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 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 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. 00:00:26.470 --> 00:00:28.790 LORNA GIBSON: I wanted to talk a little bit about cork 00:00:28.790 --> 00:00:30.759 partly because cork is kind of interesting. 00:00:30.759 --> 00:00:32.800 Cork has a structure that's a little bit like one 00:00:32.800 --> 00:00:34.290 of those honeycomb things. 00:00:34.290 --> 00:00:37.280 What I'm going to do is I'm just going to talk and go 00:00:37.280 --> 00:00:38.600 through the slides. 00:00:38.600 --> 00:00:40.630 I'm not going to write the notes on the board. 00:00:40.630 --> 00:00:42.070 There's only a few pages of notes, 00:00:42.070 --> 00:00:43.427 and it's in the Stellar site. 00:00:43.427 --> 00:00:46.010 I'm not going to write notes for this because it's just really 00:00:46.010 --> 00:00:47.140 for fun. 00:00:47.140 --> 00:00:50.900 This first slide starts off with historical uses of cork. 00:00:50.900 --> 00:00:53.120 Cork was used by the Romans. 00:00:53.120 --> 00:00:55.850 They used it for the soles of sandals, the same as we do. 00:00:55.850 --> 00:00:58.360 And they used it for stopping bottles 00:00:58.360 --> 00:00:59.985 of wine, the same as we do. 00:00:59.985 --> 00:01:02.360 But they didn't realize that you could just use the cork. 00:01:02.360 --> 00:01:04.640 They would take the cork, put it in the wine bottle, 00:01:04.640 --> 00:01:06.380 and then they would use pitch which 00:01:06.380 --> 00:01:09.450 is a tarry stuff from a tree. 00:01:09.450 --> 00:01:13.370 They would use the pitch and seal the bottle with the pitch. 00:01:13.370 --> 00:01:15.704 In the 1600s, there were some Benedictine monks 00:01:15.704 --> 00:01:17.620 that realized that you could just use the cork 00:01:17.620 --> 00:01:19.040 and not use the pitch. 00:01:19.040 --> 00:01:21.460 They were the ones who really perfected the use of corks 00:01:21.460 --> 00:01:23.590 in wine bottles for sealing it. 00:01:23.590 --> 00:01:25.640 So what is cork? 00:01:25.640 --> 00:01:27.570 Cork is the bark of a tree called 00:01:27.570 --> 00:01:29.800 Quercus suber, the cork oak. 00:01:29.800 --> 00:01:32.950 Here's a piece of the cork bark. 00:01:32.950 --> 00:01:35.727 All trees have a layer of cork in them. 00:01:35.727 --> 00:01:37.810 But the thing that's different about Quercus suber 00:01:37.810 --> 00:01:40.240 is that it's very thick and the corks are 00:01:40.240 --> 00:01:42.600 obtained from this thick layer. 00:01:42.600 --> 00:01:44.900 Quercus suber is a Mediterranean type of tree. 00:01:44.900 --> 00:01:47.560 It grows, Portugal is the main place 00:01:47.560 --> 00:01:52.190 that exports cork, but also places like Algeria, Spain. 00:01:52.190 --> 00:01:54.830 You can grow it in California. 00:01:54.830 --> 00:01:57.390 The cork is kind of unusual because-- let 00:01:57.390 --> 00:02:01.000 me scoot onto the next slide-- it's kind of unusual. 00:02:01.000 --> 00:02:02.130 So here's a little picture. 00:02:02.130 --> 00:02:04.210 I went to Portugal when I was a graduate student doing 00:02:04.210 --> 00:02:04.850 this project. 00:02:04.850 --> 00:02:05.990 Here's the cork tree here. 00:02:05.990 --> 00:02:08.680 Here's the little mini we rented. 00:02:08.680 --> 00:02:11.500 Here's the cork being harvested here. 00:02:11.500 --> 00:02:14.650 Cork's unusual because you can remove the bark from a cork oak 00:02:14.650 --> 00:02:15.780 tree and it regrows. 00:02:15.780 --> 00:02:18.580 So most trees if you did this, you would kill the tree. 00:02:18.580 --> 00:02:22.430 But cork doesn't get killed by doing this. 00:02:22.430 --> 00:02:24.950 What happens is they plant trees. 00:02:24.950 --> 00:02:28.530 You have to wait something like 10 or 15 years for the tree 00:02:28.530 --> 00:02:30.460 to get big enough to harvest the cork. 00:02:30.460 --> 00:02:33.810 And then the first harvest is poor quality 00:02:33.810 --> 00:02:34.820 and they don't use that. 00:02:34.820 --> 00:02:36.819 And then you have to wait another 10 years or so 00:02:36.819 --> 00:02:39.060 before you can actually harvest the cork. 00:02:39.060 --> 00:02:43.610 So you can imagine if a cork orchard or forest gets chopped 00:02:43.610 --> 00:02:46.320 down to build a skyscraper, or apartment buildings, 00:02:46.320 --> 00:02:50.810 or something, it's not an economically feasible thing 00:02:50.810 --> 00:02:52.170 to plant cork trees these days. 00:02:52.170 --> 00:02:55.470 So when the trees are cut down, they 00:02:55.470 --> 00:02:57.910 don't tend to get replanted at this point. 00:02:57.910 --> 00:03:00.200 And there's a number of artificial substitutes 00:03:00.200 --> 00:03:00.700 for corks. 00:03:00.700 --> 00:03:04.070 You've probably seen wine bundles with foam plastic corks 00:03:04.070 --> 00:03:06.600 in them too. 00:03:06.600 --> 00:03:08.000 OK. 00:03:08.000 --> 00:03:09.660 The reason it's called Quercus suber 00:03:09.660 --> 00:03:15.220 is that the cell walls in this particular type of cork oak 00:03:15.220 --> 00:03:18.880 are covered with a waxy substance called "suberine." 00:03:18.880 --> 00:03:21.520 That's where the Quercus suber-- "Quercus" is "oak." 00:03:21.520 --> 00:03:25.790 Every oak is Quercus something. "Quercus alba" is white oak. 00:03:25.790 --> 00:03:28.130 Quercus just means oak. 00:03:28.130 --> 00:03:30.410 If we look at the structure of the cork, 00:03:30.410 --> 00:03:36.030 we can see that it's got these different views 00:03:36.030 --> 00:03:38.170 of the cork that are seen here. 00:03:38.170 --> 00:03:40.950 This is a drawing by Robert Hooke in the 1600s 00:03:40.950 --> 00:03:42.510 from his book Micrographia. 00:03:42.510 --> 00:03:46.240 He was the first person to really draw cork like this. 00:03:46.240 --> 00:03:49.990 You can see he drew sort of boxy cells on this side. 00:03:49.990 --> 00:03:53.020 And then, the other perpendicular plane, the cells 00:03:53.020 --> 00:03:55.570 have this structure here, sort of more rounded. 00:03:55.570 --> 00:03:59.210 And here's a little sketch he's got of the cork tree. 00:03:59.210 --> 00:04:00.910 Over here are SEM pictures. 00:04:00.910 --> 00:04:04.870 These two planes here correspond to this plane 00:04:04.870 --> 00:04:06.120 in Hooke's drawing. 00:04:06.120 --> 00:04:09.270 This plane here corresponds to this plane here. 00:04:09.270 --> 00:04:11.373 This over here is Hooke's actual microscope. 00:04:14.330 --> 00:04:17.880 I think the Royal Society still has that microscope 00:04:17.880 --> 00:04:20.779 that Hooke used in the 1600s. 00:04:20.779 --> 00:04:22.250 Some of you know that Hooke wrote 00:04:22.250 --> 00:04:23.500 this book called Micrographia. 00:04:23.500 --> 00:04:25.790 He got one of the first microscopes. 00:04:25.790 --> 00:04:27.540 He looked at a lot of different materials, 00:04:27.540 --> 00:04:29.350 and he drew these beautiful drawings. 00:04:29.350 --> 00:04:32.640 He wrote a page or two about each of the drawings. 00:04:32.640 --> 00:04:35.770 Harvard has a first edition of Micrographia, 00:04:35.770 --> 00:04:37.480 and I made a little video on it. 00:04:37.480 --> 00:04:41.110 So I thought I'd show you the video because the drawings are 00:04:41.110 --> 00:04:42.000 beautiful. 00:04:42.000 --> 00:04:46.160 But the url's on the slides, and so you can watch it yourself. 00:04:46.160 --> 00:04:48.330 There's more on this and on how he 00:04:48.330 --> 00:04:52.410 came to be so good at making scientific apparatus, 00:04:52.410 --> 00:04:54.870 how he came to do the Micrographia book. 00:04:54.870 --> 00:04:56.780 At the end of it, there's a comparison 00:04:56.780 --> 00:04:59.590 of a number of his drawings with modern SEM 00:04:59.590 --> 00:05:00.840 images of the same thing. 00:05:00.840 --> 00:05:04.230 He had this very famous picture where he draws a flea. 00:05:04.230 --> 00:05:06.540 Don Galler, the person who runs the SEM for me, 00:05:06.540 --> 00:05:08.070 I had him put a flea in. 00:05:08.070 --> 00:05:11.302 And he has essentially the same kind of image. 00:05:11.302 --> 00:05:12.760 The thing that's spectacular is you 00:05:12.760 --> 00:05:15.560 can see how much of the detail that Hooke was able to capture 00:05:15.560 --> 00:05:16.250 in his drawings. 00:05:16.250 --> 00:05:17.700 There's some really beautiful drawings. 00:05:17.700 --> 00:05:18.220 OK. 00:05:18.220 --> 00:05:20.480 So let's go back to cork. 00:05:20.480 --> 00:05:23.550 Hooke was the first person to use the word "cell" to describe 00:05:23.550 --> 00:05:28.020 biological cells, and he described the cell in cork. 00:05:28.020 --> 00:05:32.110 That's the structure looking at the SEM micrographs 00:05:32.110 --> 00:05:35.110 and his optical micrographs. 00:05:35.110 --> 00:05:37.110 These are just some more higher resolution, 00:05:37.110 --> 00:05:39.324 higher magnification, images. 00:05:39.324 --> 00:05:40.740 One of the things that you can see 00:05:40.740 --> 00:05:43.331 is that the cork has these little corrugations on the cell 00:05:43.331 --> 00:05:43.830 walls. 00:05:43.830 --> 00:05:45.110 See those little wrinkles? 00:05:45.110 --> 00:05:47.080 All the cell walls have those little wrinkles. 00:05:47.080 --> 00:05:49.100 This plane here is the perpendicular plane. 00:05:49.100 --> 00:05:51.400 If you look down into the cells, you 00:05:51.400 --> 00:05:53.200 can see those blurry things. 00:05:53.200 --> 00:05:56.850 Those are the corrugations in the cell walls. 00:05:56.850 --> 00:05:58.610 That's the structure. 00:05:58.610 --> 00:06:00.150 Here's a schematic. 00:06:00.150 --> 00:06:01.510 Here's the cork tree. 00:06:01.510 --> 00:06:04.520 The cork is the layer just beneath the bark. 00:06:04.520 --> 00:06:07.400 This is a picture of how the cells are oriented 00:06:07.400 --> 00:06:10.130 relative to their radial, and tangential, and axial 00:06:10.130 --> 00:06:11.400 directions. 00:06:11.400 --> 00:06:13.240 You can think of them as roughly hexagonal. 00:06:13.240 --> 00:06:14.823 They've got these little corrugations. 00:06:14.823 --> 00:06:18.860 This is a schematic of an individual cell. 00:06:18.860 --> 00:06:20.779 We measured the dimensions of the cells, 00:06:20.779 --> 00:06:22.570 and these are some average dimensions here. 00:06:22.570 --> 00:06:25.470 Typically, the cells are tens of microns long, 00:06:25.470 --> 00:06:27.330 and the cell wall is about a micron thick. 00:06:27.330 --> 00:06:28.320 Something like that. 00:06:28.320 --> 00:06:29.321 OK? 00:06:29.321 --> 00:06:30.820 One of the things we're going to see 00:06:30.820 --> 00:06:33.260 is that corrugated structure gives rise 00:06:33.260 --> 00:06:36.970 to some of the interesting properties of cork. 00:06:36.970 --> 00:06:39.920 If we load cork and just do mechanical tests on it-- 00:06:39.920 --> 00:06:42.792 this is just a uniaxial stress-strain curve. 00:06:42.792 --> 00:06:44.500 You can see the stress-strain curve looks 00:06:44.500 --> 00:06:46.750 like all these other curves we've seen for honeycombs. 00:06:46.750 --> 00:06:48.540 There's a linear elastic part here. 00:06:48.540 --> 00:06:50.190 There's a stress plateau here. 00:06:50.190 --> 00:06:52.480 And then there's a densification part here. 00:06:52.480 --> 00:06:55.760 Typically, the relative density of cork is around 0.15. 00:06:55.760 --> 00:06:59.040 Something like that. 00:06:59.040 --> 00:07:03.510 It densifies at a strain of something less than 0.85. 00:07:03.510 --> 00:07:05.200 It's a stress-strain curve. 00:07:05.200 --> 00:07:07.600 And when we did our little project on cork, 00:07:07.600 --> 00:07:10.990 we measured the properties in the three directions 00:07:10.990 --> 00:07:14.670 because in one direction, it's roughly hexagonal cells. 00:07:14.670 --> 00:07:16.780 That plane is isotropic. 00:07:16.780 --> 00:07:20.270 The e1,e2 plane is isotropic. 00:07:20.270 --> 00:07:22.570 This compares the measured values 00:07:22.570 --> 00:07:24.640 of the properties versus what we calculated 00:07:24.640 --> 00:07:26.380 from the honeycomb model. 00:07:26.380 --> 00:07:28.450 And really, we just used the sorts of equations 00:07:28.450 --> 00:07:30.533 that we talked about in class over the last couple 00:07:30.533 --> 00:07:33.182 of lectures, apart from loading in the x3 direction 00:07:33.182 --> 00:07:34.640 because in the x3 direction, you've 00:07:34.640 --> 00:07:35.972 got those corrugated walls. 00:07:35.972 --> 00:07:38.180 And you have to take those corrugations into account. 00:07:38.180 --> 00:07:40.794 So there's another complication that I'm not going to go into. 00:07:40.794 --> 00:07:42.210 But there's a sort of modification 00:07:42.210 --> 00:07:43.780 you can do to account for that. 00:07:43.780 --> 00:07:46.238 But you can see there's actually pretty good agreement here 00:07:46.238 --> 00:07:48.560 between the elastic moduli that we measured 00:07:48.560 --> 00:07:49.930 and what we calculated. 00:07:49.930 --> 00:07:52.880 The compressive strengths down here are not quite so good. 00:07:52.880 --> 00:07:55.870 They're off by a factor of two, more or less. 00:07:55.870 --> 00:08:01.090 But they're in the right ballpark for the cork. 00:08:01.090 --> 00:08:03.370 So those are some of the structures, 00:08:03.370 --> 00:08:04.584 some of the properties. 00:08:04.584 --> 00:08:06.250 One of the interesting things about cork 00:08:06.250 --> 00:08:07.860 is what it's used for. 00:08:07.860 --> 00:08:11.980 The uses of cork really exploit the mechanical properties. 00:08:11.980 --> 00:08:14.070 Obviously, it's used for stoppers for bottles. 00:08:14.070 --> 00:08:16.297 I brought a champagne cork along with me, 00:08:16.297 --> 00:08:18.755 and I brought a couple of other little pieces of cork here. 00:08:18.755 --> 00:08:20.960 I'll pass those around in a minute. 00:08:20.960 --> 00:08:22.460 One of the things to look at is just 00:08:22.460 --> 00:08:25.600 the still wine cork, which is the one on the right, 00:08:25.600 --> 00:08:28.550 and the champagne cork, which is the one on the left. 00:08:28.550 --> 00:08:30.210 If you notice the still wine cork 00:08:30.210 --> 00:08:32.080 is just made of one piece of cork 00:08:32.080 --> 00:08:34.980 that's cored out from the bark. 00:08:34.980 --> 00:08:36.650 And if you notice these little channels. 00:08:36.650 --> 00:08:39.580 These little channels here are called "lenticels." 00:08:39.580 --> 00:08:41.809 On the still wine cork, they go this way. 00:08:41.809 --> 00:08:44.140 And on the champagne cork, they go that way. 00:08:44.140 --> 00:08:46.390 They're oriented perpendicular. 00:08:46.390 --> 00:08:51.720 It turns out that they are normal to the isotropic plane 00:08:51.720 --> 00:08:53.480 in the cork. 00:08:53.480 --> 00:08:57.130 If you look at the champagne cork, 00:08:57.130 --> 00:08:59.360 this plane here is the isotropic plane. 00:08:59.360 --> 00:09:01.877 If you think of that being put into a bottle, 00:09:01.877 --> 00:09:03.460 I think part of the reason they orient 00:09:03.460 --> 00:09:05.120 it this way is because it gives you 00:09:05.120 --> 00:09:08.150 a uniform compression against the neck of the bottle 00:09:08.150 --> 00:09:09.940 and gives a nice seal. 00:09:09.940 --> 00:09:12.932 So that's one of the things about corks. 00:09:12.932 --> 00:09:14.390 Another thing that's interesting is 00:09:14.390 --> 00:09:20.500 that cork has a Poisson's ratio equal to zero if you load it 00:09:20.500 --> 00:09:22.930 in that direction. 00:09:22.930 --> 00:09:23.450 Let's see. 00:09:23.450 --> 00:09:24.700 Did I not bring that picture? 00:09:24.700 --> 00:09:26.116 Maybe I didn't bring that picture. 00:09:26.116 --> 00:09:27.280 Hang on a sec. 00:09:27.280 --> 00:09:27.780 Nope. 00:09:27.780 --> 00:09:28.488 I guess I didn't. 00:09:28.488 --> 00:09:29.030 Sorry. 00:09:29.030 --> 00:09:30.488 I thought I brought a picture where 00:09:30.488 --> 00:09:34.811 I had the deformation of the cells when you load it 00:09:34.811 --> 00:09:35.560 in that direction. 00:09:35.560 --> 00:09:37.100 When you load it-- so say the cells 00:09:37.100 --> 00:09:38.224 are corrugated this way on. 00:09:38.224 --> 00:09:41.140 When you load it that way on, it's like having a bellows 00:09:41.140 --> 00:09:43.514 and folding up a bellows, or unfolding a bellows. 00:09:43.514 --> 00:09:45.180 So when you load it that way on, there's 00:09:45.180 --> 00:09:47.460 no expansion or contraction this way on. 00:09:47.460 --> 00:09:49.910 And so you get zero Poisson's ratio. 00:09:49.910 --> 00:09:52.710 And if you think of trying to get the cork into the bottle, 00:09:52.710 --> 00:09:55.254 it's rather convenient to have zero Poisson's ratio 00:09:55.254 --> 00:09:56.920 because you don't get as much expansion. 00:09:56.920 --> 00:09:58.461 As you're pushing it into the bottle, 00:09:58.461 --> 00:10:01.040 you don't get as much expansion that way out, 00:10:01.040 --> 00:10:02.450 pressing against it. 00:10:02.450 --> 00:10:05.680 In fact, if you compare wine corks with rubber stoppers, 00:10:05.680 --> 00:10:08.550 this is a kind of typical rubber stopper. 00:10:08.550 --> 00:10:10.420 Wine corks are always just cylinders. 00:10:10.420 --> 00:10:11.870 In fact, even the champagne corks 00:10:11.870 --> 00:10:13.350 are cylinders when they start off. 00:10:13.350 --> 00:10:14.725 When they put it into the bottle, 00:10:14.725 --> 00:10:16.030 it's just a straight cylinder. 00:10:16.030 --> 00:10:17.790 It gets deformed like that from being 00:10:17.790 --> 00:10:19.650 in the bottle for some time. 00:10:19.650 --> 00:10:22.140 They have straight sides and you can just squeeze them. 00:10:22.140 --> 00:10:24.580 There's these funnel things that wine makers 00:10:24.580 --> 00:10:27.140 have for putting the cork in the bottle. 00:10:27.140 --> 00:10:29.100 You can just squeeze them into the bottle top. 00:10:29.100 --> 00:10:31.640 You can do that because the Poisson's ratio is zero 00:10:31.640 --> 00:10:34.000 and because the Young's modulus and the bulk modulus 00:10:34.000 --> 00:10:35.140 are both small. 00:10:35.140 --> 00:10:36.640 But if you look at a rubber stopper, 00:10:36.640 --> 00:10:40.070 rubber stoppers always have these tapered sides to them. 00:10:40.070 --> 00:10:43.810 And that's because the Poisson's ratio of the rubber is 0.5. 00:10:43.810 --> 00:10:46.912 As you squeeze it in, it's trying to move out this way. 00:10:46.912 --> 00:10:48.620 You couldn't get the stopper in unless it 00:10:48.620 --> 00:10:50.180 had those tapered sides. 00:10:50.180 --> 00:10:52.690 So that's sort of an interesting thing about cork. 00:10:52.690 --> 00:10:53.190 Let's see. 00:10:53.190 --> 00:10:54.680 Another application of cork is it's 00:10:54.680 --> 00:10:57.050 used for gaskets for the same sorts of reasons. 00:10:57.050 --> 00:10:59.630 It's relatively compliant. 00:10:59.630 --> 00:11:02.120 It takes up any slack between two pieces that you 00:11:02.120 --> 00:11:03.840 want to press together. 00:11:03.840 --> 00:11:05.800 It's often used for musical instruments 00:11:05.800 --> 00:11:06.804 that come in pieces. 00:11:06.804 --> 00:11:08.220 Things like clarinets, there'll be 00:11:08.220 --> 00:11:10.481 a piece of cork-- you a clarinet player? 00:11:10.481 --> 00:11:10.980 Yeah? 00:11:10.980 --> 00:11:11.880 Yeah. 00:11:11.880 --> 00:11:14.640 One of the interesting things about the clarinet 00:11:14.640 --> 00:11:16.431 is that, if you can see here at the ends, 00:11:16.431 --> 00:11:17.680 there's a piece of cork there. 00:11:17.680 --> 00:11:19.638 And I think there's a piece of cork down there. 00:11:19.638 --> 00:11:22.370 And the other pieces mate up with that. 00:11:22.370 --> 00:11:24.570 The cork provides a seal. 00:11:24.570 --> 00:11:27.410 And the way the cork is cut, it's cut in such a way 00:11:27.410 --> 00:11:30.650 that those lenticels go radially out like this, which 00:11:30.650 --> 00:11:35.340 means that the plane of isotropy and the direction that's 00:11:35.340 --> 00:11:41.040 got the zero Poisson's ratio is that radial direction. 00:11:41.040 --> 00:11:45.140 When you're squeezing, say, the second part onto it, 00:11:45.140 --> 00:11:48.142 the cork does not expand circumferentially. 00:11:48.142 --> 00:11:49.600 So as you're squeezing it this way, 00:11:49.600 --> 00:11:50.770 it doesn't expand that way. 00:11:50.770 --> 00:11:53.620 It doesn't wrinkle or anything on your other part. 00:11:53.620 --> 00:11:56.150 They use the cork in a particular orientation 00:11:56.150 --> 00:11:57.500 for that reason, I think. 00:11:57.500 --> 00:11:59.500 So it's used for gaskets 00:11:59.500 --> 00:12:03.590 It's also used because it's got a good friction property. 00:12:03.590 --> 00:12:07.650 It's got a property that is taken advantage of in things 00:12:07.650 --> 00:12:09.510 like flooring and shoes. 00:12:09.510 --> 00:12:13.190 Cork has a high friction even if it gets wet. 00:12:13.190 --> 00:12:15.410 Some sources of friction are from adhesion, 00:12:15.410 --> 00:12:16.770 from a surface effect. 00:12:16.770 --> 00:12:20.450 Then if, say, the floor gets wet, then you break that, 00:12:20.450 --> 00:12:22.520 and it could be slippery. 00:12:22.520 --> 00:12:24.430 But the source of friction in cork 00:12:24.430 --> 00:12:27.690 is from an energy loss and dissipation 00:12:27.690 --> 00:12:29.050 as you're deforming it. 00:12:29.050 --> 00:12:31.160 Imagine you have a wheel here. 00:12:31.160 --> 00:12:33.820 The wheel is rotating on this cork floor. 00:12:33.820 --> 00:12:36.600 And here, a piece of cork is getting deformed as the wheel 00:12:36.600 --> 00:12:38.240 rolls over it. 00:12:38.240 --> 00:12:41.747 As it gets deformed, there's some histeresis loop. 00:12:41.747 --> 00:12:43.330 Cork has quite a lot of damping in it. 00:12:43.330 --> 00:12:46.550 There's quite a lot of energy lost in that histeresis loop. 00:12:46.550 --> 00:12:50.020 What that means is that's characteristic of the cork 00:12:50.020 --> 00:12:50.590 itself. 00:12:50.590 --> 00:12:52.390 It's not a surface effect. 00:12:52.390 --> 00:12:57.330 That means that if you use it for floors or for shoes, 00:12:57.330 --> 00:13:01.547 it doesn't lose that friction and damping when it gets wet. 00:13:01.547 --> 00:13:04.130 Here's some measurements we did of the coefficient of friction 00:13:04.130 --> 00:13:07.310 for cork versus doing it dry and doing it 00:13:07.310 --> 00:13:10.330 with a liquid, soapy surface. 00:13:10.330 --> 00:13:12.850 You can see the soap doesn't make any difference. 00:13:12.850 --> 00:13:15.430 Cork is seen as a very attractive material 00:13:15.430 --> 00:13:17.244 for things like flooring. 00:13:17.244 --> 00:13:18.660 It's actually not a cheap material 00:13:18.660 --> 00:13:22.170 to make your floors out of, but it's an attractive material 00:13:22.170 --> 00:13:22.846 for flooring. 00:13:22.846 --> 00:13:24.470 Part of the reason it's used for floors 00:13:24.470 --> 00:13:26.761 and for the soles of shoes is because of these friction 00:13:26.761 --> 00:13:27.980 properties. 00:13:27.980 --> 00:13:29.610 Another feature of cork is that it's 00:13:29.610 --> 00:13:32.850 got very small cells compared to a lot of engineering polymers. 00:13:32.850 --> 00:13:35.080 The cells are on the order of tens of microns, 00:13:35.080 --> 00:13:37.610 whereas many polymer foams, the cells 00:13:37.610 --> 00:13:40.480 are hundreds of microns or millimeters. 00:13:40.480 --> 00:13:42.500 We'll get into this later, but this plot here 00:13:42.500 --> 00:13:45.030 is really saying that the thermal conductivity 00:13:45.030 --> 00:13:49.350 of a cellular material depends, in part, on the cell size. 00:13:49.350 --> 00:13:51.760 The cell size for foam plastics is in here. 00:13:51.760 --> 00:13:54.150 And that for cork is down here. 00:13:54.150 --> 00:13:55.860 Because it has a smaller cell size, 00:13:55.860 --> 00:13:58.500 it has a lower thermal conductivity. 00:13:58.500 --> 00:14:01.090 Cork was at one point used to some extent 00:14:01.090 --> 00:14:02.590 for thermal insulation. 00:14:02.590 --> 00:14:05.020 If you go to Portugal, where cork comes from, 00:14:05.020 --> 00:14:06.240 there's hermit caves. 00:14:06.240 --> 00:14:08.930 There were these old hermit, religious people 00:14:08.930 --> 00:14:10.720 who had holed up in a cave. 00:14:10.720 --> 00:14:12.840 And they would line the caves with cork 00:14:12.840 --> 00:14:16.950 to try to make it a little more insulated, a little more warm. 00:14:16.950 --> 00:14:20.130 The other place you see this is if you look at cigarettes, 00:14:20.130 --> 00:14:22.920 you know cigarettes have that little brown tip on the part 00:14:22.920 --> 00:14:24.760 that touches your lips? 00:14:24.760 --> 00:14:26.550 That's meant to look like cork. 00:14:26.550 --> 00:14:28.300 And if you look, it has little dots on it. 00:14:28.300 --> 00:14:30.360 The little dots are the little lenticels. 00:14:30.360 --> 00:14:33.220 Apparently, they used cork originally in cigarettes 00:14:33.220 --> 00:14:35.520 as a sort of thermal insulation between the cigarette 00:14:35.520 --> 00:14:36.940 and your mouth. 00:14:36.940 --> 00:14:39.177 So it was used for that too. 00:14:39.177 --> 00:14:40.260 And then, one final thing. 00:14:40.260 --> 00:14:43.150 Cork's also used, obviously, for bulletin boards. 00:14:43.150 --> 00:14:46.330 If you push a pin into cork, then you 00:14:46.330 --> 00:14:48.350 get this local deformation here. 00:14:48.350 --> 00:14:49.170 Here's our pin. 00:14:49.170 --> 00:14:51.215 And here's cells locally deformed. 00:14:51.215 --> 00:14:52.590 When you pull the pin out, you'll 00:14:52.590 --> 00:14:54.930 recover some of that deformation because the deformation 00:14:54.930 --> 00:14:55.590 is elastic. 00:14:55.590 --> 00:14:57.550 So the hole will partly close. 00:14:57.550 --> 00:14:59.070 So that's my little spiel on cork. 00:14:59.070 --> 00:15:01.770 And that's just because it's interesting. 00:15:01.770 --> 00:15:04.160 There's no test on cork or anything like that. 00:15:04.160 --> 00:15:05.140 OK. 00:15:05.140 --> 00:15:06.410 So are we good with cork? 00:15:06.410 --> 00:15:07.680 Any questions? 00:15:07.680 --> 00:15:08.770 We're OK? 00:15:08.770 --> 00:15:10.000 OK. 00:15:10.000 --> 00:15:12.470 Let me scoot out of there. 00:15:12.470 --> 00:15:14.640 Then the next part of the course, 00:15:14.640 --> 00:15:16.125 I wanted to talk about foams. 00:15:19.220 --> 00:15:21.390 Let me just park the cork thing. 00:15:21.390 --> 00:15:22.760 Let me pass these corks around. 00:15:22.760 --> 00:15:25.290 So you can play with those too. 00:15:25.290 --> 00:15:26.590 Oops. 00:15:26.590 --> 00:15:28.660 There's little bits of cork. 00:15:28.660 --> 00:15:29.619 There you go. 00:15:29.619 --> 00:15:32.160 There's the champagne cork, the rubber cork, some little cork 00:15:32.160 --> 00:15:32.660 layers. 00:15:37.640 --> 00:15:38.140 OK. 00:15:38.140 --> 00:15:41.820 So the next part of the course, I wanted to talk about foams. 00:15:41.820 --> 00:15:44.940 And I want to talk about how we model the mechanical behavior 00:15:44.940 --> 00:15:46.750 of foams. 00:15:46.750 --> 00:15:49.530 If we look at the stress-strain curve for foams, 00:15:49.530 --> 00:15:52.270 these are some examples for foams made out 00:15:52.270 --> 00:15:55.340 of different materials with different characteristics. 00:15:55.340 --> 00:15:57.130 The polyurethane and the polyethylene 00:15:57.130 --> 00:16:01.840 here are examples of elastomeric foams, really. 00:16:01.840 --> 00:16:03.570 This one here is an open-celled foam. 00:16:03.570 --> 00:16:05.390 This one's a closed-cell foam. 00:16:05.390 --> 00:16:10.830 Polymethacrylamide is a polymer that has a yield point. 00:16:10.830 --> 00:16:12.370 Mullite is a ceramic. 00:16:12.370 --> 00:16:14.000 You can see the shapes of these curves 00:16:14.000 --> 00:16:16.230 resemble the shapes that we saw for the honeycombs. 00:16:16.230 --> 00:16:16.730 Right? 00:16:16.730 --> 00:16:18.700 They look exactly the same, in fact. 00:16:18.700 --> 00:16:21.720 And the mechanisms of deformation in the foams 00:16:21.720 --> 00:16:24.100 are very similar to the honeycombs. 00:16:24.100 --> 00:16:27.180 Even though the foams have a much more complicated geometry, 00:16:27.180 --> 00:16:29.650 we can use some of the ideas from the honeycombs 00:16:29.650 --> 00:16:31.230 to understand how the foams behave. 00:16:31.230 --> 00:16:33.070 So that was part of the reason for doing 00:16:33.070 --> 00:16:36.440 the honeycomb analysis. 00:16:36.440 --> 00:16:37.770 Let me back up. 00:16:37.770 --> 00:16:39.820 These curves here were all in compression. 00:16:39.820 --> 00:16:41.570 These curves here were all in tension. 00:16:41.570 --> 00:16:43.120 So again, these ones in tension also 00:16:43.120 --> 00:16:46.010 look like the curves for the honeycombs. 00:16:46.010 --> 00:16:49.034 Remember, in tension, we don't get any elastic buckling. 00:16:49.034 --> 00:16:50.700 So if the foam was made of an elastomer, 00:16:50.700 --> 00:16:53.210 we don't see any stress plateau. 00:16:53.210 --> 00:16:56.330 If the foam is made of material with a yield stress, 00:16:56.330 --> 00:16:59.040 then we get a very short yield plateau 00:16:59.040 --> 00:17:00.910 because of a slight geometrical difference 00:17:00.910 --> 00:17:03.870 between pulling and compressing the foam. 00:17:03.870 --> 00:17:06.369 And if it's a brittle material, then we just get fracturing. 00:17:06.369 --> 00:17:08.452 There's going to be some fracture toughness that's 00:17:08.452 --> 00:17:09.800 going to characterize it. 00:17:09.800 --> 00:17:14.660 We can look at the deformation and the failure in these foams 00:17:14.660 --> 00:17:15.990 and look at the mechanisms. 00:17:15.990 --> 00:17:19.280 And what we're going to do is model the mechanisms 00:17:19.280 --> 00:17:21.730 and not worry too much about the cell geometry. 00:17:21.730 --> 00:17:24.390 So we're going to use dimensional arguments here. 00:17:24.390 --> 00:17:26.060 Here's a foam in compression. 00:17:26.060 --> 00:17:28.460 It was compressed from the top to the bottom. 00:17:28.460 --> 00:17:32.260 And you can see this strut that's circled in red. 00:17:32.260 --> 00:17:33.134 This is unloaded. 00:17:33.134 --> 00:17:34.550 And then, this is after some load. 00:17:34.550 --> 00:17:36.340 You can see this has bent somewhat. 00:17:36.340 --> 00:17:38.440 And then, you can see this vertical strut here. 00:17:38.440 --> 00:17:41.880 As the load gets larger, you can see that strut's buckled. 00:17:41.880 --> 00:17:44.500 In an elastomeric foam, you get bending and buckling 00:17:44.500 --> 00:17:48.250 just the same as we did in the honeycomb. 00:17:48.250 --> 00:17:50.640 Then here's a metal foam. 00:17:50.640 --> 00:17:52.690 You form plastic hinges in the metal foam. 00:17:52.690 --> 00:17:55.050 So here's a cell wall here. 00:17:55.050 --> 00:17:57.250 And it's a little bent to start with at zero load. 00:17:57.250 --> 00:18:00.620 But you can see it becomes more bent in this image over here. 00:18:00.620 --> 00:18:03.240 And here's a cell wall that's more or less vertical. 00:18:03.240 --> 00:18:05.520 And you can see that wall buckles. 00:18:05.520 --> 00:18:07.220 It's a plastic buckling in this case. 00:18:07.220 --> 00:18:10.280 There's a permanent deformation there. 00:18:10.280 --> 00:18:11.590 Here's a brittle foam. 00:18:11.590 --> 00:18:14.530 And you can see cell walls in this foam fracture. 00:18:14.530 --> 00:18:17.704 So this region here is equivalent to this region here. 00:18:17.704 --> 00:18:20.120 That little glitch there is the same as that little glitch 00:18:20.120 --> 00:18:20.752 there. 00:18:20.752 --> 00:18:22.710 You can see there's a couple of cell walls here 00:18:22.710 --> 00:18:23.520 that are fractured. 00:18:23.520 --> 00:18:24.890 So we get fracture. 00:18:24.890 --> 00:18:26.720 The idea is is that the mechanisms 00:18:26.720 --> 00:18:28.630 of deformation in the foams parallel 00:18:28.630 --> 00:18:31.050 those in the honeycombs. 00:18:31.050 --> 00:18:32.611 OK? 00:18:32.611 --> 00:18:33.110 All right. 00:18:33.110 --> 00:18:36.322 So let me write some of this stuff on the board. 00:18:36.322 --> 00:18:37.780 We're going to start off by talking 00:18:37.780 --> 00:18:39.990 about open-celled foams, so foams where there's 00:18:39.990 --> 00:18:42.630 just solid in the edges, but not in the faces 00:18:42.630 --> 00:18:43.910 of the polyhedral cells. 00:18:43.910 --> 00:18:45.960 Then we'll talk about close-cell foams 00:18:45.960 --> 00:18:47.810 where there's solid in the faces, as well. 00:18:47.810 --> 00:18:49.130 But the open-celled ones are easier. 00:18:49.130 --> 00:18:50.171 So we'll start with that. 00:19:08.540 --> 00:19:11.980 In compression, we see the same three regimes 00:19:11.980 --> 00:19:13.480 as we did before for the honeycombs. 00:19:22.030 --> 00:19:24.410 There's a linear elastic regime that corresponds 00:19:24.410 --> 00:19:26.110 to bending of the cell walls. 00:19:30.800 --> 00:19:31.937 There's a stress plateau. 00:19:34.860 --> 00:19:39.650 And for elastomeric foams, that corresponds to buckling. 00:19:45.960 --> 00:19:48.390 For metal foams, that corresponds to the formation 00:19:48.390 --> 00:19:49.520 of plastic hinges. 00:19:53.590 --> 00:20:01.180 And then, for ceramic or brittle foams, 00:20:01.180 --> 00:20:03.490 that corresponds to brittle crushing, so 00:20:03.490 --> 00:20:04.840 fracturing of the cell walls. 00:20:10.830 --> 00:20:14.337 Then, if you load the foam up to higher strains 00:20:14.337 --> 00:20:16.712 and higher stresses, eventually you get to densification. 00:20:24.590 --> 00:20:28.050 And in tension, just like the honeycombs, 00:20:28.050 --> 00:20:30.650 for the elastomeric materials there is no buckling. 00:20:36.420 --> 00:20:45.531 We can get a stress plateau from plastic hinges 00:20:45.531 --> 00:20:48.330 if there's, say, a metal foam. 00:20:48.330 --> 00:20:56.420 And for a brittle foam, we would get a fracture toughness 00:20:56.420 --> 00:20:59.630 and brittle fracture in tension. 00:21:10.270 --> 00:21:13.040 So the idea is the mechanisms of deformation and failure 00:21:13.040 --> 00:21:17.849 just parallel what we've seen in the honeycombs. 00:21:17.849 --> 00:21:20.015 So we'll start off with the linear elastic behavior. 00:21:27.260 --> 00:21:29.178 And we'll start with open-cell foams. 00:21:42.080 --> 00:21:44.350 The initial linear elasticity is due to bending 00:21:44.350 --> 00:21:45.140 of the cell walls. 00:21:56.110 --> 00:21:59.340 And if the thickness of the cell edges relative to the length 00:21:59.340 --> 00:22:03.450 is small, the bending dominates the deformation. 00:22:03.450 --> 00:22:06.260 But as the thickness to length ratio increases, 00:22:06.260 --> 00:22:09.020 then axial deformations can become important too. 00:22:30.609 --> 00:22:32.650 What we're going to do is we're going to consider 00:22:32.650 --> 00:22:33.750 dimensional arguments. 00:22:41.987 --> 00:22:43.820 We're going to set the dimensional arguments 00:22:43.820 --> 00:22:47.460 up so that we replicate the mechanisms of deformation 00:22:47.460 --> 00:22:47.977 and failure. 00:22:47.977 --> 00:22:50.143 But we don't worry too much about the cell geometry. 00:23:27.430 --> 00:23:29.990 What I'm going to start with is considering a cubic cell. 00:23:34.710 --> 00:23:38.700 And I've arranged the cubic cell so that the cell 00:23:38.700 --> 00:23:39.735 edges are staggered. 00:23:39.735 --> 00:23:41.235 That's going to give me the bending. 00:23:45.600 --> 00:23:51.130 The edge length is going to be L. 00:23:51.130 --> 00:23:55.000 I'm going to say we have a square cross-section, 00:23:55.000 --> 00:23:55.690 t squared. 00:24:06.880 --> 00:24:12.790 Here's our idealized model here with a cubic cell. 00:24:12.790 --> 00:24:14.350 All the members have a length, l. 00:24:14.350 --> 00:24:17.460 All of them have a square cross-section, t squared. 00:24:17.460 --> 00:24:18.770 That's an open-cell model. 00:24:18.770 --> 00:24:22.040 We've got just solid on the edges and nothing on the faces. 00:24:22.040 --> 00:24:24.150 The idea is that if I bend that, or if I 00:24:24.150 --> 00:24:28.100 load that in compression, so I apply, say, a stress out here 00:24:28.100 --> 00:24:29.820 that puts forces on those members 00:24:29.820 --> 00:24:32.820 there, because I've staggered these cell 00:24:32.820 --> 00:24:35.650 walls with these ones here, we're 00:24:35.650 --> 00:24:38.700 going to get bending in this cell edge here. 00:24:38.700 --> 00:24:41.680 That bending is going to be what we model. 00:25:15.960 --> 00:25:19.170 I'm going to set this up so that one thing is 00:25:19.170 --> 00:25:21.350 proportional to something else. 00:25:21.350 --> 00:25:24.130 These relationships are going to be true regardless of the cell 00:25:24.130 --> 00:25:24.720 geometry. 00:25:24.720 --> 00:25:26.997 So I could have picked a tetrakaidecahedra 00:25:26.997 --> 00:25:29.580 if I wanted to, and I would have had these same relationships. 00:25:29.580 --> 00:25:31.663 I'm just picking a cubic thing because it's easier 00:25:31.663 --> 00:25:33.720 to think about. 00:25:33.720 --> 00:25:37.540 So first of all, we look at the relative density. 00:25:37.540 --> 00:25:41.290 Remember, the relative density is the volume fraction 00:25:41.290 --> 00:25:41.840 of solid. 00:25:44.480 --> 00:25:48.080 So it's the volume of the solid over the total volume. 00:25:48.080 --> 00:25:55.410 And that's going to go as t squared l over l cubed, or just 00:25:55.410 --> 00:25:58.894 t over l all squared. 00:25:58.894 --> 00:26:01.060 You remember for the honeycomb, the relative density 00:26:01.060 --> 00:26:02.950 went linearly with t over l. 00:26:02.950 --> 00:26:06.069 For the open-celled foam, it goes as t over l squared. 00:26:10.380 --> 00:26:12.680 The moment of inertia in this case 00:26:12.680 --> 00:26:14.790 is going to go as t to the fourth. 00:26:14.790 --> 00:26:17.850 Remember, we have a square section, t squared. 00:26:17.850 --> 00:26:21.810 So if it's bh cubed over 12, b is t, h is t. 00:26:21.810 --> 00:26:24.290 It's going as t to the fourth. 00:26:24.290 --> 00:26:26.050 Then what I'm going to say is the stress 00:26:26.050 --> 00:26:29.780 is going to go as F over an area length squared. 00:26:29.780 --> 00:26:30.830 OK? 00:26:30.830 --> 00:26:33.820 So if I look at my little square thing here, 00:26:33.820 --> 00:26:35.480 I look at having my force here. 00:26:35.480 --> 00:26:36.920 Here we have a force f. 00:26:36.920 --> 00:26:38.420 And it's acting over an area that's 00:26:38.420 --> 00:26:40.310 somehow related to l squared. 00:26:40.310 --> 00:26:41.779 Right? 00:26:41.779 --> 00:26:43.570 I don't know exactly what that constant is, 00:26:43.570 --> 00:26:45.840 and I'm going to not try to calculate that. 00:26:45.840 --> 00:26:47.900 But it goes as F over l squared. 00:26:47.900 --> 00:26:50.240 Similarly, I can write that the strain 00:26:50.240 --> 00:26:52.680 is going to go as delta over l. 00:26:52.680 --> 00:26:56.180 So the strain is going to go as this bending deflection 00:26:56.180 --> 00:27:00.320 here, that delta divided by the height of the cell. 00:27:00.320 --> 00:27:03.200 And that's also l. 00:27:03.200 --> 00:27:05.570 Then I also know from structural mechanics 00:27:05.570 --> 00:27:12.340 that delta is going to go as Fl cubed over E of the solid 00:27:12.340 --> 00:27:18.054 and I. 00:27:18.054 --> 00:27:20.220 Then I'm just going to put all these things together 00:27:20.220 --> 00:27:21.940 to get the modulus. 00:27:21.940 --> 00:27:24.100 The modulus of the foam is going to go 00:27:24.100 --> 00:27:27.300 as the stress over the strain. 00:27:27.300 --> 00:27:31.450 If I plug in what I have for the stress, it's F over l squared. 00:27:31.450 --> 00:27:34.390 If I plug in what I have for the strain, it's delta over l. 00:27:37.800 --> 00:27:41.290 So this is F over l and delta. 00:27:41.290 --> 00:27:46.822 I'm going to replace delta by Fl cubed over Es. 00:27:46.822 --> 00:27:48.280 I'm going to use, instead of I, I'm 00:27:48.280 --> 00:27:51.160 going to use t to the fourth. 00:27:51.160 --> 00:27:52.744 Then the F's are going to cancel out. 00:27:57.100 --> 00:28:00.950 I've got that the modulus goes as the modulus 00:28:00.950 --> 00:28:05.120 of the solid times t over l to the fourth power. 00:28:05.120 --> 00:28:07.328 Then I can put that in terms of the relative density. 00:28:14.370 --> 00:28:18.960 It's going to go as the relative density squared. 00:28:18.960 --> 00:28:20.660 So I can summarize all of this by saying 00:28:20.660 --> 00:28:22.880 that the Young's modulus of the foam 00:28:22.880 --> 00:28:26.320 is some constant C1, I'm going to call it, 00:28:26.320 --> 00:28:29.420 times the modulus of the solid times 00:28:29.420 --> 00:28:30.790 the relative density squared. 00:28:36.890 --> 00:28:37.390 OK. 00:28:37.390 --> 00:28:39.240 So this has the same kind of form 00:28:39.240 --> 00:28:41.530 as those equations we had for the honeycombs. 00:28:41.530 --> 00:28:42.480 Right? 00:28:42.480 --> 00:28:44.310 There's a solid-cell wall property. 00:28:44.310 --> 00:28:46.270 The solid module's here. 00:28:46.270 --> 00:28:48.570 For the honeycombs, I put it in terms of t over l. 00:28:48.570 --> 00:28:51.670 But the t over l was related to the relative density. 00:28:51.670 --> 00:28:53.910 How much solid you've got is reflected 00:28:53.910 --> 00:28:56.830 in the relative density. 00:28:56.830 --> 00:28:59.710 And then, this constant C1 wraps up 00:28:59.710 --> 00:29:01.640 all the geometrical constants that I've said, 00:29:01.640 --> 00:29:03.234 one thing's proportional to another, 00:29:03.234 --> 00:29:05.150 and something else is proportional to another. 00:29:05.150 --> 00:29:07.090 C1 just wraps up all of those. 00:29:07.090 --> 00:29:08.520 OK? 00:29:08.520 --> 00:29:10.560 I'm just going to say here C1 includes 00:29:10.560 --> 00:29:11.930 all the geometrical constants. 00:29:21.310 --> 00:29:25.109 We have to get C1 by looking at data. 00:29:25.109 --> 00:29:26.900 If we look at data for the Young's modulus, 00:29:26.900 --> 00:29:30.129 we find that C1 is just about equal to one. 00:29:35.030 --> 00:29:38.020 People have also done more sophisticated analyses 00:29:38.020 --> 00:29:38.621 than this. 00:29:38.621 --> 00:29:40.120 There's a group of people who looked 00:29:40.120 --> 00:29:43.190 at doing a full-scale, structural analysis 00:29:43.190 --> 00:29:45.980 of an open-celled tetrakaidecahedral cell. 00:29:45.980 --> 00:29:47.690 Remember, I said they pack to fill space. 00:29:47.690 --> 00:29:49.640 So you can look at a unit cell. 00:29:49.640 --> 00:29:52.750 They also made their cells such that the thickness 00:29:52.750 --> 00:29:55.165 along the length of the cell was not constant. 00:29:55.165 --> 00:29:57.290 The thickness varied as something called a "plateau 00:29:57.290 --> 00:29:58.340 border." 00:29:58.340 --> 00:30:01.210 If you have a foam that's made by surface tension, 00:30:01.210 --> 00:30:03.610 the edges will tend to have these plateau borders. 00:30:03.610 --> 00:30:06.000 And the thickness will vary along the length of the edge. 00:30:06.000 --> 00:30:08.270 So when they did all this whole, complicated thing, 00:30:08.270 --> 00:30:10.580 they could calculate a value for C1. 00:30:10.580 --> 00:30:12.140 They calculated 0.98. 00:30:12.140 --> 00:30:15.746 So it's very close to 1. 00:30:15.746 --> 00:30:30.870 I'll say analysis of open-cell tetrakaidecahedron 00:30:30.870 --> 00:30:46.980 cells with these plateau borders give C1 equal to 0.98. 00:30:46.980 --> 00:30:47.480 OK. 00:30:47.480 --> 00:30:48.730 So that's the Young's modulus. 00:30:53.860 --> 00:30:56.021 We can also look at the shear modulus. 00:30:56.021 --> 00:30:58.020 The shear modulus is also going to be controlled 00:30:58.020 --> 00:30:59.950 by bending of the cell walls. 00:30:59.950 --> 00:31:02.210 And so the shear modulus is just going 00:31:02.210 --> 00:31:05.340 to be some other constant times Es 00:31:05.340 --> 00:31:08.380 times the relative density squared, so 00:31:08.380 --> 00:31:10.270 a similar kind of relationship. 00:31:10.270 --> 00:31:11.930 It's just a different constant. 00:31:11.930 --> 00:31:17.380 And if the foam's isotropic, and if the Poisson's ratio 00:31:17.380 --> 00:31:25.151 is a third, then C2 is equal to 3/8. 00:31:29.670 --> 00:31:35.290 Remember, if we have isotropy, then the shear modulus 00:31:35.290 --> 00:31:40.270 is equal to E over 2 1 plus nu. 00:31:40.270 --> 00:31:43.530 And so you can get the C2 from that if you say nu is a third. 00:31:52.950 --> 00:31:56.420 Then we can also get Poisson's ratio for the foam. 00:31:56.420 --> 00:32:01.950 If the foam is isotropic, so we'll say for an isotropic foam 00:32:01.950 --> 00:32:11.530 here, nu is equal to E over 2G minus 1. 00:32:11.530 --> 00:32:14.920 That's just rearranging this expression here. 00:32:14.920 --> 00:32:17.620 And because E and G both depend on the relative density 00:32:17.620 --> 00:32:19.600 squared, they both depend on Es squared, 00:32:19.600 --> 00:32:21.590 that's all going to cancel out. 00:32:21.590 --> 00:32:26.970 So this is going to be equal to C1 over 2 C2 minus 1. 00:32:26.970 --> 00:32:30.760 So that's going to equal to a constant. 00:32:30.760 --> 00:32:32.980 That constant's going to be independent of whatever 00:32:32.980 --> 00:32:36.385 material the foam is made from and the relative density. 00:32:53.430 --> 00:32:55.490 The constant just depends on the cell geometry. 00:33:07.500 --> 00:33:09.500 Remember, in honeycombs we found the same thing. 00:33:09.500 --> 00:33:11.900 The Poisson's ratio for the honeycombs 00:33:11.900 --> 00:33:13.472 only depended on the cell geometry. 00:33:13.472 --> 00:33:15.180 It didn't depend on the solid properties. 00:33:15.180 --> 00:33:17.180 It didn't depend on the relative density. 00:33:17.180 --> 00:33:20.357 So this is an exactly parallel thing here for the foams. 00:33:33.030 --> 00:33:33.662 Yeah? 00:33:33.662 --> 00:33:36.000 AUDIENCE: I have a silly kind of question. 00:33:36.000 --> 00:33:39.640 What is the difference between foam and the honeycombs? 00:33:39.640 --> 00:33:40.780 LORNA GIBSON: Oh. 00:33:40.780 --> 00:33:44.240 The honeycombs have cells in a plane, 00:33:44.240 --> 00:33:46.720 and they're prismatic in the third direction. 00:33:46.720 --> 00:33:50.510 And the foams have polyhedral cells. 00:33:50.510 --> 00:33:53.140 You know what a tetrakaidecahedron, a 3D, 00:33:53.140 --> 00:33:54.730 polyhedral cell. 00:33:54.730 --> 00:33:56.500 OK? 00:33:56.500 --> 00:33:58.320 The honeycombs are prismatic. 00:33:58.320 --> 00:34:01.150 And the foams have polyhedral cells. 00:34:01.150 --> 00:34:02.110 OK? 00:34:02.110 --> 00:34:02.740 Are we good? 00:34:30.831 --> 00:34:31.330 OK. 00:34:39.178 --> 00:34:40.969 So there's a couple more interesting things 00:34:40.969 --> 00:34:42.340 about Poisson's ratio. 00:34:42.340 --> 00:34:43.810 The same way we can make honeycombs 00:34:43.810 --> 00:34:45.429 with negative Poisson's ratios, we 00:34:45.429 --> 00:34:47.689 can also make foams with negative Poisson's ratios. 00:35:05.530 --> 00:35:09.780 They do it the same way as for the honeycombs, really. 00:35:09.780 --> 00:35:12.700 The honeycombs had negative Poisson's ratios 00:35:12.700 --> 00:35:15.730 if the cell walls looked like this, this sort of arrangement. 00:35:24.730 --> 00:35:27.520 So that sort of a thing. 00:35:27.520 --> 00:35:29.960 We said that theta was negative for the honeycombs. 00:35:29.960 --> 00:35:32.360 And if you invert the cell walls on a foam, 00:35:32.360 --> 00:35:34.820 you also get negative Poisson's ratios. 00:35:34.820 --> 00:35:38.316 And the way they do that is they take a thermoplastic foam, 00:35:38.316 --> 00:35:39.940 and then, they load it hydrostatically. 00:35:39.940 --> 00:35:42.010 So they compress it in all three directions. 00:35:42.010 --> 00:35:44.091 And they smush the cells in on each other. 00:35:44.091 --> 00:35:45.840 And then, they heat it up to a temperature 00:35:45.840 --> 00:35:47.580 above the glass transition temperature 00:35:47.580 --> 00:35:49.234 while it's still smushed. 00:35:49.234 --> 00:35:50.400 And then, they cool it down. 00:35:50.400 --> 00:35:54.088 So they end up with that structure frozen in. 00:35:54.088 --> 00:35:56.463 And if they do that, they get a negative Poisson's ratio. 00:36:01.610 --> 00:36:04.600 I'll just say they invert the cell angles 00:36:04.600 --> 00:36:05.845 analogous to the honeycomb. 00:36:33.890 --> 00:36:36.760 They load the foam hydrostatically 00:36:36.760 --> 00:36:42.370 and heat to a temperature above Tg. 00:36:42.370 --> 00:36:43.880 And then, they cool and release it. 00:37:10.480 --> 00:37:14.150 I have a photograph here of a foam with a negative Poisson's 00:37:14.150 --> 00:37:14.690 ratio. 00:37:14.690 --> 00:37:18.040 You can see how the cells have been smushed in. 00:37:18.040 --> 00:37:21.720 It's the equivalent of the way it's done for the honeycomb. 00:37:21.720 --> 00:37:22.350 OK? 00:37:22.350 --> 00:37:24.191 Are we good? 00:37:24.191 --> 00:37:24.690 OK. 00:37:38.390 --> 00:37:41.740 That's the linear elastic moduli for open-celled foams. 00:37:41.740 --> 00:37:44.830 The next thing I wanted to do was closed-cell foams. 00:37:44.830 --> 00:37:47.240 If we look at a closed-cell foam, 00:37:47.240 --> 00:37:51.820 we can idealize it in this kind of a way here. 00:37:51.820 --> 00:37:53.690 I've set it up so that the edge thickness is 00:37:53.690 --> 00:37:55.590 different from the face thickness. 00:37:55.590 --> 00:37:58.600 That's really representing the fact that in foams, many foams 00:37:58.600 --> 00:38:02.570 are made using a liquid, and the foaming 00:38:02.570 --> 00:38:04.680 is controlled by surface tension. 00:38:04.680 --> 00:38:07.880 Often, the surface tension draws material into the edges 00:38:07.880 --> 00:38:08.930 and away from the faces. 00:38:08.930 --> 00:38:12.600 So the faces tend to be thinner than the edges. 00:38:12.600 --> 00:38:16.720 When we have deformation of the closed-cell foam, 00:38:16.720 --> 00:38:18.650 we've got bending of the edges the same 00:38:18.650 --> 00:38:20.410 as we did for the open-celled foam. 00:38:20.410 --> 00:38:22.140 But the faces can stretch. 00:38:22.140 --> 00:38:25.120 So they can have an axial stretching. 00:38:25.120 --> 00:38:28.300 You can think of that as a cell membrane stretching here. 00:38:28.300 --> 00:38:31.960 So imagine if I either pull on the foam or I compress it, 00:38:31.960 --> 00:38:35.540 there's going to be some axial load in the faces. 00:38:35.540 --> 00:38:37.760 So when we analyze them, we have to account 00:38:37.760 --> 00:38:40.770 for both bending of the edges and axial deformation 00:38:40.770 --> 00:38:42.980 in the faces. 00:38:42.980 --> 00:38:53.618 I'll just say we have edge bending as in open-cell foams. 00:38:58.170 --> 00:38:59.757 And we also get a face stretching. 00:39:07.640 --> 00:39:10.080 Another thing that can happen in the closed-cell foams 00:39:10.080 --> 00:39:12.194 is we can get compression of the gas. 00:39:12.194 --> 00:39:14.360 In an open-cell foam, the gas can move from one cell 00:39:14.360 --> 00:39:14.990 to the next. 00:39:14.990 --> 00:39:17.460 But in a closed-cell foam, the gas is trapped. 00:39:17.460 --> 00:39:20.590 And as the volume of the cell changes, 00:39:20.590 --> 00:39:21.723 the gas gets compressed. 00:39:26.610 --> 00:39:28.030 So we have another effect here. 00:39:35.400 --> 00:39:40.530 So we'll say for polymer foams, surface tension 00:39:40.530 --> 00:39:43.113 tends to draw material to the edges during processing. 00:40:07.610 --> 00:40:11.160 We define two thicknesses, one for the edge 00:40:11.160 --> 00:40:13.960 and one for the face. 00:40:13.960 --> 00:40:17.275 And then we apply a force to this cubic structure. 00:40:24.360 --> 00:40:27.000 And we can do an analysis a little bit like what 00:40:27.000 --> 00:40:28.450 we did for the open-cell foam. 00:40:32.430 --> 00:40:33.697 Let me rub all this off. 00:41:08.750 --> 00:41:09.360 OK. 00:41:09.360 --> 00:41:10.735 I'm going to set this up a little 00:41:10.735 --> 00:41:13.490 bit differently than for the open-celled foam. 00:41:13.490 --> 00:41:15.200 We're going to do a work argument. 00:41:15.200 --> 00:41:17.460 We're going to look at the external work done 00:41:17.460 --> 00:41:20.370 by the force F going through a deformation, delta. 00:41:20.370 --> 00:41:22.750 And that's going to have to equal the internal work done 00:41:22.750 --> 00:41:25.530 by the edges bending and by the faces stretching. 00:41:25.530 --> 00:41:27.950 So let me set that up. 00:41:27.950 --> 00:41:35.410 We're going to say the external work done, 00:41:35.410 --> 00:41:38.355 that's going to be proportional to F times delta. 00:41:38.355 --> 00:41:43.810 So delta is how much the whole thing is going to deform. 00:41:43.810 --> 00:41:47.795 Then, I've got internal work from bending of the edges. 00:41:55.480 --> 00:42:01.440 That internal work is going to be proportional to F 00:42:01.440 --> 00:42:05.537 over delta times delta squared. 00:42:05.537 --> 00:42:07.120 I'm going to end up with an expression 00:42:07.120 --> 00:42:08.620 where everything's in delta squared. 00:42:08.620 --> 00:42:11.010 So I want to keep the delta squared there for now. 00:42:11.010 --> 00:42:13.520 And F over delta is the stiffness. 00:42:13.520 --> 00:42:18.700 That's going to go as E of the solid times I over l cubed. 00:42:22.160 --> 00:42:24.340 I is going to go is Te to the fourth 00:42:24.340 --> 00:42:28.330 here because it's I of the edges. 00:42:28.330 --> 00:42:31.110 I've also got internal work from stretching of the faces. 00:42:42.230 --> 00:42:44.336 That internal work is going to be-- I'm 00:42:44.336 --> 00:42:46.085 going to run into my other equations here. 00:42:46.085 --> 00:42:48.230 Let me put it down a little. 00:42:48.230 --> 00:42:52.060 That's going to go as the stress on the face 00:42:52.060 --> 00:42:56.940 times the strain in the face times the volume of the face. 00:42:56.940 --> 00:42:59.970 Or I could write that, instead of stress of the face, 00:42:59.970 --> 00:43:02.080 I can put it in terms of Hooke's law 00:43:02.080 --> 00:43:04.740 and say it's E of the solid times 00:43:04.740 --> 00:43:09.750 the strain in the face squared times the volume of the face. 00:43:09.750 --> 00:43:12.750 And I can replace the strain in the face by delta over l. 00:43:18.700 --> 00:43:20.120 So it's delta over l squared. 00:43:20.120 --> 00:43:22.120 And then, the volume of the face is 00:43:22.120 --> 00:43:25.800 going to be t of the face times l squared. 00:43:33.990 --> 00:43:35.910 Then I want to balance the internal work 00:43:35.910 --> 00:43:37.880 and the external work. 00:43:37.880 --> 00:43:41.540 I can say F times delta is going to equal 00:43:41.540 --> 00:43:44.980 some constant I'm going to call "alpha" times 00:43:44.980 --> 00:43:49.090 E of the solid times t of the edge to the fourth, 00:43:49.090 --> 00:43:54.780 that's this guy up here, over l cubed 00:43:54.780 --> 00:43:58.140 times delta squared plus some other constant I'm 00:43:58.140 --> 00:44:05.110 going to call "beta" times E of the solid times delta over l 00:44:05.110 --> 00:44:08.437 squared tf l squared. 00:44:13.200 --> 00:44:16.190 So far, I've got this in terms of the force I'm applying. 00:44:16.190 --> 00:44:19.250 But I want to get a modulus of the foam out of this. 00:44:19.250 --> 00:44:23.070 I want to relate this force here to the modulus of the foam. 00:44:23.070 --> 00:44:25.600 I can say the modulus of the foam 00:44:25.600 --> 00:44:30.440 is going to be related to F over l squared, that's 00:44:30.440 --> 00:44:35.240 the stress, divided by the strain, delta over l. 00:44:35.240 --> 00:44:41.590 I can write the force is proportional to the modulus 00:44:41.590 --> 00:44:47.060 times the deflection, delta, and times the member length, l. 00:44:47.060 --> 00:44:50.120 And then, I'm going to plug that guy into this expression 00:44:50.120 --> 00:44:51.140 up here. 00:44:51.140 --> 00:44:53.339 And I'm going to get a delta squared on the left. 00:44:53.339 --> 00:44:55.130 And I'm going to get delta squareds in each 00:44:55.130 --> 00:44:56.160 of the right-hand terms. 00:44:56.160 --> 00:44:57.909 And so I'm going to cancel out the deltas. 00:45:02.440 --> 00:45:04.980 If I put all that together, I have 00:45:04.980 --> 00:45:10.090 the modulus in the foam times delta squared times l. 00:45:10.090 --> 00:45:12.090 There's a delta here, and there's a delta there. 00:45:12.090 --> 00:45:13.256 That gives me delta squared. 00:45:16.020 --> 00:45:20.630 And that's going to equal alpha Es Te 00:45:20.630 --> 00:45:29.350 to the fourth over l cubed times delta squared plus beta 00:45:29.350 --> 00:45:36.480 times Es times delta squared tf, if I cancel out 00:45:36.480 --> 00:45:38.424 one of those l squareds. 00:45:38.424 --> 00:45:39.840 So here I can get rid of the delta 00:45:39.840 --> 00:45:41.090 squareds in all these terms. 00:45:43.860 --> 00:45:46.200 And if I just divide by l, I'm going 00:45:46.200 --> 00:45:48.150 to have the modulus of the foam. 00:45:55.050 --> 00:45:59.400 I get a term here that it goes as alpha E of the solid times 00:45:59.400 --> 00:46:05.030 t of the edge over l to the fourth power plus beta times 00:46:05.030 --> 00:46:10.330 E of the solid times tf over l. 00:46:14.300 --> 00:46:17.300 Are we good? 00:46:17.300 --> 00:46:18.451 OK. 00:46:18.451 --> 00:46:20.450 And what I'd like to do is instead of putting it 00:46:20.450 --> 00:46:22.690 in terms of te and tf, I'd like to put it in terms 00:46:22.690 --> 00:46:25.240 of the relative density. 00:46:25.240 --> 00:46:26.752 I'm going to look at two limits. 00:46:29.550 --> 00:46:32.010 I'm going to say if we just had open cells, if there were 00:46:32.010 --> 00:46:34.920 no faces on the membranes, if we just had open cells, 00:46:34.920 --> 00:46:41.690 and we had a uniform t, then the relative density 00:46:41.690 --> 00:46:43.580 would go as t over l squared. 00:46:46.740 --> 00:46:55.480 If I just had closed cells, and I had a uniform t, 00:46:55.480 --> 00:47:02.660 then the relative density just goes linearly with t over l. 00:47:02.660 --> 00:47:07.130 The relative density is the volume fraction of solids. 00:47:07.130 --> 00:47:10.840 It's the volume of the solid over the total volume. 00:47:10.840 --> 00:47:13.250 For a closed-cell foam with a uniform t, 00:47:13.250 --> 00:47:16.390 the volume of the solid is going to be t times l squared. 00:47:16.390 --> 00:47:18.400 And then, the volume total is l cubed. 00:47:18.400 --> 00:47:21.850 So it's t over l. 00:47:21.850 --> 00:47:23.940 Now, I'm going to define one more thing. 00:47:23.940 --> 00:47:26.350 If I say that phi is equal to the volume 00:47:26.350 --> 00:47:45.080 fraction of the solid in the edges, then I can say te over l 00:47:45.080 --> 00:47:48.950 is some constant times phi to the 1/2 times 00:47:48.950 --> 00:47:50.310 the relative density to the 1/2. 00:47:53.120 --> 00:47:59.410 And I can say tf over l is equal to some other constant, 00:47:59.410 --> 00:48:03.880 c prime, I'll call it, times 1 minus phi, that's how much 00:48:03.880 --> 00:48:06.490 is in the faces, times the relative density. 00:48:09.670 --> 00:48:13.050 Then I could combine all of this. 00:48:13.050 --> 00:48:14.880 Can I put it in here? 00:48:14.880 --> 00:48:16.420 Maybe I'll just stick it down here. 00:48:20.140 --> 00:48:20.640 Hang on. 00:48:20.640 --> 00:48:21.140 OK. 00:48:23.730 --> 00:48:24.440 Put it up here. 00:48:45.710 --> 00:48:48.850 This is my final expression here. 00:48:48.850 --> 00:48:52.130 And this term here arises from the edge bending. 00:48:55.880 --> 00:48:58.000 This term here arises from the face stretching. 00:49:07.486 --> 00:49:09.610 So I think I should wait a bit for you to catch up. 00:49:14.180 --> 00:49:15.630 The idea is the edges are bending. 00:49:15.630 --> 00:49:17.240 The faces are stretching. 00:49:17.240 --> 00:49:20.080 We're looking at the work done by deforming 00:49:20.080 --> 00:49:22.340 the edges and the faces and equating that 00:49:22.340 --> 00:49:25.264 to the work done by the externally applied load, f. 00:49:35.462 --> 00:49:35.962 OK? 00:49:44.780 --> 00:49:47.100 That gives us two terms, one that 00:49:47.100 --> 00:49:49.337 accounts for the edge bending, and one that accounts 00:49:49.337 --> 00:49:50.336 for the face stretching. 00:49:53.880 --> 00:49:56.300 To be comprehensive, we want to take into account 00:49:56.300 --> 00:49:58.600 the compression of the gas, as well. 00:49:58.600 --> 00:50:00.864 So there's one more term I'm going to add on to that. 00:50:57.890 --> 00:51:02.090 Typically the gas effect is only significant 00:51:02.090 --> 00:51:03.670 for elastomeric foams. 00:51:03.670 --> 00:51:06.004 If you had a metal foam or a ceramic foam that 00:51:06.004 --> 00:51:08.170 was closed-cell, it wouldn't really contribute much. 00:51:08.170 --> 00:51:10.512 But just to be complete, we'll go through this. 00:51:39.720 --> 00:51:41.230 The idea here is we say we've got 00:51:41.230 --> 00:51:43.460 a cubic element of the foam. 00:51:43.460 --> 00:51:46.020 Initially, it has a volume, v0. 00:51:46.020 --> 00:51:50.780 If we apply a stress, a uniaxial stress, 00:51:50.780 --> 00:51:53.110 there's some change in the volume of the foam. 00:51:53.110 --> 00:51:56.590 So there's some volume, v, after we compress it 00:51:56.590 --> 00:51:59.940 by some strain, epsilon. 00:51:59.940 --> 00:52:01.960 If we can figure out what the volume is 00:52:01.960 --> 00:52:03.720 relative to the initial volume, we 00:52:03.720 --> 00:52:07.230 can figure out how the change of the amount of gas 00:52:07.230 --> 00:52:10.320 goes from the initial state to the compressed state. 00:52:10.320 --> 00:52:11.850 Then we can use Boyle's law. 00:52:11.850 --> 00:52:15.330 The idea is P1V1 equals to P2V2. 00:52:15.330 --> 00:52:17.700 Then we can figure out, using Boyle's law, what 00:52:17.700 --> 00:52:18.630 the pressure must be. 00:52:18.630 --> 00:52:20.005 And then, that pressure is what's 00:52:20.005 --> 00:52:23.040 contributing to the modulus. 00:52:23.040 --> 00:52:25.750 When I do this, I'm going to write down some equations that 00:52:25.750 --> 00:52:27.509 just have the main results. 00:52:27.509 --> 00:52:29.050 There's a whole bunch of algebra just 00:52:29.050 --> 00:52:30.310 to get from one to the other. 00:52:30.310 --> 00:52:32.426 And I'm not going to write them all down. 00:52:32.426 --> 00:52:33.800 When I write the equations, don't 00:52:33.800 --> 00:52:36.320 panic if it's not obvious how you get from one to the other. 00:52:36.320 --> 00:52:38.150 In the notes that I'm going to put online, 00:52:38.150 --> 00:52:42.520 there's all the details of how you get from one to the other. 00:52:42.520 --> 00:52:49.818 The idea is that we start with a cubic element of foam. 00:52:56.380 --> 00:53:01.410 Initially, before it's loaded, it has a volume V0. 00:53:01.410 --> 00:53:03.600 Then we apply a uniaxial stress. 00:53:11.760 --> 00:53:14.847 And we say the axial strain in the direction of the stress, 00:53:14.847 --> 00:53:15.930 I'm going to call epsilon. 00:53:28.690 --> 00:53:31.935 Just from the geometry, you can calculate the deformed volume. 00:53:35.840 --> 00:53:39.400 So after you load it, the volume is V. 00:53:39.400 --> 00:53:43.340 And that volume on loading, V, divided by the initial volume, 00:53:43.340 --> 00:53:46.484 V0, you can show. 00:53:46.484 --> 00:53:47.650 It's fairly straightforward. 00:53:47.650 --> 00:53:51.610 It's 1 minus epsilon times 1 minus 2 times 00:53:51.610 --> 00:53:53.920 the Poisson's ratio of the foam. 00:53:53.920 --> 00:53:56.490 Here, I'm taking compressive strain as positive. 00:54:06.200 --> 00:54:09.490 And if you do this whole volume thing, 00:54:09.490 --> 00:54:13.030 you'll get terms in epsilon squared and epsilon cubed. 00:54:13.030 --> 00:54:14.720 But because if it's linear elastic, 00:54:14.720 --> 00:54:16.140 epsilon is going to be small. 00:54:16.140 --> 00:54:18.050 I'm going to ignore the epsilon squared 00:54:18.050 --> 00:54:19.663 and the epsilon cubed terms. 00:54:24.530 --> 00:54:27.180 That's the total volume of the foam 00:54:27.180 --> 00:54:30.700 after and before the compression. 00:54:30.700 --> 00:54:32.766 And then, what I want is the volume of the gas. 00:54:32.766 --> 00:54:34.140 And the volume of the gas is just 00:54:34.140 --> 00:54:37.580 going to be the volume minus the volume of the solid. 00:54:37.580 --> 00:54:39.420 And I can get that by just subtracting off 00:54:39.420 --> 00:54:41.240 the relative density. 00:54:41.240 --> 00:54:44.010 So Vg over Vg0. 00:54:44.010 --> 00:54:47.740 Again, Vg0 is the volume of the gas initially. 00:54:47.740 --> 00:54:51.360 Vg is the volume of the gas when I'm compressing it. 00:54:59.502 --> 00:55:01.710 Remember, the relative density is the volume fraction 00:55:01.710 --> 00:55:02.310 of solids. 00:55:02.310 --> 00:55:04.748 So I'm essentially subtracting out the amount of solid 00:55:04.748 --> 00:55:05.789 to get the amount of gas. 00:55:34.660 --> 00:55:36.190 Then we can use Boyle's law. 00:55:49.090 --> 00:55:50.610 Here, p is going to be the pressure 00:55:50.610 --> 00:55:53.540 after the strain and p0 is going to be the initial pressure. 00:56:11.720 --> 00:56:15.132 The building seems rather making unhappy noises for some reason. 00:56:15.132 --> 00:56:16.590 I'm not sure what that's all about. 00:56:22.260 --> 00:56:25.230 There'll be some initial pressure in the gas 00:56:25.230 --> 00:56:28.160 even before the strain, or before the stress is applied. 00:56:28.160 --> 00:56:29.270 That's p0. 00:56:29.270 --> 00:56:32.140 So the pressure we need to overcome is p minus p0. 00:56:49.450 --> 00:56:51.510 Again, I'm missing out a bunch of steps. 00:56:51.510 --> 00:56:55.890 But using that expression and this expression 00:56:55.890 --> 00:56:57.600 and this expression, you could find 00:56:57.600 --> 00:57:01.100 that p prime is equal to p0 times 00:57:01.100 --> 00:57:09.210 epsilon times 1 minus 2 nu divided by 1 minus epsilon 1 00:57:09.210 --> 00:57:19.204 minus 2 nu minus the relative density. 00:57:23.220 --> 00:57:25.774 Then the contribution of the gas to the modulus 00:57:25.774 --> 00:57:28.190 you can get by just taking the derivative of that pressure 00:57:28.190 --> 00:57:29.620 with respect to the strain. 00:57:29.620 --> 00:57:31.936 So remember, modulus is stress over strain. 00:57:31.936 --> 00:57:33.060 It's the same kind of idea. 00:57:50.690 --> 00:57:53.400 So I'm going to call it E star g for the contribution 00:57:53.400 --> 00:57:54.900 from the gas. 00:57:54.900 --> 00:57:59.930 So that's going to dp prime d epsilon. 00:57:59.930 --> 00:58:03.894 And that's going to equal p0 times 1 minus 2 nu 00:58:03.894 --> 00:58:05.310 over 1 minus the relative density. 00:58:09.990 --> 00:58:10.490 OK? 00:58:13.540 --> 00:58:17.510 As I said, I'll scan the pages that have the details 00:58:17.510 --> 00:58:18.480 and put it online. 00:58:57.170 --> 00:59:01.390 The final expression we get combines all of these things. 00:59:01.390 --> 00:59:05.170 The modulus of the foam relative to the solid 00:59:05.170 --> 00:59:14.400 is phi squared rho over rho s squared plus 1 minus phi, 00:59:14.400 --> 00:59:17.230 that's the amount of material in the faces, 00:59:17.230 --> 00:59:20.776 times the relative density and then plus this gas compression 00:59:20.776 --> 00:59:21.275 term. 01:00:07.840 --> 01:00:10.950 Like I said, for most foams, the gas compression is negligible. 01:00:10.950 --> 01:00:19.650 So if p0 is equal to the atmospheric pressure, 01:00:19.650 --> 01:00:22.630 so about 0.1 megapascal, then the gas term's 01:00:22.630 --> 01:00:24.664 negligible, except for elastomeric foams. 01:00:49.025 --> 01:00:50.530 So that's the Young's modulus. 01:00:58.520 --> 01:01:02.120 Then we can do a similar thing for the shear modulus. 01:01:02.120 --> 01:01:04.010 And the thing to notice in shear is 01:01:04.010 --> 01:01:06.370 that when you shear something, there's no volume change. 01:01:06.370 --> 01:01:08.370 So if you shear it and there's no volume change, 01:01:08.370 --> 01:01:09.680 there's no gas compression. 01:01:09.680 --> 01:01:11.513 There's no pressure built up because there's 01:01:11.513 --> 01:01:12.717 no change in the volume. 01:01:12.717 --> 01:01:15.216 So for the shear modulus, you just have the first two terms. 01:01:52.070 --> 01:01:54.540 And if the foam is isotropic and you use a third, 01:01:54.540 --> 01:01:56.380 then this constant here is going to be 3/8. 01:02:23.506 --> 01:02:24.880 And then, Poisson's ratio is just 01:02:24.880 --> 01:02:26.940 going to be a function of the cell geometry, again. 01:02:26.940 --> 01:02:29.356 And roughly, we could say it's going to be around a third. 01:02:44.674 --> 01:02:45.174 OK. 01:03:11.000 --> 01:03:12.475 Are we good? 01:03:12.475 --> 01:03:13.547 Let me wait a little bit. 01:03:26.230 --> 01:03:29.639 So the idea is we're looking at the deformation of the bending 01:03:29.639 --> 01:03:31.680 in the cell edges, and the stretching in the cell 01:03:31.680 --> 01:03:33.060 faces, and the gas compression. 01:03:33.060 --> 01:03:35.860 And we're accounting for those different terms. 01:03:35.860 --> 01:03:38.120 The next thing is to compare these model equations 01:03:38.120 --> 01:03:39.910 with some data. 01:03:39.910 --> 01:03:42.277 And here's data for Young's modulus. 01:03:42.277 --> 01:03:44.360 Here, we're plotting the relative Young's modulus, 01:03:44.360 --> 01:03:46.992 so the modulus of the foam divided by the solid 01:03:46.992 --> 01:03:48.200 against the relative density. 01:03:48.200 --> 01:03:51.450 Again, these are log-log scales. 01:03:51.450 --> 01:03:54.330 On this plot, everything with open symbols 01:03:54.330 --> 01:03:56.750 is an open-cell foam, and everything with filled symbols 01:03:56.750 --> 01:03:58.570 is a closed-cell foam. 01:03:58.570 --> 01:04:01.890 Here's our equation for the open-celled foams, the simplest 01:04:01.890 --> 01:04:04.450 thing that goes with density squared. 01:04:04.450 --> 01:04:06.400 And that's that thick line there. 01:04:06.400 --> 01:04:08.380 And you can see these are all open-cell foams. 01:04:08.380 --> 01:04:10.320 There's some more up here. 01:04:10.320 --> 01:04:13.870 So that gives a reasonable description of that data. 01:04:13.870 --> 01:04:16.370 And then, these are two lines for closed-cell foams 01:04:16.370 --> 01:04:18.590 for different values of phi, so different values 01:04:18.590 --> 01:04:20.600 of the amount of solid in the edges. 01:04:20.600 --> 01:04:22.950 And you can see all the filled symbols 01:04:22.950 --> 01:04:24.380 are the closed-cell foams. 01:04:24.380 --> 01:04:27.160 And they're in between this line here and this line here, 01:04:27.160 --> 01:04:28.260 basically. 01:04:28.260 --> 01:04:30.350 So that gives a fairly good description 01:04:30.350 --> 01:04:32.800 for the modulus of the foams considering 01:04:32.800 --> 01:04:35.437 how crude this model is. 01:04:35.437 --> 01:04:37.520 We're not trying to account for the cell geometry. 01:04:37.520 --> 01:04:38.936 We're just modeling the mechanisms 01:04:38.936 --> 01:04:40.850 of deformation and failure. 01:04:40.850 --> 01:04:44.910 And then, here's a similar plot for the shear modulus. 01:04:44.910 --> 01:04:50.150 Here's for open-cells, 3/8 times the relative density squared. 01:04:50.150 --> 01:04:52.820 These are open-celled foams down here. 01:04:52.820 --> 01:04:55.560 These are closed-cell foams up here. 01:04:55.560 --> 01:05:00.220 If the amount of material in the faces is small, 01:05:00.220 --> 01:05:03.390 then you would just get the shear 01:05:03.390 --> 01:05:06.060 modulus varying with the relative density squared. 01:05:06.060 --> 01:05:08.227 Then here's data for Poisson's ratio. 01:05:08.227 --> 01:05:09.810 We don't really know what the constant 01:05:09.810 --> 01:05:11.184 is because we don't know what all 01:05:11.184 --> 01:05:12.930 these geometrical parameters are. 01:05:12.930 --> 01:05:14.444 But here's a value of a third. 01:05:14.444 --> 01:05:16.360 And you can see there's a lot of scatter here. 01:05:16.360 --> 01:05:18.420 This is like less than 0.2. 01:05:18.420 --> 01:05:20.900 This is more than 0.5. 01:05:20.900 --> 01:05:23.550 And the scatter really represents differences 01:05:23.550 --> 01:05:24.790 in the cell geometry. 01:05:24.790 --> 01:05:26.840 If the foams were, say, anisotropic, 01:05:26.840 --> 01:05:29.970 and the cells were stretched in one direction, 01:05:29.970 --> 01:05:33.070 then you would get different values of the Poisson's ratios. 01:05:33.070 --> 01:05:36.230 So that's the Poisson's ratios there. 01:05:36.230 --> 01:05:37.900 I'm thinking I'm going to stop there 01:05:37.900 --> 01:05:40.360 because that seems like enough equations for today. 01:05:40.360 --> 01:05:44.270 And then, next time we'll start doing the stress plateau 01:05:44.270 --> 01:05:46.709 and we'll figure out the elastic collapse stress 01:05:46.709 --> 01:05:48.500 from buckling and a plastic collapse stress 01:05:48.500 --> 01:05:50.600 from yielding, and so on, and so on. 01:05:50.600 --> 01:05:54.320 We'll finish doing the modeling next time. 01:05:54.320 --> 01:05:57.300 And we'll probably start doing a little bit of other stuff 01:05:57.300 --> 01:05:59.600 on foams next time.