:41:08.300 --> 00:41:12.070 AUDIENCE: For the closed-cell foams-- because we can't really 00:41:12.070 --> 00:41:17.580 see it without cutting it, is it that all of the faces 00:41:17.580 --> 00:41:19.260 are closed? 00:41:19.260 --> 00:41:22.300 Or is it like some fraction of the faces are closed? 00:41:22.300 --> 00:41:25.180 LORNA GIBSON: If you look at this one on the top right here, 00:41:25.180 --> 00:41:26.550 they're pretty much all closed. 00:41:26.550 --> 00:41:28.300 But the reason we have this little picture 00:41:28.300 --> 00:41:30.660 down here is some of them are closed and some of them 00:41:30.660 --> 00:41:31.160 are open. 00:41:31.160 --> 00:41:32.880 So you can get ones that are in between. 00:41:32.880 --> 00:41:34.820 But typically-- this is kind of unusual. 00:41:34.820 --> 00:41:36.778 Usually, they're either all open or all closed. 00:42:14.500 --> 00:42:16.240 If we look at the mechanical properties 00:42:16.240 --> 00:42:19.460 of cellular materials, typically the cell geometry 00:42:19.460 --> 00:42:20.960 doesn't have that much of an effect. 00:42:20.960 --> 00:42:23.370 The relative density is much more important. 00:42:23.370 --> 00:42:25.420 The relative density, we define as the density 00:42:25.420 --> 00:42:27.020 of the cellular solid. 00:42:27.020 --> 00:42:30.650 And when I use a parameter like rho or e or something, 00:42:30.650 --> 00:42:32.650 if it's got a star, it's for the cellular thing 00:42:32.650 --> 00:42:35.017 and if it's got an s, it's for the solid. 00:42:35.017 --> 00:42:36.600 So rho star is going to be the density 00:42:36.600 --> 00:42:37.770 of the cellular material. 00:42:46.590 --> 00:42:48.680 And rho s is going to be the density of the solid 00:42:48.680 --> 00:42:49.440 it's made from. 00:43:01.570 --> 00:43:05.282 And so the relative density is just rho star over rho s. 00:43:05.282 --> 00:43:06.990 And I just wanted to show you how this is 00:43:06.990 --> 00:43:08.920 the volume fraction of solids. 00:43:08.920 --> 00:43:11.800 So rho star is going to be the mass of solid 00:43:11.800 --> 00:43:13.047 over the total volume. 00:43:13.047 --> 00:43:15.380 Imagine you've got a honeycomb or a foam and you've got, 00:43:15.380 --> 00:43:17.350 say, a unit cube of it, the sum total 00:43:17.350 --> 00:43:19.375 volume of the whole thing-- the density 00:43:19.375 --> 00:43:20.750 of the cellular material is going 00:43:20.750 --> 00:43:22.775 to the mass of the solid over the whole volume. 00:43:22.775 --> 00:43:24.150 The density of the solid is going 00:43:24.150 --> 00:43:28.130 to be the mass of the solid over the volume of the solid. 00:43:28.130 --> 00:43:30.937 This is really just equivalent to the volume fraction 00:43:30.937 --> 00:43:32.520 of solids, how much solids you've got. 00:43:41.610 --> 00:43:43.980 And that's also n equal to 1 minus the porosity. 00:43:51.720 --> 00:43:54.820 Typical values for cellular materials-- 00:43:54.820 --> 00:43:59.350 I think last time I passed around one of those collagen 00:43:59.350 --> 00:44:01.300 scaffolds-- those tissue enineering scaffolds. 00:44:01.300 --> 00:44:03.680 It was in a little plastic bag. 00:44:03.680 --> 00:44:09.920 That collagen scaffold has a relative density of 0.005, 00:44:09.920 --> 00:44:14.290 so its 0.5% solid and 99.5% air. 00:44:14.290 --> 00:44:20.950 And if we look at typical polymer foams, 00:44:20.950 --> 00:44:22.510 the relative density is typically 00:44:22.510 --> 00:44:25.505 between about 2% and 20%. 00:44:31.920 --> 00:44:34.610 And if we look at something like softwoods-- wood 00:44:34.610 --> 00:44:36.510 is a cellular material. 00:44:36.510 --> 00:44:39.210 And we look at softwoods, the relative density 00:44:39.210 --> 00:44:43.860 is usually between about 15% and about 40%. 00:44:43.860 --> 00:44:44.711 Something like that. 00:44:44.711 --> 00:44:45.210 OK? 00:44:49.550 --> 00:44:51.370 As the relative density increases, 00:44:51.370 --> 00:44:53.410 you get more material on the cell edges, 00:44:53.410 --> 00:44:56.310 and if it's closed-cell foam, on the cell faces. 00:44:56.310 --> 00:44:58.480 And the pore volume decreases. 00:44:58.480 --> 00:45:00.550 And you can think of some limit. 00:45:00.550 --> 00:45:03.100 If you keep increasing the relative density more and more 00:45:03.100 --> 00:45:05.550 and more, eventually you've got-- it's not really 00:45:05.550 --> 00:45:06.950 a cellular material anymore. 00:45:06.950 --> 00:45:10.380 It's more like a solid with little isolated pores in it. 00:45:10.380 --> 00:45:11.991 And so there's two bounds. 00:45:11.991 --> 00:45:14.490 And the density has to be less than a certain amount for you 00:45:14.490 --> 00:45:16.740 to consider it a cellular material in the models 00:45:16.740 --> 00:45:19.690 that we're going to derive to be valid. 00:45:19.690 --> 00:45:23.040 And if the relevant density is more than a certain amount, 00:45:23.040 --> 00:45:25.373 people model it as a solid with isolated holes. 00:46:06.450 --> 00:46:10.627 If I have a unit square of material, 00:46:10.627 --> 00:46:12.210 if it's a cellular material, you might 00:46:12.210 --> 00:46:15.390 expect that you've got pores that would look like this. 00:46:15.390 --> 00:46:18.850 And you've got relatively thin cell walls, 00:46:18.850 --> 00:46:21.360 relative to the length of the material. 00:46:21.360 --> 00:46:25.730 And for a cellular material, typically, the relative density 00:46:25.730 --> 00:46:31.290 is less than about 0.3. 00:46:31.290 --> 00:46:34.220 And when we come to the modeling for the honeycombs 00:46:34.220 --> 00:46:38.410 and the foams, we're going to see that the cell walls deform, 00:46:38.410 --> 00:46:40.610 in many cases, by bending. 00:46:40.610 --> 00:46:42.260 And that you can model the deformation 00:46:42.260 --> 00:46:43.780 by modeling the bending. 00:46:43.780 --> 00:46:46.340 And that the bending dominates the behavior 00:46:46.340 --> 00:46:48.870 if the density is less than about that. 00:46:48.870 --> 00:46:52.230 And at the other extreme, you can 00:46:52.230 --> 00:46:54.110 imagine if you had just little teeny pores. 00:46:54.110 --> 00:46:55.760 I have a little pore here and one here 00:46:55.760 --> 00:46:58.620 and one there and one there. 00:46:58.620 --> 00:47:00.350 That's not really a cellular material. 00:47:00.350 --> 00:47:02.750 It's just got a teeny weeny little bit of pores. 00:47:02.750 --> 00:47:07.900 And that could be modeled as isolated pores in a solid. 00:47:07.900 --> 00:47:09.704 Each one is acting independently. 00:47:12.760 --> 00:47:16.510 And people have found that that is appropriate 00:47:16.510 --> 00:47:19.440 if the relative density is greater than about 0.8. 00:47:19.440 --> 00:47:21.430 And then, in between, there's a transition 00:47:21.430 --> 00:47:23.670 in behavior between the cellular solid 00:47:23.670 --> 00:47:26.830 and the isolated pores in the solid. 00:47:26.830 --> 00:47:28.020 OK? 00:47:28.020 --> 00:47:28.691 Are we OK? 00:48:04.512 --> 00:48:06.720 The next thing I wanted to talk about was unit cells. 00:48:09.310 --> 00:48:12.440 Especially for honeycombs, people often use unit cells. 00:48:12.440 --> 00:48:14.450 A hexagonal cell is an obvious one 00:48:14.450 --> 00:48:18.150 to use to model this kind of behavior. 00:48:18.150 --> 00:48:20.880 For honeycomb materials, you can have unit cells 00:48:20.880 --> 00:48:22.420 and you can have different ones. 00:48:22.420 --> 00:48:25.190 On the left here, we've got triangles, in the middle, 00:48:25.190 --> 00:48:26.610 I've got squares. 00:48:26.610 --> 00:48:28.850 On the right-hand side, I've got hexagons. 00:48:28.850 --> 00:48:31.130 And you can see, even if you have a certain unit cell, 00:48:31.130 --> 00:48:32.920 there's also different ways to stack it. 00:48:32.920 --> 00:48:36.920 So the number of edges that meet at a vertex 00:48:36.920 --> 00:48:40.200 is different for, say, this example on the top left 00:48:40.200 --> 00:48:42.950 and this example on the bottom left. 00:48:42.950 --> 00:48:46.510 Here, we've got six members coming into each vertex, 00:48:46.510 --> 00:48:48.040 and here, we've got four. 00:48:48.040 --> 00:48:51.440 And again, this stacking for the two square cells 00:48:51.440 --> 00:48:53.700 is also different. 00:48:53.700 --> 00:48:58.070 So you can have different numbers of edges per vertex. 00:48:58.070 --> 00:49:00.800 Another thing to note that's kind of interesting-- 00:49:00.800 --> 00:49:03.730 if you look at the honeycomb cells here, 00:49:03.730 --> 00:49:06.660 this one on the top left-- this equilateral triangle 00:49:06.660 --> 00:49:10.040 one with the stacking-- and this one on the top right-- 00:49:10.040 --> 00:49:13.790 the regular hexagonal cells-- those two are isotropic 00:49:13.790 --> 00:49:16.500 for linear elastic behavior, whereas all the other ones are 00:49:16.500 --> 00:49:18.650 not. 00:49:18.650 --> 00:49:22.890 So we have 2D honeycomb unit cells. 00:49:26.960 --> 00:49:38.550 We can have triangles, squares, and hexagons. 00:49:38.550 --> 00:49:40.656 They can be stacked in more than one way. 00:49:49.670 --> 00:49:53.377 And that gives different numbers of edges per vertex. 00:50:01.110 --> 00:50:04.370 And in that figure, a and e are isotropic, 00:50:04.370 --> 00:50:06.160 for linear elasticity. 00:50:16.730 --> 00:50:17.300 OK. 00:50:17.300 --> 00:50:19.160 When we come to modeling the honeycombs, 00:50:19.160 --> 00:50:21.780 we're going to focus on the hexagonal cells. 00:50:21.780 --> 00:50:24.300 We'll talk a little bit about the square and triangular 00:50:24.300 --> 00:50:26.490 cells, as well. 00:50:26.490 --> 00:50:28.371 And then, for foams, when you look 00:50:28.371 --> 00:50:30.120 at the structure of a foam, it's obviously 00:50:30.120 --> 00:50:33.080 not a unit cell that repeats over and over again. 00:50:33.080 --> 00:50:36.450 But people started off trying to model the mechanical behavior 00:50:36.450 --> 00:50:40.950 of foams by looking at periodic repeating polyhedral cells. 00:50:40.950 --> 00:50:43.540 And there's three cells here that are prismatic. 00:50:43.540 --> 00:50:46.050 We're not really going to talk about those beyond this. 00:50:46.050 --> 00:50:49.800 So they're not really physically realistic or interesting. 00:50:49.800 --> 00:50:53.310 But people would use these two cells here in initial attempts 00:50:53.310 --> 00:50:54.630 to model foams. 00:50:54.630 --> 00:50:58.220 And this one here is called the rhombic dodecahedra. 00:50:58.220 --> 00:50:59.990 Rhombic because each of the faces 00:50:59.990 --> 00:51:05.770 has four sides and dodecahedra because each polyhedra has 12 00:51:05.770 --> 00:51:07.770 faces. 00:51:07.770 --> 00:51:10.640 I forget if I've bored you with my Latin already. 00:51:10.640 --> 00:51:14.160 Hedron means face in-- oh, this is Greek, I think. 00:51:14.160 --> 00:51:15.870 Hedron means face. 00:51:15.870 --> 00:51:18.600 Do is two, deca is 10. 00:51:18.600 --> 00:51:20.530 So dodeca is two plus 10. 00:51:20.530 --> 00:51:21.930 It's got 12 faces. 00:51:21.930 --> 00:51:22.700 OK? 00:51:22.700 --> 00:51:25.870 So that's the rhombic dodecahedra over here. 00:51:25.870 --> 00:51:29.536 And then this bottom one down here is a tetrakaidecahedra. 00:51:29.536 --> 00:51:30.410 It's a similar thing. 00:51:30.410 --> 00:51:32.930 Tetra's four, kai mean and. 00:51:32.930 --> 00:51:37.000 Four and 10-- tetra kai deca-- it's got 14 faces. 00:51:37.000 --> 00:51:37.910 OK? 00:51:37.910 --> 00:51:40.310 And those two pack to fill space. 00:51:40.310 --> 00:51:44.320 I think those are the only uniform polyhedra 00:51:44.320 --> 00:51:47.900 that pack to fill space. 00:51:47.900 --> 00:51:51.545 Here is the 3D foams. 00:51:54.910 --> 00:52:02.640 We have the rhombic dodecahedron and the tetrakaidecahedron. 00:52:34.510 --> 00:52:38.920 And the tetrakaidecahedron packs in a bcc packing. 00:52:56.020 --> 00:53:00.030 Initial models for foams-- they took these two unit cells. 00:53:00.030 --> 00:53:02.760 And what they would do is have an infinite array of them 00:53:02.760 --> 00:53:04.060 to make up the whole material. 00:53:04.060 --> 00:53:05.900 And then they would isolate a unit cell. 00:53:05.900 --> 00:53:08.900 And they would apply loads-- some say compressive stress, 00:53:08.900 --> 00:53:09.860 for example. 00:53:09.860 --> 00:53:12.420 And then they would figure out what the load, or force, was 00:53:12.420 --> 00:53:15.360 in every single member, and how much that member deformed. 00:53:15.360 --> 00:53:17.024 And they would figure out the component 00:53:17.024 --> 00:53:18.690 of the deformation in the same direction 00:53:18.690 --> 00:53:19.820 that they were putting the load on. 00:53:19.820 --> 00:53:22.153 And they would figure out things like a Young's modulus, 00:53:22.153 --> 00:53:24.870 or they would figure out when there was some failure of one 00:53:24.870 --> 00:53:26.620 of these struts, and they would figure out 00:53:26.620 --> 00:53:27.690 a strength for the foam. 00:53:27.690 --> 00:53:30.070 But you can kind of imagine, geometrically, 00:53:30.070 --> 00:53:31.740 not that easy to keep straight. 00:53:31.740 --> 00:53:33.630 A little bit complicated. 00:53:33.630 --> 00:53:36.590 So one way to model foams is by using these unit cells. 00:53:36.590 --> 00:53:38.530 But we're going to talk about a different way 00:53:38.530 --> 00:53:40.610 to do it, as well. 00:53:40.610 --> 00:53:41.340 OK. 00:53:41.340 --> 00:53:42.375 So those are unit cells. 00:53:47.770 --> 00:53:50.930 When they make foams, as we just talked about, 00:53:50.930 --> 00:53:54.380 one way to make a foam is by blowing a gas into a liquid. 00:53:54.380 --> 00:53:57.040 And if you blow a gas into a liquid, 00:53:57.040 --> 00:53:58.720 then the surface tension is going 00:53:58.720 --> 00:54:00.420 to have an effect on the cell geometry 00:54:00.420 --> 00:54:02.430 and on the shape of the cells. 00:54:02.430 --> 00:54:04.890 And if the surface tension is isotropic-- 00:54:04.890 --> 00:54:08.700 if it's the same in all directions-- then the structure 00:54:08.700 --> 00:54:10.940 that you get is one that minimizes the surface 00:54:10.940 --> 00:54:12.920 area per unit volume. 00:54:12.920 --> 00:54:15.560 And so people were interested in what sort of cell shape 00:54:15.560 --> 00:54:18.440 minimizes the surface area per unit volume. 00:54:18.440 --> 00:54:21.530 And Lord Kelvin, in the 1800s, was 00:54:21.530 --> 00:54:22.890 the person who worked this out. 00:54:22.890 --> 00:54:26.550 And this is called the Kelvin tetrakaidecahedron. 00:54:26.550 --> 00:54:29.310 And it's not just a straight tetrakaidecahedron. 00:54:29.310 --> 00:54:33.160 There's a slight curvature to the cells here, to the faces. 00:54:33.160 --> 00:54:35.990 And you can kind of see it in some of the edges here. 00:54:35.990 --> 00:54:37.820 Like if we-- let me get my little pointer. 00:54:37.820 --> 00:54:39.979 If you look at that edge, it's not straight. 00:54:39.979 --> 00:54:41.270 This edge here is not straight. 00:54:41.270 --> 00:54:43.120 It's got a little bit of a curvature to it. 00:54:43.120 --> 00:54:47.360 But this minimizes the surface area per unit volume. 00:54:47.360 --> 00:54:49.800 And then more recently, in the 1990's, there 00:54:49.800 --> 00:54:53.840 were two people-- Dennis Weaire and Robert Phelan-- discovered 00:54:53.840 --> 00:54:57.030 that this structure here-- which isn't a single polyhedron, 00:54:57.030 --> 00:55:00.410 but it's made up of a few polyhedra. 00:55:00.410 --> 00:55:03.560 That has a slightly smaller surface area per unit volume. 00:55:03.560 --> 00:55:05.790 Smaller by 0.3%. 00:55:05.790 --> 00:55:07.580 So, a tiny bit smaller. 00:55:07.580 --> 00:55:09.810 OK. 00:55:09.810 --> 00:55:10.370 Let's see. 00:55:10.370 --> 00:55:15.360 What I'll say here is that foams are often made 00:55:15.360 --> 00:55:17.115 by blowing a gas into a liquid. 00:55:33.190 --> 00:55:40.600 And if the surface tension controls and it's isotropic, 00:55:40.600 --> 00:55:42.960 then the structure will minimize the surface area 00:55:42.960 --> 00:55:43.680 per unit volume. 00:57:55.190 --> 00:57:55.690 OK. 00:58:09.430 --> 00:58:12.480 That's relevant if the foam is made 00:58:12.480 --> 00:58:16.820 by blowing a gas into a liquid and surface tension 00:58:16.820 --> 00:58:18.430 is the controlling factor. 00:58:18.430 --> 00:58:21.620 Sometimes foams are made by supersaturating a liquid 00:58:21.620 --> 00:58:23.720 with a gas, and then you nucleate bubbles, 00:58:23.720 --> 00:58:25.020 and then the bubbles grow. 00:58:25.020 --> 00:58:26.811 So there's a nucleation and growth process. 00:58:26.811 --> 00:58:28.400 So that's a little bit different. 00:58:28.400 --> 00:58:31.050 And if you have a nucleation and growth process, 00:58:31.050 --> 00:58:34.700 you get a structure that is similar to something 00:58:34.700 --> 00:58:36.940 called a Voronoi structure. 00:58:36.940 --> 00:58:38.810 In an idealized case, imagine that you 00:58:38.810 --> 00:58:42.090 have random points that are nucleation points 00:58:42.090 --> 00:58:45.180 and that you start to grow bubbles 00:58:45.180 --> 00:58:47.880 at those nucleation points. 00:58:47.880 --> 00:58:49.850 So you start off with these random points. 00:58:49.850 --> 00:58:52.150 And the bubbles all start to grow at the same time 00:58:52.150 --> 00:58:54.460 and they all grow at the same linear rate. 00:58:54.460 --> 00:58:57.110 If you have that situation, then you 00:58:57.110 --> 00:58:59.280 end up with this Voronoi kind of structure. 00:58:59.280 --> 00:59:00.660 And I've shown a 2D version of it 00:59:00.660 --> 00:59:02.810 here just because it's easier to see in 2D, 00:59:02.810 --> 00:59:05.320 but you can imagine a 3D system. 00:59:05.320 --> 00:59:08.060 And in order to make one of these Voronoi honeycombs, 00:59:08.060 --> 00:59:11.571 you can imagine-- if you have random points-- here, 00:59:11.571 --> 00:59:14.070 say that little point there is one of the nucleation points, 00:59:14.070 --> 00:59:15.850 and here's another point here-- you 00:59:15.850 --> 00:59:19.910 form the structure by drawing the perpendicular bisectors 00:59:19.910 --> 00:59:21.430 between each pair of points. 00:59:21.430 --> 00:59:23.840 This is the bisector between these two points. 00:59:23.840 --> 00:59:26.100 Here's a bisector between those two points. 00:59:26.100 --> 00:59:29.260 And then you form the envelope of those lines 00:59:29.260 --> 00:59:31.490 around each nucleation point. 00:59:31.490 --> 00:59:33.660 And that, then, gives you that structure. 00:59:33.660 --> 00:59:36.770 And you can see, this structure here is kind of angular. 00:59:36.770 --> 00:59:39.430 It doesn't look that representative of something 00:59:39.430 --> 00:59:41.230 like a foam. 00:59:41.230 --> 00:59:43.450 But if you have an exclusion distance, 00:59:43.450 --> 00:59:46.250 where you say that you're not going to allow the nucleation 00:59:46.250 --> 00:59:49.660 points to be closer than some given distance-- your exclusion 00:59:49.660 --> 00:59:51.920 distance-- then you get this structure here. 00:59:51.920 --> 00:59:53.530 And this starts to look a lot more 00:59:53.530 --> 00:59:55.340 like a foamy kind of structure. 00:59:55.340 --> 00:59:58.140 So these Voronoi structures are representative 00:59:58.140 --> 01:00:00.730 of structures that are related to nucleation 01:00:00.730 --> 01:00:02.990 and growth of the bubbles, or nucleation and growth 01:00:02.990 --> 01:00:04.340 processes. 01:00:04.340 --> 01:00:07.375 Let me write down something about Voronoi things. 01:00:44.000 --> 01:00:46.470 And these Voronoi structures were first 01:00:46.470 --> 01:00:49.164 developed to look at grain growth in metals. 01:00:49.164 --> 01:00:51.080 They weren't developed for cellular materials. 01:00:51.080 --> 01:00:54.170 But you can use them to model cellular materials, 01:00:54.170 --> 01:00:57.426 as well, as long as it's a nucleation and growth process. 01:00:59.954 --> 01:01:01.370 We'll say that foams are sometimes 01:01:01.370 --> 01:01:24.030 made by supersaturating a liquid with a gas, 01:01:24.030 --> 01:01:26.590 and then reducing the pressure so that the bubbles nucleate 01:01:26.590 --> 01:01:27.340 and grow. 01:01:46.824 --> 01:01:48.990 So initially, the bubbles are going to form spheres. 01:01:57.360 --> 01:01:59.410 But as the spheres grow, they start 01:01:59.410 --> 01:02:02.315 to intersect with each other and form polyhedral cells. 01:02:29.060 --> 01:02:30.480 And you get the Voronoi structure 01:02:30.480 --> 01:02:34.480 by thinking about an idealized case in which you randomly 01:02:34.480 --> 01:02:38.760 nucleate the-- you have nucleation points at a randomly 01:02:38.760 --> 01:02:40.542 distributed space. 01:02:40.542 --> 01:02:42.000 They start to grow at the same time 01:02:42.000 --> 01:02:43.720 and they grow at the same linear rate. 01:04:00.750 --> 01:04:01.250 OK. 01:04:01.250 --> 01:04:13.570 The Voronoi honeycomb, or the foam-- 01:04:13.570 --> 01:04:16.680 you can form that by drawing the perpendicular bisectors 01:04:16.680 --> 01:04:17.775 between the random points. 01:05:22.125 --> 01:05:24.530 So each cell contains the points that 01:05:24.530 --> 01:05:30.050 are closer to the nucleation point than any other point-- 01:05:30.050 --> 01:05:31.490 or any other nucleation point. 01:06:06.162 --> 01:06:13.070 And if we just do this process as I've described here, 01:06:13.070 --> 01:06:15.530 you end up with a Voronoi structure, 01:06:15.530 --> 01:06:17.850 where the cells appear kind of angular. 01:06:17.850 --> 01:06:19.920 And if you specify an exclusion distance, 01:06:19.920 --> 01:06:22.140 where you say the nucleation points can't be closer 01:06:22.140 --> 01:06:24.890 than a certain distance, then the cells 01:06:24.890 --> 01:06:27.905 become less angular, and of more similar size. 01:07:19.201 --> 01:07:19.700 OK. 01:07:19.700 --> 01:07:22.830 So are we good with the Voronoi honeycomb nucleation and growth 01:07:22.830 --> 01:07:23.330 idea? 01:07:26.124 --> 01:07:26.624 Alrighty. 01:08:42.778 --> 01:08:43.630 All right. 01:08:43.630 --> 01:08:46.100 If we think about cell shape-- if we start with honeycombs 01:08:46.100 --> 01:08:50.000 and we just think about it hexagonal honeycombs, if I have 01:08:50.000 --> 01:08:53.120 a regular hexagonal honeycomb so that all the edges are 01:08:53.120 --> 01:08:58.140 the same length and this angle here is 30 degrees, 01:08:58.140 --> 01:09:01.319 then that is an isotropic material 01:09:01.319 --> 01:09:03.483 in the plane in the linear elastic regime. 01:09:26.342 --> 01:09:27.800 One of the things we're going to do 01:09:27.800 --> 01:09:30.279 is calculate-- if I loan it this way on, 01:09:30.279 --> 01:09:31.430 what's the Young's modulus? 01:09:31.430 --> 01:09:33.660 If I load it that way on, what's the Young's modulus? 01:09:33.660 --> 01:09:35.300 And we're going to find they're the same, in fact, no matter 01:09:35.300 --> 01:09:36.399 which way on I loaded it. 01:09:36.399 --> 01:09:37.560 It would be the same. 01:09:37.560 --> 01:09:41.109 But if I now have my honeycomb, and imagine that I stretched it 01:09:41.109 --> 01:09:43.670 out-- and I'm kind of exaggerating how much we 01:09:43.670 --> 01:09:45.370 might stretch it out. 01:09:45.370 --> 01:09:47.100 But if we did something like that, 01:09:47.100 --> 01:09:49.220 it wouldn't be too surprising to think that the properties are 01:09:49.220 --> 01:09:51.220 going to be different if I loaded it this way on 01:09:51.220 --> 01:09:53.080 and that way on. 01:09:53.080 --> 01:09:55.810 And in terms of the cell geometry, 01:09:55.810 --> 01:10:00.520 I'm going to call that vertical cell edge length h. 01:10:00.520 --> 01:10:04.860 And I'm going to call this one-- the inclined one-- of length l. 01:10:04.860 --> 01:10:08.140 I'm going to say that angle is the angle theta. 01:10:08.140 --> 01:10:11.160 And the cell shape can be defined by the ratio of h 01:10:11.160 --> 01:10:13.070 over l and that angle theta. 01:10:18.030 --> 01:10:18.840 OK. 01:10:18.840 --> 01:10:21.770 When we derive equations for the mechanical properties 01:10:21.770 --> 01:10:24.250 of the honeycombs, we're going to find that they depend 01:10:24.250 --> 01:10:25.970 on some solid properties. 01:10:25.970 --> 01:10:27.880 Say, the modulus of the honeycombs 01:10:27.880 --> 01:10:29.615 can depend on the modulus of the solid. 01:10:29.615 --> 01:10:31.740 It's going to depend on the relative density raised 01:10:31.740 --> 01:10:32.490 to some power. 01:10:32.490 --> 01:10:34.100 And we're going to figure out what that is. 01:10:34.100 --> 01:10:35.599 And then it's going to depend, also, 01:10:35.599 --> 01:10:37.910 on some function of h over l and theta. 01:10:37.910 --> 01:10:40.210 And that function really represents the contribution 01:10:40.210 --> 01:10:44.680 of the cell geometry to the mechanical properties. 01:10:44.680 --> 01:10:45.180 OK. 01:10:45.180 --> 01:10:47.300 That's the honeycombs. 01:10:47.300 --> 01:10:51.230 It's fairly straightforward to characterize the shell shape 01:10:51.230 --> 01:10:53.420 for the honeycombs. 01:10:53.420 --> 01:11:00.000 It's a little more involved to do it for the foams. 01:11:00.000 --> 01:11:02.880 And the technique that's used is called the mean intercept 01:11:02.880 --> 01:11:03.380 length. 01:11:03.380 --> 01:11:05.545 At least, that's one technique that's used. 01:11:05.545 --> 01:11:06.920 Let me wait until you've finished 01:11:06.920 --> 01:11:08.300 writing because I want you to see 01:11:08.300 --> 01:11:11.270 the picture as I talk about it. 01:11:11.270 --> 01:11:13.410 OK? 01:11:13.410 --> 01:11:13.910 OK. 01:11:13.910 --> 01:11:14.770 Here's-- whoops. 01:11:14.770 --> 01:11:16.730 My pointer keeps disappearing. 01:11:16.730 --> 01:11:20.850 This top left picture here shows an SEM image of a foam. 01:11:20.850 --> 01:11:23.770 And you can see, you've got some big cells and little cells 01:11:23.770 --> 01:11:28.530 and there's no obvious way to characterize the cell shape. 01:11:28.530 --> 01:11:31.620 And what people do to calculate this mean intercept length 01:11:31.620 --> 01:11:34.170 is they would take an image. 01:11:34.170 --> 01:11:38.090 They would then sketch out just the cell edges 01:11:38.090 --> 01:11:39.780 that touch a plane's surface. 01:11:39.780 --> 01:11:42.210 So all these black lines are just the-- 01:11:42.210 --> 01:11:44.620 if you took your-- if you put ink on your foam 01:11:44.620 --> 01:11:47.810 and you just put it on a pad and put it on a piece of paper, 01:11:47.810 --> 01:11:50.600 you would get this outline of the edges of the cells, where 01:11:50.600 --> 01:11:52.300 they intersect that plane. 01:11:52.300 --> 01:11:54.480 And then what people do is they draw test circles. 01:11:54.480 --> 01:11:56.680 Here's the test circle here. 01:11:56.680 --> 01:12:01.870 And they draw parallel equiaxed, or equidistant lines. 01:12:01.870 --> 01:12:03.420 So the lines here are parallel. 01:12:03.420 --> 01:12:05.050 They're all at, say, zero degrees. 01:12:05.050 --> 01:12:07.110 And they're all the same distance apart. 01:12:07.110 --> 01:12:09.970 And then they count the number of intercepts. 01:12:09.970 --> 01:12:13.060 They count-- say we went out here. 01:12:13.060 --> 01:12:14.530 The cell wall intercepts here. 01:12:14.530 --> 01:12:16.157 There's one that intercepts here. 01:12:16.157 --> 01:12:17.240 And then, it'd go up here. 01:12:17.240 --> 01:12:18.330 Here's another intercept. 01:12:18.330 --> 01:12:19.371 Here's another intercept. 01:12:19.371 --> 01:12:22.470 So they count the number of intercepts of the cell wall 01:12:22.470 --> 01:12:23.830 with the lines. 01:12:23.830 --> 01:12:26.150 And then they get a mean intercept length, 01:12:26.150 --> 01:12:30.177 which is characteristic of the cell dimension. 01:12:30.177 --> 01:12:32.010 And then what they do-- because this is just 01:12:32.010 --> 01:12:34.240 in one orientation-- you would then 01:12:34.240 --> 01:12:37.289 rotate those parallel lines by, say, 5 degrees 01:12:37.289 --> 01:12:38.330 and do it all over again. 01:12:38.330 --> 01:12:40.121 And get another length at 5 degrees and one 01:12:40.121 --> 01:12:41.760 at 10 degrees one at 15. 01:12:41.760 --> 01:12:44.290 And so you get different lengths for the intercepts 01:12:44.290 --> 01:12:46.860 as you rotate your parallel lines around. 01:12:46.860 --> 01:12:48.610 And then you make a polar plot, and that's 01:12:48.610 --> 01:12:50.680 what the thing is down at the bottom here. 01:12:50.680 --> 01:12:53.240 And so you plot your mean intercept length 01:12:53.240 --> 01:12:56.480 as a function of the angle that you measured it at. 01:12:56.480 --> 01:12:58.399 And you can fit it to an ellipse. 01:12:58.399 --> 01:12:59.940 And if you do it in three dimensions, 01:12:59.940 --> 01:13:01.890 you fit it to an ellipsoid. 01:13:01.890 --> 01:13:06.660 And the major and minor axes of that ellipse, or ellipsoid, 01:13:06.660 --> 01:13:09.816 are characteristic of how elongated the cell is 01:13:09.816 --> 01:13:10.982 in the different directions. 01:13:10.982 --> 01:13:13.000 And the orientation of that ellipsoid 01:13:13.000 --> 01:13:16.450 is characteristic of the orientation of the cells. 01:13:16.450 --> 01:13:19.490 Those of you who took 303, too, you remember Mohr's circles? 01:13:19.490 --> 01:13:21.200 Is this beginning to look familiar? 01:13:21.200 --> 01:13:23.810 It's the same kind of idea as Mohr's circles. 01:13:23.810 --> 01:13:26.340 Same way we have principal stresses and orientation 01:13:26.340 --> 01:13:29.160 of principal stresses, now we have principal dimensions 01:13:29.160 --> 01:13:31.160 and the orientation of the principal dimensions. 01:13:31.160 --> 01:13:32.830 So it's the same kind of idea. 01:13:32.830 --> 01:13:35.230 OK? 01:13:35.230 --> 01:13:36.230 Let's see. 01:13:36.230 --> 01:13:39.410 I feel like I'm getting to the end here. 01:13:39.410 --> 01:13:42.300 Maybe I'll stop there for today. 01:13:42.300 --> 01:13:44.840 But next time, I'll write down the whole technique 01:13:44.840 --> 01:13:48.030 about how we get these mean intercepts 01:13:48.030 --> 01:13:49.770 and get this ellipsoid. 01:13:49.770 --> 01:13:54.524 And I'm going to write the mean intercepts down as a matrix. 01:13:54.524 --> 01:13:56.190 But you could also write it as a tensor. 01:13:56.190 --> 01:13:57.815 And there's something called the fabric 01:13:57.815 --> 01:14:00.860 tensor, which characterizes the shape of the cells. 01:14:00.860 --> 01:14:03.280 And as you might imagine, the same is with the honeycomb. 01:14:03.280 --> 01:14:05.770 If you have equiaxed cells in the foams, 01:14:05.770 --> 01:14:09.010 you might expect you would get isotropic properties. 01:14:09.010 --> 01:14:12.650 If you have cells that are stretched out in some way-- 01:14:12.650 --> 01:14:16.930 so you've got different principal dimensions for them-- 01:14:16.930 --> 01:14:19.130 then you've got anisotropic material. 01:14:19.130 --> 01:14:21.730 And you can relate how much anisotropy to the shape 01:14:21.730 --> 01:14:22.681 of the cells. 01:14:22.681 --> 01:14:23.180 OK. 01:14:23.180 --> 01:14:26.040 I'm going to stop there for today. 01:14:26.040 --> 01:14:27.100 I'll see you tomorrow. 01:14:27.100 --> 01:14:29.110 Seems very sudden. 01:14:29.110 --> 01:14:30.820 I'll see you tomorrow. 01:14:30.820 --> 01:14:34.010 I'll pick up and I'll finish this section on the structure. 01:14:34.010 --> 01:14:35.159 We've got a bit more to do. 01:14:35.159 --> 01:14:37.450 And then we'll start looking at honeycombs and modeling 01:14:37.450 --> 01:14:38.480 honeycombs. 01:14:38.480 --> 01:14:39.950 The honeycombs are simpler to model 01:14:39.950 --> 01:14:42.480 just because they have this nice simple unit cell. 01:14:42.480 --> 01:14:44.490 So we'll start with that, and then we'll 01:14:44.490 --> 01:14:46.000 move from there to the foams. 01:14:46.000 --> 01:14:47.550 OK?