1 00:00:00,030 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:04,000 Commons license. 3 00:00:04,000 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,025 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,025 --> 00:00:17,650 at ocw.mit.edu. 8 00:00:26,470 --> 00:00:28,790 LORNA GIBSON: I wanted to talk a little bit about cork 9 00:00:28,790 --> 00:00:30,759 partly because cork is kind of interesting. 10 00:00:30,759 --> 00:00:32,800 Cork has a structure that's a little bit like one 11 00:00:32,800 --> 00:00:34,290 of those honeycomb things. 12 00:00:34,290 --> 00:00:37,280 What I'm going to do is I'm just going to talk and go 13 00:00:37,280 --> 00:00:38,600 through the slides. 14 00:00:38,600 --> 00:00:40,630 I'm not going to write the notes on the board. 15 00:00:40,630 --> 00:00:42,070 There's only a few pages of notes, 16 00:00:42,070 --> 00:00:43,427 and it's in the Stellar site. 17 00:00:43,427 --> 00:00:46,010 I'm not going to write notes for this because it's just really 18 00:00:46,010 --> 00:00:47,140 for fun. 19 00:00:47,140 --> 00:00:50,900 This first slide starts off with historical uses of cork. 20 00:00:50,900 --> 00:00:53,120 Cork was used by the Romans. 21 00:00:53,120 --> 00:00:55,850 They used it for the soles of sandals, the same as we do. 22 00:00:55,850 --> 00:00:58,360 And they used it for stopping bottles 23 00:00:58,360 --> 00:00:59,985 of wine, the same as we do. 24 00:00:59,985 --> 00:01:02,360 But they didn't realize that you could just use the cork. 25 00:01:02,360 --> 00:01:04,640 They would take the cork, put it in the wine bottle, 26 00:01:04,640 --> 00:01:06,380 and then they would use pitch which 27 00:01:06,380 --> 00:01:09,450 is a tarry stuff from a tree. 28 00:01:09,450 --> 00:01:13,370 They would use the pitch and seal the bottle with the pitch. 29 00:01:13,370 --> 00:01:15,704 In the 1600s, there were some Benedictine monks 30 00:01:15,704 --> 00:01:17,620 that realized that you could just use the cork 31 00:01:17,620 --> 00:01:19,040 and not use the pitch. 32 00:01:19,040 --> 00:01:21,460 They were the ones who really perfected the use of corks 33 00:01:21,460 --> 00:01:23,590 in wine bottles for sealing it. 34 00:01:23,590 --> 00:01:25,640 So what is cork? 35 00:01:25,640 --> 00:01:27,570 Cork is the bark of a tree called 36 00:01:27,570 --> 00:01:29,800 Quercus suber, the cork oak. 37 00:01:29,800 --> 00:01:32,950 Here's a piece of the cork bark. 38 00:01:32,950 --> 00:01:35,727 All trees have a layer of cork in them. 39 00:01:35,727 --> 00:01:37,810 But the thing that's different about Quercus suber 40 00:01:37,810 --> 00:01:40,240 is that it's very thick and the corks are 41 00:01:40,240 --> 00:01:42,600 obtained from this thick layer. 42 00:01:42,600 --> 00:01:44,900 Quercus suber is a Mediterranean type of tree. 43 00:01:44,900 --> 00:01:47,560 It grows, Portugal is the main place 44 00:01:47,560 --> 00:01:52,190 that exports cork, but also places like Algeria, Spain. 45 00:01:52,190 --> 00:01:54,830 You can grow it in California. 46 00:01:54,830 --> 00:01:57,390 The cork is kind of unusual because-- let 47 00:01:57,390 --> 00:02:01,000 me scoot onto the next slide-- it's kind of unusual. 48 00:02:01,000 --> 00:02:02,130 So here's a little picture. 49 00:02:02,130 --> 00:02:04,210 I went to Portugal when I was a graduate student doing 50 00:02:04,210 --> 00:02:04,850 this project. 51 00:02:04,850 --> 00:02:05,990 Here's the cork tree here. 52 00:02:05,990 --> 00:02:08,680 Here's the little mini we rented. 53 00:02:08,680 --> 00:02:11,500 Here's the cork being harvested here. 54 00:02:11,500 --> 00:02:14,650 Cork's unusual because you can remove the bark from a cork oak 55 00:02:14,650 --> 00:02:15,780 tree and it regrows. 56 00:02:15,780 --> 00:02:18,580 So most trees if you did this, you would kill the tree. 57 00:02:18,580 --> 00:02:22,430 But cork doesn't get killed by doing this. 58 00:02:22,430 --> 00:02:24,950 What happens is they plant trees. 59 00:02:24,950 --> 00:02:28,530 You have to wait something like 10 or 15 years for the tree 60 00:02:28,530 --> 00:02:30,460 to get big enough to harvest the cork. 61 00:02:30,460 --> 00:02:33,810 And then the first harvest is poor quality 62 00:02:33,810 --> 00:02:34,820 and they don't use that. 63 00:02:34,820 --> 00:02:36,819 And then you have to wait another 10 years or so 64 00:02:36,819 --> 00:02:39,060 before you can actually harvest the cork. 65 00:02:39,060 --> 00:02:43,610 So you can imagine if a cork orchard or forest gets chopped 66 00:02:43,610 --> 00:02:46,320 down to build a skyscraper, or apartment buildings, 67 00:02:46,320 --> 00:02:50,810 or something, it's not an economically feasible thing 68 00:02:50,810 --> 00:02:52,170 to plant cork trees these days. 69 00:02:52,170 --> 00:02:55,470 So when the trees are cut down, they 70 00:02:55,470 --> 00:02:57,910 don't tend to get replanted at this point. 71 00:02:57,910 --> 00:03:00,200 And there's a number of artificial substitutes 72 00:03:00,200 --> 00:03:00,700 for corks. 73 00:03:00,700 --> 00:03:04,070 You've probably seen wine bundles with foam plastic corks 74 00:03:04,070 --> 00:03:06,600 in them too. 75 00:03:06,600 --> 00:03:08,000 OK. 76 00:03:08,000 --> 00:03:09,660 The reason it's called Quercus suber 77 00:03:09,660 --> 00:03:15,220 is that the cell walls in this particular type of cork oak 78 00:03:15,220 --> 00:03:18,880 are covered with a waxy substance called "suberine." 79 00:03:18,880 --> 00:03:21,520 That's where the Quercus suber-- "Quercus" is "oak." 80 00:03:21,520 --> 00:03:25,790 Every oak is Quercus something. "Quercus alba" is white oak. 81 00:03:25,790 --> 00:03:28,130 Quercus just means oak. 82 00:03:28,130 --> 00:03:30,410 If we look at the structure of the cork, 83 00:03:30,410 --> 00:03:36,030 we can see that it's got these different views 84 00:03:36,030 --> 00:03:38,170 of the cork that are seen here. 85 00:03:38,170 --> 00:03:40,950 This is a drawing by Robert Hooke in the 1600s 86 00:03:40,950 --> 00:03:42,510 from his book Micrographia. 87 00:03:42,510 --> 00:03:46,240 He was the first person to really draw cork like this. 88 00:03:46,240 --> 00:03:49,990 You can see he drew sort of boxy cells on this side. 89 00:03:49,990 --> 00:03:53,020 And then, the other perpendicular plane, the cells 90 00:03:53,020 --> 00:03:55,570 have this structure here, sort of more rounded. 91 00:03:55,570 --> 00:03:59,210 And here's a little sketch he's got of the cork tree. 92 00:03:59,210 --> 00:04:00,910 Over here are SEM pictures. 93 00:04:00,910 --> 00:04:04,870 These two planes here correspond to this plane 94 00:04:04,870 --> 00:04:06,120 in Hooke's drawing. 95 00:04:06,120 --> 00:04:09,270 This plane here corresponds to this plane here. 96 00:04:09,270 --> 00:04:11,373 This over here is Hooke's actual microscope. 97 00:04:14,330 --> 00:04:17,880 I think the Royal Society still has that microscope 98 00:04:17,880 --> 00:04:20,779 that Hooke used in the 1600s. 99 00:04:20,779 --> 00:04:22,250 Some of you know that Hooke wrote 100 00:04:22,250 --> 00:04:23,500 this book called Micrographia. 101 00:04:23,500 --> 00:04:25,790 He got one of the first microscopes. 102 00:04:25,790 --> 00:04:27,540 He looked at a lot of different materials, 103 00:04:27,540 --> 00:04:29,350 and he drew these beautiful drawings. 104 00:04:29,350 --> 00:04:32,640 He wrote a page or two about each of the drawings. 105 00:04:32,640 --> 00:04:35,770 Harvard has a first edition of Micrographia, 106 00:04:35,770 --> 00:04:37,480 and I made a little video on it. 107 00:04:37,480 --> 00:04:41,110 So I thought I'd show you the video because the drawings are 108 00:04:41,110 --> 00:04:42,000 beautiful. 109 00:04:42,000 --> 00:04:46,160 But the url's on the slides, and so you can watch it yourself. 110 00:04:46,160 --> 00:04:48,330 There's more on this and on how he 111 00:04:48,330 --> 00:04:52,410 came to be so good at making scientific apparatus, 112 00:04:52,410 --> 00:04:54,870 how he came to do the Micrographia book. 113 00:04:54,870 --> 00:04:56,780 At the end of it, there's a comparison 114 00:04:56,780 --> 00:04:59,590 of a number of his drawings with modern SEM 115 00:04:59,590 --> 00:05:00,840 images of the same thing. 116 00:05:00,840 --> 00:05:04,230 He had this very famous picture where he draws a flea. 117 00:05:04,230 --> 00:05:06,540 Don Galler, the person who runs the SEM for me, 118 00:05:06,540 --> 00:05:08,070 I had him put a flea in. 119 00:05:08,070 --> 00:05:11,302 And he has essentially the same kind of image. 120 00:05:11,302 --> 00:05:12,760 The thing that's spectacular is you 121 00:05:12,760 --> 00:05:15,560 can see how much of the detail that Hooke was able to capture 122 00:05:15,560 --> 00:05:16,250 in his drawings. 123 00:05:16,250 --> 00:05:17,700 There's some really beautiful drawings. 124 00:05:17,700 --> 00:05:18,220 OK. 125 00:05:18,220 --> 00:05:20,480 So let's go back to cork. 126 00:05:20,480 --> 00:05:23,550 Hooke was the first person to use the word "cell" to describe 127 00:05:23,550 --> 00:05:28,020 biological cells, and he described the cell in cork. 128 00:05:28,020 --> 00:05:32,110 That's the structure looking at the SEM micrographs 129 00:05:32,110 --> 00:05:35,110 and his optical micrographs. 130 00:05:35,110 --> 00:05:37,110 These are just some more higher resolution, 131 00:05:37,110 --> 00:05:39,324 higher magnification, images. 132 00:05:39,324 --> 00:05:40,740 One of the things that you can see 133 00:05:40,740 --> 00:05:43,331 is that the cork has these little corrugations on the cell 134 00:05:43,331 --> 00:05:43,830 walls. 135 00:05:43,830 --> 00:05:45,110 See those little wrinkles? 136 00:05:45,110 --> 00:05:47,080 All the cell walls have those little wrinkles. 137 00:05:47,080 --> 00:05:49,100 This plane here is the perpendicular plane. 138 00:05:49,100 --> 00:05:51,400 If you look down into the cells, you 139 00:05:51,400 --> 00:05:53,200 can see those blurry things. 140 00:05:53,200 --> 00:05:56,850 Those are the corrugations in the cell walls. 141 00:05:56,850 --> 00:05:58,610 That's the structure. 142 00:05:58,610 --> 00:06:00,150 Here's a schematic. 143 00:06:00,150 --> 00:06:01,510 Here's the cork tree. 144 00:06:01,510 --> 00:06:04,520 The cork is the layer just beneath the bark. 145 00:06:04,520 --> 00:06:07,400 This is a picture of how the cells are oriented 146 00:06:07,400 --> 00:06:10,130 relative to their radial, and tangential, and axial 147 00:06:10,130 --> 00:06:11,400 directions. 148 00:06:11,400 --> 00:06:13,240 You can think of them as roughly hexagonal. 149 00:06:13,240 --> 00:06:14,823 They've got these little corrugations. 150 00:06:14,823 --> 00:06:18,860 This is a schematic of an individual cell. 151 00:06:18,860 --> 00:06:20,779 We measured the dimensions of the cells, 152 00:06:20,779 --> 00:06:22,570 and these are some average dimensions here. 153 00:06:22,570 --> 00:06:25,470 Typically, the cells are tens of microns long, 154 00:06:25,470 --> 00:06:27,330 and the cell wall is about a micron thick. 155 00:06:27,330 --> 00:06:28,320 Something like that. 156 00:06:28,320 --> 00:06:29,321 OK? 157 00:06:29,321 --> 00:06:30,820 One of the things we're going to see 158 00:06:30,820 --> 00:06:33,260 is that corrugated structure gives rise 159 00:06:33,260 --> 00:06:36,970 to some of the interesting properties of cork. 160 00:06:36,970 --> 00:06:39,920 If we load cork and just do mechanical tests on it-- 161 00:06:39,920 --> 00:06:42,792 this is just a uniaxial stress-strain curve. 162 00:06:42,792 --> 00:06:44,500 You can see the stress-strain curve looks 163 00:06:44,500 --> 00:06:46,750 like all these other curves we've seen for honeycombs. 164 00:06:46,750 --> 00:06:48,540 There's a linear elastic part here. 165 00:06:48,540 --> 00:06:50,190 There's a stress plateau here. 166 00:06:50,190 --> 00:06:52,480 And then there's a densification part here. 167 00:06:52,480 --> 00:06:55,760 Typically, the relative density of cork is around 0.15. 168 00:06:55,760 --> 00:06:59,040 Something like that. 169 00:06:59,040 --> 00:07:03,510 It densifies at a strain of something less than 0.85. 170 00:07:03,510 --> 00:07:05,200 It's a stress-strain curve. 171 00:07:05,200 --> 00:07:07,600 And when we did our little project on cork, 172 00:07:07,600 --> 00:07:10,990 we measured the properties in the three directions 173 00:07:10,990 --> 00:07:14,670 because in one direction, it's roughly hexagonal cells. 174 00:07:14,670 --> 00:07:16,780 That plane is isotropic. 175 00:07:16,780 --> 00:07:20,270 The e1,e2 plane is isotropic. 176 00:07:20,270 --> 00:07:22,570 This compares the measured values 177 00:07:22,570 --> 00:07:24,640 of the properties versus what we calculated 178 00:07:24,640 --> 00:07:26,380 from the honeycomb model. 179 00:07:26,380 --> 00:07:28,450 And really, we just used the sorts of equations 180 00:07:28,450 --> 00:07:30,533 that we talked about in class over the last couple 181 00:07:30,533 --> 00:07:33,182 of lectures, apart from loading in the x3 direction 182 00:07:33,182 --> 00:07:34,640 because in the x3 direction, you've 183 00:07:34,640 --> 00:07:35,972 got those corrugated walls. 184 00:07:35,972 --> 00:07:38,180 And you have to take those corrugations into account. 185 00:07:38,180 --> 00:07:40,794 So there's another complication that I'm not going to go into. 186 00:07:40,794 --> 00:07:42,210 But there's a sort of modification 187 00:07:42,210 --> 00:07:43,780 you can do to account for that. 188 00:07:43,780 --> 00:07:46,238 But you can see there's actually pretty good agreement here 189 00:07:46,238 --> 00:07:48,560 between the elastic moduli that we measured 190 00:07:48,560 --> 00:07:49,930 and what we calculated. 191 00:07:49,930 --> 00:07:52,880 The compressive strengths down here are not quite so good. 192 00:07:52,880 --> 00:07:55,870 They're off by a factor of two, more or less. 193 00:07:55,870 --> 00:08:01,090 But they're in the right ballpark for the cork. 194 00:08:01,090 --> 00:08:03,370 So those are some of the structures, 195 00:08:03,370 --> 00:08:04,584 some of the properties. 196 00:08:04,584 --> 00:08:06,250 One of the interesting things about cork 197 00:08:06,250 --> 00:08:07,860 is what it's used for. 198 00:08:07,860 --> 00:08:11,980 The uses of cork really exploit the mechanical properties. 199 00:08:11,980 --> 00:08:14,070 Obviously, it's used for stoppers for bottles. 200 00:08:14,070 --> 00:08:16,297 I brought a champagne cork along with me, 201 00:08:16,297 --> 00:08:18,755 and I brought a couple of other little pieces of cork here. 202 00:08:18,755 --> 00:08:20,960 I'll pass those around in a minute. 203 00:08:20,960 --> 00:08:22,460 One of the things to look at is just 204 00:08:22,460 --> 00:08:25,600 the still wine cork, which is the one on the right, 205 00:08:25,600 --> 00:08:28,550 and the champagne cork, which is the one on the left. 206 00:08:28,550 --> 00:08:30,210 If you notice the still wine cork 207 00:08:30,210 --> 00:08:32,080 is just made of one piece of cork 208 00:08:32,080 --> 00:08:34,980 that's cored out from the bark. 209 00:08:34,980 --> 00:08:36,650 And if you notice these little channels. 210 00:08:36,650 --> 00:08:39,580 These little channels here are called "lenticels." 211 00:08:39,580 --> 00:08:41,809 On the still wine cork, they go this way. 212 00:08:41,809 --> 00:08:44,140 And on the champagne cork, they go that way. 213 00:08:44,140 --> 00:08:46,390 They're oriented perpendicular. 214 00:08:46,390 --> 00:08:51,720 It turns out that they are normal to the isotropic plane 215 00:08:51,720 --> 00:08:53,480 in the cork. 216 00:08:53,480 --> 00:08:57,130 If you look at the champagne cork, 217 00:08:57,130 --> 00:08:59,360 this plane here is the isotropic plane. 218 00:08:59,360 --> 00:09:01,877 If you think of that being put into a bottle, 219 00:09:01,877 --> 00:09:03,460 I think part of the reason they orient 220 00:09:03,460 --> 00:09:05,120 it this way is because it gives you 221 00:09:05,120 --> 00:09:08,150 a uniform compression against the neck of the bottle 222 00:09:08,150 --> 00:09:09,940 and gives a nice seal. 223 00:09:09,940 --> 00:09:12,932 So that's one of the things about corks. 224 00:09:12,932 --> 00:09:14,390 Another thing that's interesting is 225 00:09:14,390 --> 00:09:20,500 that cork has a Poisson's ratio equal to zero if you load it 226 00:09:20,500 --> 00:09:22,930 in that direction. 227 00:09:22,930 --> 00:09:23,450 Let's see. 228 00:09:23,450 --> 00:09:24,700 Did I not bring that picture? 229 00:09:24,700 --> 00:09:26,116 Maybe I didn't bring that picture. 230 00:09:26,116 --> 00:09:27,280 Hang on a sec. 231 00:09:27,280 --> 00:09:27,780 Nope. 232 00:09:27,780 --> 00:09:28,488 I guess I didn't. 233 00:09:28,488 --> 00:09:29,030 Sorry. 234 00:09:29,030 --> 00:09:30,488 I thought I brought a picture where 235 00:09:30,488 --> 00:09:34,811 I had the deformation of the cells when you load it 236 00:09:34,811 --> 00:09:35,560 in that direction. 237 00:09:35,560 --> 00:09:37,100 When you load it-- so say the cells 238 00:09:37,100 --> 00:09:38,224 are corrugated this way on. 239 00:09:38,224 --> 00:09:41,140 When you load it that way on, it's like having a bellows 240 00:09:41,140 --> 00:09:43,514 and folding up a bellows, or unfolding a bellows. 241 00:09:43,514 --> 00:09:45,180 So when you load it that way on, there's 242 00:09:45,180 --> 00:09:47,460 no expansion or contraction this way on. 243 00:09:47,460 --> 00:09:49,910 And so you get zero Poisson's ratio. 244 00:09:49,910 --> 00:09:52,710 And if you think of trying to get the cork into the bottle, 245 00:09:52,710 --> 00:09:55,254 it's rather convenient to have zero Poisson's ratio 246 00:09:55,254 --> 00:09:56,920 because you don't get as much expansion. 247 00:09:56,920 --> 00:09:58,461 As you're pushing it into the bottle, 248 00:09:58,461 --> 00:10:01,040 you don't get as much expansion that way out, 249 00:10:01,040 --> 00:10:02,450 pressing against it. 250 00:10:02,450 --> 00:10:05,680 In fact, if you compare wine corks with rubber stoppers, 251 00:10:05,680 --> 00:10:08,550 this is a kind of typical rubber stopper. 252 00:10:08,550 --> 00:10:10,420 Wine corks are always just cylinders. 253 00:10:10,420 --> 00:10:11,870 In fact, even the champagne corks 254 00:10:11,870 --> 00:10:13,350 are cylinders when they start off. 255 00:10:13,350 --> 00:10:14,725 When they put it into the bottle, 256 00:10:14,725 --> 00:10:16,030 it's just a straight cylinder. 257 00:10:16,030 --> 00:10:17,790 It gets deformed like that from being 258 00:10:17,790 --> 00:10:19,650 in the bottle for some time. 259 00:10:19,650 --> 00:10:22,140 They have straight sides and you can just squeeze them. 260 00:10:22,140 --> 00:10:24,580 There's these funnel things that wine makers 261 00:10:24,580 --> 00:10:27,140 have for putting the cork in the bottle. 262 00:10:27,140 --> 00:10:29,100 You can just squeeze them into the bottle top. 263 00:10:29,100 --> 00:10:31,640 You can do that because the Poisson's ratio is zero 264 00:10:31,640 --> 00:10:34,000 and because the Young's modulus and the bulk modulus 265 00:10:34,000 --> 00:10:35,140 are both small. 266 00:10:35,140 --> 00:10:36,640 But if you look at a rubber stopper, 267 00:10:36,640 --> 00:10:40,070 rubber stoppers always have these tapered sides to them. 268 00:10:40,070 --> 00:10:43,810 And that's because the Poisson's ratio of the rubber is 0.5. 269 00:10:43,810 --> 00:10:46,912 As you squeeze it in, it's trying to move out this way. 270 00:10:46,912 --> 00:10:48,620 You couldn't get the stopper in unless it 271 00:10:48,620 --> 00:10:50,180 had those tapered sides. 272 00:10:50,180 --> 00:10:52,690 So that's sort of an interesting thing about cork. 273 00:10:52,690 --> 00:10:53,190 Let's see. 274 00:10:53,190 --> 00:10:54,680 Another application of cork is it's 275 00:10:54,680 --> 00:10:57,050 used for gaskets for the same sorts of reasons. 276 00:10:57,050 --> 00:10:59,630 It's relatively compliant. 277 00:10:59,630 --> 00:11:02,120 It takes up any slack between two pieces that you 278 00:11:02,120 --> 00:11:03,840 want to press together. 279 00:11:03,840 --> 00:11:05,800 It's often used for musical instruments 280 00:11:05,800 --> 00:11:06,804 that come in pieces. 281 00:11:06,804 --> 00:11:08,220 Things like clarinets, there'll be 282 00:11:08,220 --> 00:11:10,481 a piece of cork-- you a clarinet player? 283 00:11:10,481 --> 00:11:10,980 Yeah? 284 00:11:10,980 --> 00:11:11,880 Yeah. 285 00:11:11,880 --> 00:11:14,640 One of the interesting things about the clarinet 286 00:11:14,640 --> 00:11:16,431 is that, if you can see here at the ends, 287 00:11:16,431 --> 00:11:17,680 there's a piece of cork there. 288 00:11:17,680 --> 00:11:19,638 And I think there's a piece of cork down there. 289 00:11:19,638 --> 00:11:22,370 And the other pieces mate up with that. 290 00:11:22,370 --> 00:11:24,570 The cork provides a seal. 291 00:11:24,570 --> 00:11:27,410 And the way the cork is cut, it's cut in such a way 292 00:11:27,410 --> 00:11:30,650 that those lenticels go radially out like this, which 293 00:11:30,650 --> 00:11:35,340 means that the plane of isotropy and the direction that's 294 00:11:35,340 --> 00:11:41,040 got the zero Poisson's ratio is that radial direction. 295 00:11:41,040 --> 00:11:45,140 When you're squeezing, say, the second part onto it, 296 00:11:45,140 --> 00:11:48,142 the cork does not expand circumferentially. 297 00:11:48,142 --> 00:11:49,600 So as you're squeezing it this way, 298 00:11:49,600 --> 00:11:50,770 it doesn't expand that way. 299 00:11:50,770 --> 00:11:53,620 It doesn't wrinkle or anything on your other part. 300 00:11:53,620 --> 00:11:56,150 They use the cork in a particular orientation 301 00:11:56,150 --> 00:11:57,500 for that reason, I think. 302 00:11:57,500 --> 00:11:59,500 So it's used for gaskets 303 00:11:59,500 --> 00:12:03,590 It's also used because it's got a good friction property. 304 00:12:03,590 --> 00:12:07,650 It's got a property that is taken advantage of in things 305 00:12:07,650 --> 00:12:09,510 like flooring and shoes. 306 00:12:09,510 --> 00:12:13,190 Cork has a high friction even if it gets wet. 307 00:12:13,190 --> 00:12:15,410 Some sources of friction are from adhesion, 308 00:12:15,410 --> 00:12:16,770 from a surface effect. 309 00:12:16,770 --> 00:12:20,450 Then if, say, the floor gets wet, then you break that, 310 00:12:20,450 --> 00:12:22,520 and it could be slippery. 311 00:12:22,520 --> 00:12:24,430 But the source of friction in cork 312 00:12:24,430 --> 00:12:27,690 is from an energy loss and dissipation 313 00:12:27,690 --> 00:12:29,050 as you're deforming it. 314 00:12:29,050 --> 00:12:31,160 Imagine you have a wheel here. 315 00:12:31,160 --> 00:12:33,820 The wheel is rotating on this cork floor. 316 00:12:33,820 --> 00:12:36,600 And here, a piece of cork is getting deformed as the wheel 317 00:12:36,600 --> 00:12:38,240 rolls over it. 318 00:12:38,240 --> 00:12:41,747 As it gets deformed, there's some histeresis loop. 319 00:12:41,747 --> 00:12:43,330 Cork has quite a lot of damping in it. 320 00:12:43,330 --> 00:12:46,550 There's quite a lot of energy lost in that histeresis loop. 321 00:12:46,550 --> 00:12:50,020 What that means is that's characteristic of the cork 322 00:12:50,020 --> 00:12:50,590 itself. 323 00:12:50,590 --> 00:12:52,390 It's not a surface effect. 324 00:12:52,390 --> 00:12:57,330 That means that if you use it for floors or for shoes, 325 00:12:57,330 --> 00:13:01,547 it doesn't lose that friction and damping when it gets wet. 326 00:13:01,547 --> 00:13:04,130 Here's some measurements we did of the coefficient of friction 327 00:13:04,130 --> 00:13:07,310 for cork versus doing it dry and doing it 328 00:13:07,310 --> 00:13:10,330 with a liquid, soapy surface. 329 00:13:10,330 --> 00:13:12,850 You can see the soap doesn't make any difference. 330 00:13:12,850 --> 00:13:15,430 Cork is seen as a very attractive material 331 00:13:15,430 --> 00:13:17,244 for things like flooring. 332 00:13:17,244 --> 00:13:18,660 It's actually not a cheap material 333 00:13:18,660 --> 00:13:22,170 to make your floors out of, but it's an attractive material 334 00:13:22,170 --> 00:13:22,846 for flooring. 335 00:13:22,846 --> 00:13:24,470 Part of the reason it's used for floors 336 00:13:24,470 --> 00:13:26,761 and for the soles of shoes is because of these friction 337 00:13:26,761 --> 00:13:27,980 properties. 338 00:13:27,980 --> 00:13:29,610 Another feature of cork is that it's 339 00:13:29,610 --> 00:13:32,850 got very small cells compared to a lot of engineering polymers. 340 00:13:32,850 --> 00:13:35,080 The cells are on the order of tens of microns, 341 00:13:35,080 --> 00:13:37,610 whereas many polymer foams, the cells 342 00:13:37,610 --> 00:13:40,480 are hundreds of microns or millimeters. 343 00:13:40,480 --> 00:13:42,500 We'll get into this later, but this plot here 344 00:13:42,500 --> 00:13:45,030 is really saying that the thermal conductivity 345 00:13:45,030 --> 00:13:49,350 of a cellular material depends, in part, on the cell size. 346 00:13:49,350 --> 00:13:51,760 The cell size for foam plastics is in here. 347 00:13:51,760 --> 00:13:54,150 And that for cork is down here. 348 00:13:54,150 --> 00:13:55,860 Because it has a smaller cell size, 349 00:13:55,860 --> 00:13:58,500 it has a lower thermal conductivity. 350 00:13:58,500 --> 00:14:01,090 Cork was at one point used to some extent 351 00:14:01,090 --> 00:14:02,590 for thermal insulation. 352 00:14:02,590 --> 00:14:05,020 If you go to Portugal, where cork comes from, 353 00:14:05,020 --> 00:14:06,240 there's hermit caves. 354 00:14:06,240 --> 00:14:08,930 There were these old hermit, religious people 355 00:14:08,930 --> 00:14:10,720 who had holed up in a cave. 356 00:14:10,720 --> 00:14:12,840 And they would line the caves with cork 357 00:14:12,840 --> 00:14:16,950 to try to make it a little more insulated, a little more warm. 358 00:14:16,950 --> 00:14:20,130 The other place you see this is if you look at cigarettes, 359 00:14:20,130 --> 00:14:22,920 you know cigarettes have that little brown tip on the part 360 00:14:22,920 --> 00:14:24,760 that touches your lips? 361 00:14:24,760 --> 00:14:26,550 That's meant to look like cork. 362 00:14:26,550 --> 00:14:28,300 And if you look, it has little dots on it. 363 00:14:28,300 --> 00:14:30,360 The little dots are the little lenticels. 364 00:14:30,360 --> 00:14:33,220 Apparently, they used cork originally in cigarettes 365 00:14:33,220 --> 00:14:35,520 as a sort of thermal insulation between the cigarette 366 00:14:35,520 --> 00:14:36,940 and your mouth. 367 00:14:36,940 --> 00:14:39,177 So it was used for that too. 368 00:14:39,177 --> 00:14:40,260 And then, one final thing. 369 00:14:40,260 --> 00:14:43,150 Cork's also used, obviously, for bulletin boards. 370 00:14:43,150 --> 00:14:46,330 If you push a pin into cork, then you 371 00:14:46,330 --> 00:14:48,350 get this local deformation here. 372 00:14:48,350 --> 00:14:49,170 Here's our pin. 373 00:14:49,170 --> 00:14:51,215 And here's cells locally deformed. 374 00:14:51,215 --> 00:14:52,590 When you pull the pin out, you'll 375 00:14:52,590 --> 00:14:54,930 recover some of that deformation because the deformation 376 00:14:54,930 --> 00:14:55,590 is elastic. 377 00:14:55,590 --> 00:14:57,550 So the hole will partly close. 378 00:14:57,550 --> 00:14:59,070 So that's my little spiel on cork. 379 00:14:59,070 --> 00:15:01,770 And that's just because it's interesting. 380 00:15:01,770 --> 00:15:04,160 There's no test on cork or anything like that. 381 00:15:04,160 --> 00:15:05,140 OK. 382 00:15:05,140 --> 00:15:06,410 So are we good with cork? 383 00:15:06,410 --> 00:15:07,680 Any questions? 384 00:15:07,680 --> 00:15:08,770 We're OK? 385 00:15:08,770 --> 00:15:10,000 OK. 386 00:15:10,000 --> 00:15:12,470 Let me scoot out of there. 387 00:15:12,470 --> 00:15:14,640 Then the next part of the course, 388 00:15:14,640 --> 00:15:16,125 I wanted to talk about foams. 389 00:15:19,220 --> 00:15:21,390 Let me just park the cork thing. 390 00:15:21,390 --> 00:15:22,760 Let me pass these corks around. 391 00:15:22,760 --> 00:15:25,290 So you can play with those too. 392 00:15:25,290 --> 00:15:26,590 Oops. 393 00:15:26,590 --> 00:15:28,660 There's little bits of cork. 394 00:15:28,660 --> 00:15:29,619 There you go. 395 00:15:29,619 --> 00:15:32,160 There's the champagne cork, the rubber cork, some little cork 396 00:15:32,160 --> 00:15:32,660 layers. 397 00:15:37,640 --> 00:15:38,140 OK. 398 00:15:38,140 --> 00:15:41,820 So the next part of the course, I wanted to talk about foams. 399 00:15:41,820 --> 00:15:44,940 And I want to talk about how we model the mechanical behavior 400 00:15:44,940 --> 00:15:46,750 of foams. 401 00:15:46,750 --> 00:15:49,530 If we look at the stress-strain curve for foams, 402 00:15:49,530 --> 00:15:52,270 these are some examples for foams made out 403 00:15:52,270 --> 00:15:55,340 of different materials with different characteristics. 404 00:15:55,340 --> 00:15:57,130 The polyurethane and the polyethylene 405 00:15:57,130 --> 00:16:01,840 here are examples of elastomeric foams, really. 406 00:16:01,840 --> 00:16:03,570 This one here is an open-celled foam. 407 00:16:03,570 --> 00:16:05,390 This one's a closed-cell foam. 408 00:16:05,390 --> 00:16:10,830 Polymethacrylamide is a polymer that has a yield point. 409 00:16:10,830 --> 00:16:12,370 Mullite is a ceramic. 410 00:16:12,370 --> 00:16:14,000 You can see the shapes of these curves 411 00:16:14,000 --> 00:16:16,230 resemble the shapes that we saw for the honeycombs. 412 00:16:16,230 --> 00:16:16,730 Right? 413 00:16:16,730 --> 00:16:18,700 They look exactly the same, in fact. 414 00:16:18,700 --> 00:16:21,720 And the mechanisms of deformation in the foams 415 00:16:21,720 --> 00:16:24,100 are very similar to the honeycombs. 416 00:16:24,100 --> 00:16:27,180 Even though the foams have a much more complicated geometry, 417 00:16:27,180 --> 00:16:29,650 we can use some of the ideas from the honeycombs 418 00:16:29,650 --> 00:16:31,230 to understand how the foams behave. 419 00:16:31,230 --> 00:16:33,070 So that was part of the reason for doing 420 00:16:33,070 --> 00:16:36,440 the honeycomb analysis. 421 00:16:36,440 --> 00:16:37,770 Let me back up. 422 00:16:37,770 --> 00:16:39,820 These curves here were all in compression. 423 00:16:39,820 --> 00:16:41,570 These curves here were all in tension. 424 00:16:41,570 --> 00:16:43,120 So again, these ones in tension also 425 00:16:43,120 --> 00:16:46,010 look like the curves for the honeycombs. 426 00:16:46,010 --> 00:16:49,034 Remember, in tension, we don't get any elastic buckling. 427 00:16:49,034 --> 00:16:50,700 So if the foam was made of an elastomer, 428 00:16:50,700 --> 00:16:53,210 we don't see any stress plateau. 429 00:16:53,210 --> 00:16:56,330 If the foam is made of material with a yield stress, 430 00:16:56,330 --> 00:16:59,040 then we get a very short yield plateau 431 00:16:59,040 --> 00:17:00,910 because of a slight geometrical difference 432 00:17:00,910 --> 00:17:03,870 between pulling and compressing the foam. 433 00:17:03,870 --> 00:17:06,369 And if it's a brittle material, then we just get fracturing. 434 00:17:06,369 --> 00:17:08,452 There's going to be some fracture toughness that's 435 00:17:08,452 --> 00:17:09,800 going to characterize it. 436 00:17:09,800 --> 00:17:14,660 We can look at the deformation and the failure in these foams 437 00:17:14,660 --> 00:17:15,990 and look at the mechanisms. 438 00:17:15,990 --> 00:17:19,280 And what we're going to do is model the mechanisms 439 00:17:19,280 --> 00:17:21,730 and not worry too much about the cell geometry. 440 00:17:21,730 --> 00:17:24,390 So we're going to use dimensional arguments here. 441 00:17:24,390 --> 00:17:26,060 Here's a foam in compression. 442 00:17:26,060 --> 00:17:28,460 It was compressed from the top to the bottom. 443 00:17:28,460 --> 00:17:32,260 And you can see this strut that's circled in red. 444 00:17:32,260 --> 00:17:33,134 This is unloaded. 445 00:17:33,134 --> 00:17:34,550 And then, this is after some load. 446 00:17:34,550 --> 00:17:36,340 You can see this has bent somewhat. 447 00:17:36,340 --> 00:17:38,440 And then, you can see this vertical strut here. 448 00:17:38,440 --> 00:17:41,880 As the load gets larger, you can see that strut's buckled. 449 00:17:41,880 --> 00:17:44,500 In an elastomeric foam, you get bending and buckling 450 00:17:44,500 --> 00:17:48,250 just the same as we did in the honeycomb. 451 00:17:48,250 --> 00:17:50,640 Then here's a metal foam. 452 00:17:50,640 --> 00:17:52,690 You form plastic hinges in the metal foam. 453 00:17:52,690 --> 00:17:55,050 So here's a cell wall here. 454 00:17:55,050 --> 00:17:57,250 And it's a little bent to start with at zero load. 455 00:17:57,250 --> 00:18:00,620 But you can see it becomes more bent in this image over here. 456 00:18:00,620 --> 00:18:03,240 And here's a cell wall that's more or less vertical. 457 00:18:03,240 --> 00:18:05,520 And you can see that wall buckles. 458 00:18:05,520 --> 00:18:07,220 It's a plastic buckling in this case. 459 00:18:07,220 --> 00:18:10,280 There's a permanent deformation there. 460 00:18:10,280 --> 00:18:11,590 Here's a brittle foam. 461 00:18:11,590 --> 00:18:14,530 And you can see cell walls in this foam fracture. 462 00:18:14,530 --> 00:18:17,704 So this region here is equivalent to this region here. 463 00:18:17,704 --> 00:18:20,120 That little glitch there is the same as that little glitch 464 00:18:20,120 --> 00:18:20,752 there. 465 00:18:20,752 --> 00:18:22,710 You can see there's a couple of cell walls here 466 00:18:22,710 --> 00:18:23,520 that are fractured. 467 00:18:23,520 --> 00:18:24,890 So we get fracture. 468 00:18:24,890 --> 00:18:26,720 The idea is is that the mechanisms 469 00:18:26,720 --> 00:18:28,630 of deformation in the foams parallel 470 00:18:28,630 --> 00:18:31,050 those in the honeycombs. 471 00:18:31,050 --> 00:18:32,611 OK? 472 00:18:32,611 --> 00:18:33,110 All right. 473 00:18:33,110 --> 00:18:36,322 So let me write some of this stuff on the board. 474 00:18:36,322 --> 00:18:37,780 We're going to start off by talking 475 00:18:37,780 --> 00:18:39,990 about open-celled foams, so foams where there's 476 00:18:39,990 --> 00:18:42,630 just solid in the edges, but not in the faces 477 00:18:42,630 --> 00:18:43,910 of the polyhedral cells. 478 00:18:43,910 --> 00:18:45,960 Then we'll talk about close-cell foams 479 00:18:45,960 --> 00:18:47,810 where there's solid in the faces, as well. 480 00:18:47,810 --> 00:18:49,130 But the open-celled ones are easier. 481 00:18:49,130 --> 00:18:50,171 So we'll start with that. 482 00:19:08,540 --> 00:19:11,980 In compression, we see the same three regimes 483 00:19:11,980 --> 00:19:13,480 as we did before for the honeycombs. 484 00:19:22,030 --> 00:19:24,410 There's a linear elastic regime that corresponds 485 00:19:24,410 --> 00:19:26,110 to bending of the cell walls. 486 00:19:30,800 --> 00:19:31,937 There's a stress plateau. 487 00:19:34,860 --> 00:19:39,650 And for elastomeric foams, that corresponds to buckling. 488 00:19:45,960 --> 00:19:48,390 For metal foams, that corresponds to the formation 489 00:19:48,390 --> 00:19:49,520 of plastic hinges. 490 00:19:53,590 --> 00:20:01,180 And then, for ceramic or brittle foams, 491 00:20:01,180 --> 00:20:03,490 that corresponds to brittle crushing, so 492 00:20:03,490 --> 00:20:04,840 fracturing of the cell walls. 493 00:20:10,830 --> 00:20:14,337 Then, if you load the foam up to higher strains 494 00:20:14,337 --> 00:20:16,712 and higher stresses, eventually you get to densification. 495 00:20:24,590 --> 00:20:28,050 And in tension, just like the honeycombs, 496 00:20:28,050 --> 00:20:30,650 for the elastomeric materials there is no buckling. 497 00:20:36,420 --> 00:20:45,531 We can get a stress plateau from plastic hinges 498 00:20:45,531 --> 00:20:48,330 if there's, say, a metal foam. 499 00:20:48,330 --> 00:20:56,420 And for a brittle foam, we would get a fracture toughness 500 00:20:56,420 --> 00:20:59,630 and brittle fracture in tension. 501 00:21:10,270 --> 00:21:13,040 So the idea is the mechanisms of deformation and failure 502 00:21:13,040 --> 00:21:17,849 just parallel what we've seen in the honeycombs. 503 00:21:17,849 --> 00:21:20,015 So we'll start off with the linear elastic behavior. 504 00:21:27,260 --> 00:21:29,178 And we'll start with open-cell foams. 505 00:21:42,080 --> 00:21:44,350 The initial linear elasticity is due to bending 506 00:21:44,350 --> 00:21:45,140 of the cell walls. 507 00:21:56,110 --> 00:21:59,340 And if the thickness of the cell edges relative to the length 508 00:21:59,340 --> 00:22:03,450 is small, the bending dominates the deformation. 509 00:22:03,450 --> 00:22:06,260 But as the thickness to length ratio increases, 510 00:22:06,260 --> 00:22:09,020 then axial deformations can become important too. 511 00:22:30,609 --> 00:22:32,650 What we're going to do is we're going to consider 512 00:22:32,650 --> 00:22:33,750 dimensional arguments. 513 00:22:41,987 --> 00:22:43,820 We're going to set the dimensional arguments 514 00:22:43,820 --> 00:22:47,460 up so that we replicate the mechanisms of deformation 515 00:22:47,460 --> 00:22:47,977 and failure. 516 00:22:47,977 --> 00:22:50,143 But we don't worry too much about the cell geometry. 517 00:23:27,430 --> 00:23:29,990 What I'm going to start with is considering a cubic cell. 518 00:23:34,710 --> 00:23:38,700 And I've arranged the cubic cell so that the cell 519 00:23:38,700 --> 00:23:39,735 edges are staggered. 520 00:23:39,735 --> 00:23:41,235 That's going to give me the bending. 521 00:23:45,600 --> 00:23:51,130 The edge length is going to be L. 522 00:23:51,130 --> 00:23:55,000 I'm going to say we have a square cross-section, 523 00:23:55,000 --> 00:23:55,690 t squared. 524 00:24:06,880 --> 00:24:12,790 Here's our idealized model here with a cubic cell. 525 00:24:12,790 --> 00:24:14,350 All the members have a length, l. 526 00:24:14,350 --> 00:24:17,460 All of them have a square cross-section, t squared. 527 00:24:17,460 --> 00:24:18,770 That's an open-cell model. 528 00:24:18,770 --> 00:24:22,040 We've got just solid on the edges and nothing on the faces. 529 00:24:22,040 --> 00:24:24,150 The idea is that if I bend that, or if I 530 00:24:24,150 --> 00:24:28,100 load that in compression, so I apply, say, a stress out here 531 00:24:28,100 --> 00:24:29,820 that puts forces on those members 532 00:24:29,820 --> 00:24:32,820 there, because I've staggered these cell 533 00:24:32,820 --> 00:24:35,650 walls with these ones here, we're 534 00:24:35,650 --> 00:24:38,700 going to get bending in this cell edge here. 535 00:24:38,700 --> 00:24:41,680 That bending is going to be what we model. 536 00:25:15,960 --> 00:25:19,170 I'm going to set this up so that one thing is 537 00:25:19,170 --> 00:25:21,350 proportional to something else. 538 00:25:21,350 --> 00:25:24,130 These relationships are going to be true regardless of the cell 539 00:25:24,130 --> 00:25:24,720 geometry. 540 00:25:24,720 --> 00:25:26,997 So I could have picked a tetrakaidecahedra 541 00:25:26,997 --> 00:25:29,580 if I wanted to, and I would have had these same relationships. 542 00:25:29,580 --> 00:25:31,663 I'm just picking a cubic thing because it's easier 543 00:25:31,663 --> 00:25:33,720 to think about. 544 00:25:33,720 --> 00:25:37,540 So first of all, we look at the relative density. 545 00:25:37,540 --> 00:25:41,290 Remember, the relative density is the volume fraction 546 00:25:41,290 --> 00:25:41,840 of solid. 547 00:25:44,480 --> 00:25:48,080 So it's the volume of the solid over the total volume. 548 00:25:48,080 --> 00:25:55,410 And that's going to go as t squared l over l cubed, or just 549 00:25:55,410 --> 00:25:58,894 t over l all squared. 550 00:25:58,894 --> 00:26:01,060 You remember for the honeycomb, the relative density 551 00:26:01,060 --> 00:26:02,950 went linearly with t over l. 552 00:26:02,950 --> 00:26:06,069 For the open-celled foam, it goes as t over l squared. 553 00:26:10,380 --> 00:26:12,680 The moment of inertia in this case 554 00:26:12,680 --> 00:26:14,790 is going to go as t to the fourth. 555 00:26:14,790 --> 00:26:17,850 Remember, we have a square section, t squared. 556 00:26:17,850 --> 00:26:21,810 So if it's bh cubed over 12, b is t, h is t. 557 00:26:21,810 --> 00:26:24,290 It's going as t to the fourth. 558 00:26:24,290 --> 00:26:26,050 Then what I'm going to say is the stress 559 00:26:26,050 --> 00:26:29,780 is going to go as F over an area length squared. 560 00:26:29,780 --> 00:26:30,830 OK? 561 00:26:30,830 --> 00:26:33,820 So if I look at my little square thing here, 562 00:26:33,820 --> 00:26:35,480 I look at having my force here. 563 00:26:35,480 --> 00:26:36,920 Here we have a force f. 564 00:26:36,920 --> 00:26:38,420 And it's acting over an area that's 565 00:26:38,420 --> 00:26:40,310 somehow related to l squared. 566 00:26:40,310 --> 00:26:41,779 Right? 567 00:26:41,779 --> 00:26:43,570 I don't know exactly what that constant is, 568 00:26:43,570 --> 00:26:45,840 and I'm going to not try to calculate that. 569 00:26:45,840 --> 00:26:47,900 But it goes as F over l squared. 570 00:26:47,900 --> 00:26:50,240 Similarly, I can write that the strain 571 00:26:50,240 --> 00:26:52,680 is going to go as delta over l. 572 00:26:52,680 --> 00:26:56,180 So the strain is going to go as this bending deflection 573 00:26:56,180 --> 00:27:00,320 here, that delta divided by the height of the cell. 574 00:27:00,320 --> 00:27:03,200 And that's also l. 575 00:27:03,200 --> 00:27:05,570 Then I also know from structural mechanics 576 00:27:05,570 --> 00:27:12,340 that delta is going to go as Fl cubed over E of the solid 577 00:27:12,340 --> 00:27:18,054 and I. 578 00:27:18,054 --> 00:27:20,220 Then I'm just going to put all these things together 579 00:27:20,220 --> 00:27:21,940 to get the modulus. 580 00:27:21,940 --> 00:27:24,100 The modulus of the foam is going to go 581 00:27:24,100 --> 00:27:27,300 as the stress over the strain. 582 00:27:27,300 --> 00:27:31,450 If I plug in what I have for the stress, it's F over l squared. 583 00:27:31,450 --> 00:27:34,390 If I plug in what I have for the strain, it's delta over l. 584 00:27:37,800 --> 00:27:41,290 So this is F over l and delta. 585 00:27:41,290 --> 00:27:46,822 I'm going to replace delta by Fl cubed over Es. 586 00:27:46,822 --> 00:27:48,280 I'm going to use, instead of I, I'm 587 00:27:48,280 --> 00:27:51,160 going to use t to the fourth. 588 00:27:51,160 --> 00:27:52,744 Then the F's are going to cancel out. 589 00:27:57,100 --> 00:28:00,950 I've got that the modulus goes as the modulus 590 00:28:00,950 --> 00:28:05,120 of the solid times t over l to the fourth power. 591 00:28:05,120 --> 00:28:07,328 Then I can put that in terms of the relative density. 592 00:28:14,370 --> 00:28:18,960 It's going to go as the relative density squared. 593 00:28:18,960 --> 00:28:20,660 So I can summarize all of this by saying 594 00:28:20,660 --> 00:28:22,880 that the Young's modulus of the foam 595 00:28:22,880 --> 00:28:26,320 is some constant C1, I'm going to call it, 596 00:28:26,320 --> 00:28:29,420 times the modulus of the solid times 597 00:28:29,420 --> 00:28:30,790 the relative density squared. 598 00:28:36,890 --> 00:28:37,390 OK. 599 00:28:37,390 --> 00:28:39,240 So this has the same kind of form 600 00:28:39,240 --> 00:28:41,530 as those equations we had for the honeycombs. 601 00:28:41,530 --> 00:28:42,480 Right? 602 00:28:42,480 --> 00:28:44,310 There's a solid-cell wall property. 603 00:28:44,310 --> 00:28:46,270 The solid module's here. 604 00:28:46,270 --> 00:28:48,570 For the honeycombs, I put it in terms of t over l. 605 00:28:48,570 --> 00:28:51,670 But the t over l was related to the relative density. 606 00:28:51,670 --> 00:28:53,910 How much solid you've got is reflected 607 00:28:53,910 --> 00:28:56,830 in the relative density. 608 00:28:56,830 --> 00:28:59,710 And then, this constant C1 wraps up 609 00:28:59,710 --> 00:29:01,640 all the geometrical constants that I've said, 610 00:29:01,640 --> 00:29:03,234 one thing's proportional to another, 611 00:29:03,234 --> 00:29:05,150 and something else is proportional to another. 612 00:29:05,150 --> 00:29:07,090 C1 just wraps up all of those. 613 00:29:07,090 --> 00:29:08,520 OK? 614 00:29:08,520 --> 00:29:10,560 I'm just going to say here C1 includes 615 00:29:10,560 --> 00:29:11,930 all the geometrical constants. 616 00:29:21,310 --> 00:29:25,109 We have to get C1 by looking at data. 617 00:29:25,109 --> 00:29:26,900 If we look at data for the Young's modulus, 618 00:29:26,900 --> 00:29:30,129 we find that C1 is just about equal to one. 619 00:29:35,030 --> 00:29:38,020 People have also done more sophisticated analyses 620 00:29:38,020 --> 00:29:38,621 than this. 621 00:29:38,621 --> 00:29:40,120 There's a group of people who looked 622 00:29:40,120 --> 00:29:43,190 at doing a full-scale, structural analysis 623 00:29:43,190 --> 00:29:45,980 of an open-celled tetrakaidecahedral cell. 624 00:29:45,980 --> 00:29:47,690 Remember, I said they pack to fill space. 625 00:29:47,690 --> 00:29:49,640 So you can look at a unit cell. 626 00:29:49,640 --> 00:29:52,750 They also made their cells such that the thickness 627 00:29:52,750 --> 00:29:55,165 along the length of the cell was not constant. 628 00:29:55,165 --> 00:29:57,290 The thickness varied as something called a "plateau 629 00:29:57,290 --> 00:29:58,340 border." 630 00:29:58,340 --> 00:30:01,210 If you have a foam that's made by surface tension, 631 00:30:01,210 --> 00:30:03,610 the edges will tend to have these plateau borders. 632 00:30:03,610 --> 00:30:06,000 And the thickness will vary along the length of the edge. 633 00:30:06,000 --> 00:30:08,270 So when they did all this whole, complicated thing, 634 00:30:08,270 --> 00:30:10,580 they could calculate a value for C1. 635 00:30:10,580 --> 00:30:12,140 They calculated 0.98. 636 00:30:12,140 --> 00:30:15,746 So it's very close to 1. 637 00:30:15,746 --> 00:30:30,870 I'll say analysis of open-cell tetrakaidecahedron 638 00:30:30,870 --> 00:30:46,980 cells with these plateau borders give C1 equal to 0.98. 639 00:30:46,980 --> 00:30:47,480 OK. 640 00:30:47,480 --> 00:30:48,730 So that's the Young's modulus. 641 00:30:53,860 --> 00:30:56,021 We can also look at the shear modulus. 642 00:30:56,021 --> 00:30:58,020 The shear modulus is also going to be controlled 643 00:30:58,020 --> 00:30:59,950 by bending of the cell walls. 644 00:30:59,950 --> 00:31:02,210 And so the shear modulus is just going 645 00:31:02,210 --> 00:31:05,340 to be some other constant times Es 646 00:31:05,340 --> 00:31:08,380 times the relative density squared, so 647 00:31:08,380 --> 00:31:10,270 a similar kind of relationship. 648 00:31:10,270 --> 00:31:11,930 It's just a different constant. 649 00:31:11,930 --> 00:31:17,380 And if the foam's isotropic, and if the Poisson's ratio 650 00:31:17,380 --> 00:31:25,151 is a third, then C2 is equal to 3/8. 651 00:31:29,670 --> 00:31:35,290 Remember, if we have isotropy, then the shear modulus 652 00:31:35,290 --> 00:31:40,270 is equal to E over 2 1 plus nu. 653 00:31:40,270 --> 00:31:43,530 And so you can get the C2 from that if you say nu is a third. 654 00:31:52,950 --> 00:31:56,420 Then we can also get Poisson's ratio for the foam. 655 00:31:56,420 --> 00:32:01,950 If the foam is isotropic, so we'll say for an isotropic foam 656 00:32:01,950 --> 00:32:11,530 here, nu is equal to E over 2G minus 1. 657 00:32:11,530 --> 00:32:14,920 That's just rearranging this expression here. 658 00:32:14,920 --> 00:32:17,620 And because E and G both depend on the relative density 659 00:32:17,620 --> 00:32:19,600 squared, they both depend on Es squared, 660 00:32:19,600 --> 00:32:21,590 that's all going to cancel out. 661 00:32:21,590 --> 00:32:26,970 So this is going to be equal to C1 over 2 C2 minus 1. 662 00:32:26,970 --> 00:32:30,760 So that's going to equal to a constant. 663 00:32:30,760 --> 00:32:32,980 That constant's going to be independent of whatever 664 00:32:32,980 --> 00:32:36,385 material the foam is made from and the relative density. 665 00:32:53,430 --> 00:32:55,490 The constant just depends on the cell geometry. 666 00:33:07,500 --> 00:33:09,500 Remember, in honeycombs we found the same thing. 667 00:33:09,500 --> 00:33:11,900 The Poisson's ratio for the honeycombs 668 00:33:11,900 --> 00:33:13,472 only depended on the cell geometry. 669 00:33:13,472 --> 00:33:15,180 It didn't depend on the solid properties. 670 00:33:15,180 --> 00:33:17,180 It didn't depend on the relative density. 671 00:33:17,180 --> 00:33:20,357 So this is an exactly parallel thing here for the foams. 672 00:33:33,030 --> 00:33:33,662 Yeah? 673 00:33:33,662 --> 00:33:36,000 AUDIENCE: I have a silly kind of question. 674 00:33:36,000 --> 00:33:39,640 What is the difference between foam and the honeycombs? 675 00:33:39,640 --> 00:33:40,780 LORNA GIBSON: Oh. 676 00:33:40,780 --> 00:33:44,240 The honeycombs have cells in a plane, 677 00:33:44,240 --> 00:33:46,720 and they're prismatic in the third direction. 678 00:33:46,720 --> 00:33:50,510 And the foams have polyhedral cells. 679 00:33:50,510 --> 00:33:53,140 You know what a tetrakaidecahedron, a 3D, 680 00:33:53,140 --> 00:33:54,730 polyhedral cell. 681 00:33:54,730 --> 00:33:56,500 OK? 682 00:33:56,500 --> 00:33:58,320 The honeycombs are prismatic. 683 00:33:58,320 --> 00:34:01,150 And the foams have polyhedral cells. 684 00:34:01,150 --> 00:34:02,110 OK? 685 00:34:02,110 --> 00:34:02,740 Are we good? 686 00:34:30,831 --> 00:34:31,330 OK. 687 00:34:39,178 --> 00:34:40,969 So there's a couple more interesting things 688 00:34:40,969 --> 00:34:42,340 about Poisson's ratio. 689 00:34:42,340 --> 00:34:43,810 The same way we can make honeycombs 690 00:34:43,810 --> 00:34:45,429 with negative Poisson's ratios, we 691 00:34:45,429 --> 00:34:47,689 can also make foams with negative Poisson's ratios. 692 00:35:05,530 --> 00:35:09,780 They do it the same way as for the honeycombs, really. 693 00:35:09,780 --> 00:35:12,700 The honeycombs had negative Poisson's ratios 694 00:35:12,700 --> 00:35:15,730 if the cell walls looked like this, this sort of arrangement. 695 00:35:24,730 --> 00:35:27,520 So that sort of a thing. 696 00:35:27,520 --> 00:35:29,960 We said that theta was negative for the honeycombs. 697 00:35:29,960 --> 00:35:32,360 And if you invert the cell walls on a foam, 698 00:35:32,360 --> 00:35:34,820 you also get negative Poisson's ratios. 699 00:35:34,820 --> 00:35:38,316 And the way they do that is they take a thermoplastic foam, 700 00:35:38,316 --> 00:35:39,940 and then, they load it hydrostatically. 701 00:35:39,940 --> 00:35:42,010 So they compress it in all three directions. 702 00:35:42,010 --> 00:35:44,091 And they smush the cells in on each other. 703 00:35:44,091 --> 00:35:45,840 And then, they heat it up to a temperature 704 00:35:45,840 --> 00:35:47,580 above the glass transition temperature 705 00:35:47,580 --> 00:35:49,234 while it's still smushed. 706 00:35:49,234 --> 00:35:50,400 And then, they cool it down. 707 00:35:50,400 --> 00:35:54,088 So they end up with that structure frozen in. 708 00:35:54,088 --> 00:35:56,463 And if they do that, they get a negative Poisson's ratio. 709 00:36:01,610 --> 00:36:04,600 I'll just say they invert the cell angles 710 00:36:04,600 --> 00:36:05,845 analogous to the honeycomb. 711 00:36:33,890 --> 00:36:36,760 They load the foam hydrostatically 712 00:36:36,760 --> 00:36:42,370 and heat to a temperature above Tg. 713 00:36:42,370 --> 00:36:43,880 And then, they cool and release it. 714 00:37:10,480 --> 00:37:14,150 I have a photograph here of a foam with a negative Poisson's 715 00:37:14,150 --> 00:37:14,690 ratio. 716 00:37:14,690 --> 00:37:18,040 You can see how the cells have been smushed in. 717 00:37:18,040 --> 00:37:21,720 It's the equivalent of the way it's done for the honeycomb. 718 00:37:21,720 --> 00:37:22,350 OK? 719 00:37:22,350 --> 00:37:24,191 Are we good? 720 00:37:24,191 --> 00:37:24,690 OK. 721 00:37:38,390 --> 00:37:41,740 That's the linear elastic moduli for open-celled foams. 722 00:37:41,740 --> 00:37:44,830 The next thing I wanted to do was closed-cell foams. 723 00:37:44,830 --> 00:37:47,240 If we look at a closed-cell foam, 724 00:37:47,240 --> 00:37:51,820 we can idealize it in this kind of a way here. 725 00:37:51,820 --> 00:37:53,690 I've set it up so that the edge thickness is 726 00:37:53,690 --> 00:37:55,590 different from the face thickness. 727 00:37:55,590 --> 00:37:58,600 That's really representing the fact that in foams, many foams 728 00:37:58,600 --> 00:38:02,570 are made using a liquid, and the foaming 729 00:38:02,570 --> 00:38:04,680 is controlled by surface tension. 730 00:38:04,680 --> 00:38:07,880 Often, the surface tension draws material into the edges 731 00:38:07,880 --> 00:38:08,930 and away from the faces. 732 00:38:08,930 --> 00:38:12,600 So the faces tend to be thinner than the edges. 733 00:38:12,600 --> 00:38:16,720 When we have deformation of the closed-cell foam, 734 00:38:16,720 --> 00:38:18,650 we've got bending of the edges the same 735 00:38:18,650 --> 00:38:20,410 as we did for the open-celled foam. 736 00:38:20,410 --> 00:38:22,140 But the faces can stretch. 737 00:38:22,140 --> 00:38:25,120 So they can have an axial stretching. 738 00:38:25,120 --> 00:38:28,300 You can think of that as a cell membrane stretching here. 739 00:38:28,300 --> 00:38:31,960 So imagine if I either pull on the foam or I compress it, 740 00:38:31,960 --> 00:38:35,540 there's going to be some axial load in the faces. 741 00:38:35,540 --> 00:38:37,760 So when we analyze them, we have to account 742 00:38:37,760 --> 00:38:40,770 for both bending of the edges and axial deformation 743 00:38:40,770 --> 00:38:42,980 in the faces. 744 00:38:42,980 --> 00:38:53,618 I'll just say we have edge bending as in open-cell foams. 745 00:38:58,170 --> 00:38:59,757 And we also get a face stretching. 746 00:39:07,640 --> 00:39:10,080 Another thing that can happen in the closed-cell foams 747 00:39:10,080 --> 00:39:12,194 is we can get compression of the gas. 748 00:39:12,194 --> 00:39:14,360 In an open-cell foam, the gas can move from one cell 749 00:39:14,360 --> 00:39:14,990 to the next. 750 00:39:14,990 --> 00:39:17,460 But in a closed-cell foam, the gas is trapped. 751 00:39:17,460 --> 00:39:20,590 And as the volume of the cell changes, 752 00:39:20,590 --> 00:39:21,723 the gas gets compressed. 753 00:39:26,610 --> 00:39:28,030 So we have another effect here. 754 00:39:35,400 --> 00:39:40,530 So we'll say for polymer foams, surface tension 755 00:39:40,530 --> 00:39:43,113 tends to draw material to the edges during processing. 756 00:40:07,610 --> 00:40:11,160 We define two thicknesses, one for the edge 757 00:40:11,160 --> 00:40:13,960 and one for the face. 758 00:40:13,960 --> 00:40:17,275 And then we apply a force to this cubic structure. 759 00:40:24,360 --> 00:40:27,000 And we can do an analysis a little bit like what 760 00:40:27,000 --> 00:40:28,450 we did for the open-cell foam. 761 00:40:32,430 --> 00:40:33,697 Let me rub all this off. 762 00:41:08,750 --> 00:41:09,360 OK. 763 00:41:09,360 --> 00:41:10,735 I'm going to set this up a little 764 00:41:10,735 --> 00:41:13,490 bit differently than for the open-celled foam. 765 00:41:13,490 --> 00:41:15,200 We're going to do a work argument. 766 00:41:15,200 --> 00:41:17,460 We're going to look at the external work done 767 00:41:17,460 --> 00:41:20,370 by the force F going through a deformation, delta. 768 00:41:20,370 --> 00:41:22,750 And that's going to have to equal the internal work done 769 00:41:22,750 --> 00:41:25,530 by the edges bending and by the faces stretching. 770 00:41:25,530 --> 00:41:27,950 So let me set that up. 771 00:41:27,950 --> 00:41:35,410 We're going to say the external work done, 772 00:41:35,410 --> 00:41:38,355 that's going to be proportional to F times delta. 773 00:41:38,355 --> 00:41:43,810 So delta is how much the whole thing is going to deform. 774 00:41:43,810 --> 00:41:47,795 Then, I've got internal work from bending of the edges. 775 00:41:55,480 --> 00:42:01,440 That internal work is going to be proportional to F 776 00:42:01,440 --> 00:42:05,537 over delta times delta squared. 777 00:42:05,537 --> 00:42:07,120 I'm going to end up with an expression 778 00:42:07,120 --> 00:42:08,620 where everything's in delta squared. 779 00:42:08,620 --> 00:42:11,010 So I want to keep the delta squared there for now. 780 00:42:11,010 --> 00:42:13,520 And F over delta is the stiffness. 781 00:42:13,520 --> 00:42:18,700 That's going to go as E of the solid times I over l cubed. 782 00:42:22,160 --> 00:42:24,340 I is going to go is Te to the fourth 783 00:42:24,340 --> 00:42:28,330 here because it's I of the edges. 784 00:42:28,330 --> 00:42:31,110 I've also got internal work from stretching of the faces. 785 00:42:42,230 --> 00:42:44,336 That internal work is going to be-- I'm 786 00:42:44,336 --> 00:42:46,085 going to run into my other equations here. 787 00:42:46,085 --> 00:42:48,230 Let me put it down a little. 788 00:42:48,230 --> 00:42:52,060 That's going to go as the stress on the face 789 00:42:52,060 --> 00:42:56,940 times the strain in the face times the volume of the face. 790 00:42:56,940 --> 00:42:59,970 Or I could write that, instead of stress of the face, 791 00:42:59,970 --> 00:43:02,080 I can put it in terms of Hooke's law 792 00:43:02,080 --> 00:43:04,740 and say it's E of the solid times 793 00:43:04,740 --> 00:43:09,750 the strain in the face squared times the volume of the face. 794 00:43:09,750 --> 00:43:12,750 And I can replace the strain in the face by delta over l. 795 00:43:18,700 --> 00:43:20,120 So it's delta over l squared. 796 00:43:20,120 --> 00:43:22,120 And then, the volume of the face is 797 00:43:22,120 --> 00:43:25,800 going to be t of the face times l squared. 798 00:43:33,990 --> 00:43:35,910 Then I want to balance the internal work 799 00:43:35,910 --> 00:43:37,880 and the external work. 800 00:43:37,880 --> 00:43:41,540 I can say F times delta is going to equal 801 00:43:41,540 --> 00:43:44,980 some constant I'm going to call "alpha" times 802 00:43:44,980 --> 00:43:49,090 E of the solid times t of the edge to the fourth, 803 00:43:49,090 --> 00:43:54,780 that's this guy up here, over l cubed 804 00:43:54,780 --> 00:43:58,140 times delta squared plus some other constant I'm 805 00:43:58,140 --> 00:44:05,110 going to call "beta" times E of the solid times delta over l 806 00:44:05,110 --> 00:44:08,437 squared tf l squared. 807 00:44:13,200 --> 00:44:16,190 So far, I've got this in terms of the force I'm applying. 808 00:44:16,190 --> 00:44:19,250 But I want to get a modulus of the foam out of this. 809 00:44:19,250 --> 00:44:23,070 I want to relate this force here to the modulus of the foam. 810 00:44:23,070 --> 00:44:25,600 I can say the modulus of the foam 811 00:44:25,600 --> 00:44:30,440 is going to be related to F over l squared, that's 812 00:44:30,440 --> 00:44:35,240 the stress, divided by the strain, delta over l. 813 00:44:35,240 --> 00:44:41,590 I can write the force is proportional to the modulus 814 00:44:41,590 --> 00:44:47,060 times the deflection, delta, and times the member length, l. 815 00:44:47,060 --> 00:44:50,120 And then, I'm going to plug that guy into this expression 816 00:44:50,120 --> 00:44:51,140 up here. 817 00:44:51,140 --> 00:44:53,339 And I'm going to get a delta squared on the left. 818 00:44:53,339 --> 00:44:55,130 And I'm going to get delta squareds in each 819 00:44:55,130 --> 00:44:56,160 of the right-hand terms. 820 00:44:56,160 --> 00:44:57,909 And so I'm going to cancel out the deltas. 821 00:45:02,440 --> 00:45:04,980 If I put all that together, I have 822 00:45:04,980 --> 00:45:10,090 the modulus in the foam times delta squared times l. 823 00:45:10,090 --> 00:45:12,090 There's a delta here, and there's a delta there. 824 00:45:12,090 --> 00:45:13,256 That gives me delta squared. 825 00:45:16,020 --> 00:45:20,630 And that's going to equal alpha Es Te 826 00:45:20,630 --> 00:45:29,350 to the fourth over l cubed times delta squared plus beta 827 00:45:29,350 --> 00:45:36,480 times Es times delta squared tf, if I cancel out 828 00:45:36,480 --> 00:45:38,424 one of those l squareds. 829 00:45:38,424 --> 00:45:39,840 So here I can get rid of the delta 830 00:45:39,840 --> 00:45:41,090 squareds in all these terms. 831 00:45:43,860 --> 00:45:46,200 And if I just divide by l, I'm going 832 00:45:46,200 --> 00:45:48,150 to have the modulus of the foam. 833 00:45:55,050 --> 00:45:59,400 I get a term here that it goes as alpha E of the solid times 834 00:45:59,400 --> 00:46:05,030 t of the edge over l to the fourth power plus beta times 835 00:46:05,030 --> 00:46:10,330 E of the solid times tf over l. 836 00:46:14,300 --> 00:46:17,300 Are we good? 837 00:46:17,300 --> 00:46:18,451 OK. 838 00:46:18,451 --> 00:46:20,450 And what I'd like to do is instead of putting it 839 00:46:20,450 --> 00:46:22,690 in terms of te and tf, I'd like to put it in terms 840 00:46:22,690 --> 00:46:25,240 of the relative density. 841 00:46:25,240 --> 00:46:26,752 I'm going to look at two limits. 842 00:46:29,550 --> 00:46:32,010 I'm going to say if we just had open cells, if there were 843 00:46:32,010 --> 00:46:34,920 no faces on the membranes, if we just had open cells, 844 00:46:34,920 --> 00:46:41,690 and we had a uniform t, then the relative density 845 00:46:41,690 --> 00:46:43,580 would go as t over l squared. 846 00:46:46,740 --> 00:46:55,480 If I just had closed cells, and I had a uniform t, 847 00:46:55,480 --> 00:47:02,660 then the relative density just goes linearly with t over l. 848 00:47:02,660 --> 00:47:07,130 The relative density is the volume fraction of solids. 849 00:47:07,130 --> 00:47:10,840 It's the volume of the solid over the total volume. 850 00:47:10,840 --> 00:47:13,250 For a closed-cell foam with a uniform t, 851 00:47:13,250 --> 00:47:16,390 the volume of the solid is going to be t times l squared. 852 00:47:16,390 --> 00:47:18,400 And then, the volume total is l cubed. 853 00:47:18,400 --> 00:47:21,850 So it's t over l. 854 00:47:21,850 --> 00:47:23,940 Now, I'm going to define one more thing. 855 00:47:23,940 --> 00:47:26,350 If I say that phi is equal to the volume 856 00:47:26,350 --> 00:47:45,080 fraction of the solid in the edges, then I can say te over l 857 00:47:45,080 --> 00:47:48,950 is some constant times phi to the 1/2 times 858 00:47:48,950 --> 00:47:50,310 the relative density to the 1/2. 859 00:47:53,120 --> 00:47:59,410 And I can say tf over l is equal to some other constant, 860 00:47:59,410 --> 00:48:03,880 c prime, I'll call it, times 1 minus phi, that's how much 861 00:48:03,880 --> 00:48:06,490 is in the faces, times the relative density. 862 00:48:09,670 --> 00:48:13,050 Then I could combine all of this. 863 00:48:13,050 --> 00:48:14,880 Can I put it in here? 864 00:48:14,880 --> 00:48:16,420 Maybe I'll just stick it down here. 865 00:48:20,140 --> 00:48:20,640 Hang on. 866 00:48:20,640 --> 00:48:21,140 OK. 867 00:48:23,730 --> 00:48:24,440 Put it up here. 868 00:48:45,710 --> 00:48:48,850 This is my final expression here. 869 00:48:48,850 --> 00:48:52,130 And this term here arises from the edge bending. 870 00:48:55,880 --> 00:48:58,000 This term here arises from the face stretching. 871 00:49:07,486 --> 00:49:09,610 So I think I should wait a bit for you to catch up. 872 00:49:14,180 --> 00:49:15,630 The idea is the edges are bending. 873 00:49:15,630 --> 00:49:17,240 The faces are stretching. 874 00:49:17,240 --> 00:49:20,080 We're looking at the work done by deforming 875 00:49:20,080 --> 00:49:22,340 the edges and the faces and equating that 876 00:49:22,340 --> 00:49:25,264 to the work done by the externally applied load, f. 877 00:49:35,462 --> 00:49:35,962 OK? 878 00:49:44,780 --> 00:49:47,100 That gives us two terms, one that 879 00:49:47,100 --> 00:49:49,337 accounts for the edge bending, and one that accounts 880 00:49:49,337 --> 00:49:50,336 for the face stretching. 881 00:49:53,880 --> 00:49:56,300 To be comprehensive, we want to take into account 882 00:49:56,300 --> 00:49:58,600 the compression of the gas, as well. 883 00:49:58,600 --> 00:50:00,864 So there's one more term I'm going to add on to that. 884 00:50:57,890 --> 00:51:02,090 Typically the gas effect is only significant 885 00:51:02,090 --> 00:51:03,670 for elastomeric foams. 886 00:51:03,670 --> 00:51:06,004 If you had a metal foam or a ceramic foam that 887 00:51:06,004 --> 00:51:08,170 was closed-cell, it wouldn't really contribute much. 888 00:51:08,170 --> 00:51:10,512 But just to be complete, we'll go through this. 889 00:51:39,720 --> 00:51:41,230 The idea here is we say we've got 890 00:51:41,230 --> 00:51:43,460 a cubic element of the foam. 891 00:51:43,460 --> 00:51:46,020 Initially, it has a volume, v0. 892 00:51:46,020 --> 00:51:50,780 If we apply a stress, a uniaxial stress, 893 00:51:50,780 --> 00:51:53,110 there's some change in the volume of the foam. 894 00:51:53,110 --> 00:51:56,590 So there's some volume, v, after we compress it 895 00:51:56,590 --> 00:51:59,940 by some strain, epsilon. 896 00:51:59,940 --> 00:52:01,960 If we can figure out what the volume is 897 00:52:01,960 --> 00:52:03,720 relative to the initial volume, we 898 00:52:03,720 --> 00:52:07,230 can figure out how the change of the amount of gas 899 00:52:07,230 --> 00:52:10,320 goes from the initial state to the compressed state. 900 00:52:10,320 --> 00:52:11,850 Then we can use Boyle's law. 901 00:52:11,850 --> 00:52:15,330 The idea is P1V1 equals to P2V2. 902 00:52:15,330 --> 00:52:17,700 Then we can figure out, using Boyle's law, what 903 00:52:17,700 --> 00:52:18,630 the pressure must be. 904 00:52:18,630 --> 00:52:20,005 And then, that pressure is what's 905 00:52:20,005 --> 00:52:23,040 contributing to the modulus. 906 00:52:23,040 --> 00:52:25,750 When I do this, I'm going to write down some equations that 907 00:52:25,750 --> 00:52:27,509 just have the main results. 908 00:52:27,509 --> 00:52:29,050 There's a whole bunch of algebra just 909 00:52:29,050 --> 00:52:30,310 to get from one to the other. 910 00:52:30,310 --> 00:52:32,426 And I'm not going to write them all down. 911 00:52:32,426 --> 00:52:33,800 When I write the equations, don't 912 00:52:33,800 --> 00:52:36,320 panic if it's not obvious how you get from one to the other. 913 00:52:36,320 --> 00:52:38,150 In the notes that I'm going to put online, 914 00:52:38,150 --> 00:52:42,520 there's all the details of how you get from one to the other. 915 00:52:42,520 --> 00:52:49,818 The idea is that we start with a cubic element of foam. 916 00:52:56,380 --> 00:53:01,410 Initially, before it's loaded, it has a volume V0. 917 00:53:01,410 --> 00:53:03,600 Then we apply a uniaxial stress. 918 00:53:11,760 --> 00:53:14,847 And we say the axial strain in the direction of the stress, 919 00:53:14,847 --> 00:53:15,930 I'm going to call epsilon. 920 00:53:28,690 --> 00:53:31,935 Just from the geometry, you can calculate the deformed volume. 921 00:53:35,840 --> 00:53:39,400 So after you load it, the volume is V. 922 00:53:39,400 --> 00:53:43,340 And that volume on loading, V, divided by the initial volume, 923 00:53:43,340 --> 00:53:46,484 V0, you can show. 924 00:53:46,484 --> 00:53:47,650 It's fairly straightforward. 925 00:53:47,650 --> 00:53:51,610 It's 1 minus epsilon times 1 minus 2 times 926 00:53:51,610 --> 00:53:53,920 the Poisson's ratio of the foam. 927 00:53:53,920 --> 00:53:56,490 Here, I'm taking compressive strain as positive. 928 00:54:06,200 --> 00:54:09,490 And if you do this whole volume thing, 929 00:54:09,490 --> 00:54:13,030 you'll get terms in epsilon squared and epsilon cubed. 930 00:54:13,030 --> 00:54:14,720 But because if it's linear elastic, 931 00:54:14,720 --> 00:54:16,140 epsilon is going to be small. 932 00:54:16,140 --> 00:54:18,050 I'm going to ignore the epsilon squared 933 00:54:18,050 --> 00:54:19,663 and the epsilon cubed terms. 934 00:54:24,530 --> 00:54:27,180 That's the total volume of the foam 935 00:54:27,180 --> 00:54:30,700 after and before the compression. 936 00:54:30,700 --> 00:54:32,766 And then, what I want is the volume of the gas. 937 00:54:32,766 --> 00:54:34,140 And the volume of the gas is just 938 00:54:34,140 --> 00:54:37,580 going to be the volume minus the volume of the solid. 939 00:54:37,580 --> 00:54:39,420 And I can get that by just subtracting off 940 00:54:39,420 --> 00:54:41,240 the relative density. 941 00:54:41,240 --> 00:54:44,010 So Vg over Vg0. 942 00:54:44,010 --> 00:54:47,740 Again, Vg0 is the volume of the gas initially. 943 00:54:47,740 --> 00:54:51,360 Vg is the volume of the gas when I'm compressing it. 944 00:54:59,502 --> 00:55:01,710 Remember, the relative density is the volume fraction 945 00:55:01,710 --> 00:55:02,310 of solids. 946 00:55:02,310 --> 00:55:04,748 So I'm essentially subtracting out the amount of solid 947 00:55:04,748 --> 00:55:05,789 to get the amount of gas. 948 00:55:34,660 --> 00:55:36,190 Then we can use Boyle's law. 949 00:55:49,090 --> 00:55:50,610 Here, p is going to be the pressure 950 00:55:50,610 --> 00:55:53,540 after the strain and p0 is going to be the initial pressure. 951 00:56:11,720 --> 00:56:15,132 The building seems rather making unhappy noises for some reason. 952 00:56:15,132 --> 00:56:16,590 I'm not sure what that's all about. 953 00:56:22,260 --> 00:56:25,230 There'll be some initial pressure in the gas 954 00:56:25,230 --> 00:56:28,160 even before the strain, or before the stress is applied. 955 00:56:28,160 --> 00:56:29,270 That's p0. 956 00:56:29,270 --> 00:56:32,140 So the pressure we need to overcome is p minus p0. 957 00:56:49,450 --> 00:56:51,510 Again, I'm missing out a bunch of steps. 958 00:56:51,510 --> 00:56:55,890 But using that expression and this expression 959 00:56:55,890 --> 00:56:57,600 and this expression, you could find 960 00:56:57,600 --> 00:57:01,100 that p prime is equal to p0 times 961 00:57:01,100 --> 00:57:09,210 epsilon times 1 minus 2 nu divided by 1 minus epsilon 1 962 00:57:09,210 --> 00:57:19,204 minus 2 nu minus the relative density. 963 00:57:23,220 --> 00:57:25,774 Then the contribution of the gas to the modulus 964 00:57:25,774 --> 00:57:28,190 you can get by just taking the derivative of that pressure 965 00:57:28,190 --> 00:57:29,620 with respect to the strain. 966 00:57:29,620 --> 00:57:31,936 So remember, modulus is stress over strain. 967 00:57:31,936 --> 00:57:33,060 It's the same kind of idea. 968 00:57:50,690 --> 00:57:53,400 So I'm going to call it E star g for the contribution 969 00:57:53,400 --> 00:57:54,900 from the gas. 970 00:57:54,900 --> 00:57:59,930 So that's going to dp prime d epsilon. 971 00:57:59,930 --> 00:58:03,894 And that's going to equal p0 times 1 minus 2 nu 972 00:58:03,894 --> 00:58:05,310 over 1 minus the relative density. 973 00:58:09,990 --> 00:58:10,490 OK? 974 00:58:13,540 --> 00:58:17,510 As I said, I'll scan the pages that have the details 975 00:58:17,510 --> 00:58:18,480 and put it online. 976 00:58:57,170 --> 00:59:01,390 The final expression we get combines all of these things. 977 00:59:01,390 --> 00:59:05,170 The modulus of the foam relative to the solid 978 00:59:05,170 --> 00:59:14,400 is phi squared rho over rho s squared plus 1 minus phi, 979 00:59:14,400 --> 00:59:17,230 that's the amount of material in the faces, 980 00:59:17,230 --> 00:59:20,776 times the relative density and then plus this gas compression 981 00:59:20,776 --> 00:59:21,275 term. 982 01:00:07,840 --> 01:00:10,950 Like I said, for most foams, the gas compression is negligible. 983 01:00:10,950 --> 01:00:19,650 So if p0 is equal to the atmospheric pressure, 984 01:00:19,650 --> 01:00:22,630 so about 0.1 megapascal, then the gas term's 985 01:00:22,630 --> 01:00:24,664 negligible, except for elastomeric foams. 986 01:00:49,025 --> 01:00:50,530 So that's the Young's modulus. 987 01:00:58,520 --> 01:01:02,120 Then we can do a similar thing for the shear modulus. 988 01:01:02,120 --> 01:01:04,010 And the thing to notice in shear is 989 01:01:04,010 --> 01:01:06,370 that when you shear something, there's no volume change. 990 01:01:06,370 --> 01:01:08,370 So if you shear it and there's no volume change, 991 01:01:08,370 --> 01:01:09,680 there's no gas compression. 992 01:01:09,680 --> 01:01:11,513 There's no pressure built up because there's 993 01:01:11,513 --> 01:01:12,717 no change in the volume. 994 01:01:12,717 --> 01:01:15,216 So for the shear modulus, you just have the first two terms. 995 01:01:52,070 --> 01:01:54,540 And if the foam is isotropic and you use a third, 996 01:01:54,540 --> 01:01:56,380 then this constant here is going to be 3/8. 997 01:02:23,506 --> 01:02:24,880 And then, Poisson's ratio is just 998 01:02:24,880 --> 01:02:26,940 going to be a function of the cell geometry, again. 999 01:02:26,940 --> 01:02:29,356 And roughly, we could say it's going to be around a third. 1000 01:02:44,674 --> 01:02:45,174 OK. 1001 01:03:11,000 --> 01:03:12,475 Are we good? 1002 01:03:12,475 --> 01:03:13,547 Let me wait a little bit. 1003 01:03:26,230 --> 01:03:29,639 So the idea is we're looking at the deformation of the bending 1004 01:03:29,639 --> 01:03:31,680 in the cell edges, and the stretching in the cell 1005 01:03:31,680 --> 01:03:33,060 faces, and the gas compression. 1006 01:03:33,060 --> 01:03:35,860 And we're accounting for those different terms. 1007 01:03:35,860 --> 01:03:38,120 The next thing is to compare these model equations 1008 01:03:38,120 --> 01:03:39,910 with some data. 1009 01:03:39,910 --> 01:03:42,277 And here's data for Young's modulus. 1010 01:03:42,277 --> 01:03:44,360 Here, we're plotting the relative Young's modulus, 1011 01:03:44,360 --> 01:03:46,992 so the modulus of the foam divided by the solid 1012 01:03:46,992 --> 01:03:48,200 against the relative density. 1013 01:03:48,200 --> 01:03:51,450 Again, these are log-log scales. 1014 01:03:51,450 --> 01:03:54,330 On this plot, everything with open symbols 1015 01:03:54,330 --> 01:03:56,750 is an open-cell foam, and everything with filled symbols 1016 01:03:56,750 --> 01:03:58,570 is a closed-cell foam. 1017 01:03:58,570 --> 01:04:01,890 Here's our equation for the open-celled foams, the simplest 1018 01:04:01,890 --> 01:04:04,450 thing that goes with density squared. 1019 01:04:04,450 --> 01:04:06,400 And that's that thick line there. 1020 01:04:06,400 --> 01:04:08,380 And you can see these are all open-cell foams. 1021 01:04:08,380 --> 01:04:10,320 There's some more up here. 1022 01:04:10,320 --> 01:04:13,870 So that gives a reasonable description of that data. 1023 01:04:13,870 --> 01:04:16,370 And then, these are two lines for closed-cell foams 1024 01:04:16,370 --> 01:04:18,590 for different values of phi, so different values 1025 01:04:18,590 --> 01:04:20,600 of the amount of solid in the edges. 1026 01:04:20,600 --> 01:04:22,950 And you can see all the filled symbols 1027 01:04:22,950 --> 01:04:24,380 are the closed-cell foams. 1028 01:04:24,380 --> 01:04:27,160 And they're in between this line here and this line here, 1029 01:04:27,160 --> 01:04:28,260 basically. 1030 01:04:28,260 --> 01:04:30,350 So that gives a fairly good description 1031 01:04:30,350 --> 01:04:32,800 for the modulus of the foams considering 1032 01:04:32,800 --> 01:04:35,437 how crude this model is. 1033 01:04:35,437 --> 01:04:37,520 We're not trying to account for the cell geometry. 1034 01:04:37,520 --> 01:04:38,936 We're just modeling the mechanisms 1035 01:04:38,936 --> 01:04:40,850 of deformation and failure. 1036 01:04:40,850 --> 01:04:44,910 And then, here's a similar plot for the shear modulus. 1037 01:04:44,910 --> 01:04:50,150 Here's for open-cells, 3/8 times the relative density squared. 1038 01:04:50,150 --> 01:04:52,820 These are open-celled foams down here. 1039 01:04:52,820 --> 01:04:55,560 These are closed-cell foams up here. 1040 01:04:55,560 --> 01:05:00,220 If the amount of material in the faces is small, 1041 01:05:00,220 --> 01:05:03,390 then you would just get the shear 1042 01:05:03,390 --> 01:05:06,060 modulus varying with the relative density squared. 1043 01:05:06,060 --> 01:05:08,227 Then here's data for Poisson's ratio. 1044 01:05:08,227 --> 01:05:09,810 We don't really know what the constant 1045 01:05:09,810 --> 01:05:11,184 is because we don't know what all 1046 01:05:11,184 --> 01:05:12,930 these geometrical parameters are. 1047 01:05:12,930 --> 01:05:14,444 But here's a value of a third. 1048 01:05:14,444 --> 01:05:16,360 And you can see there's a lot of scatter here. 1049 01:05:16,360 --> 01:05:18,420 This is like less than 0.2. 1050 01:05:18,420 --> 01:05:20,900 This is more than 0.5. 1051 01:05:20,900 --> 01:05:23,550 And the scatter really represents differences 1052 01:05:23,550 --> 01:05:24,790 in the cell geometry. 1053 01:05:24,790 --> 01:05:26,840 If the foams were, say, anisotropic, 1054 01:05:26,840 --> 01:05:29,970 and the cells were stretched in one direction, 1055 01:05:29,970 --> 01:05:33,070 then you would get different values of the Poisson's ratios. 1056 01:05:33,070 --> 01:05:36,230 So that's the Poisson's ratios there. 1057 01:05:36,230 --> 01:05:37,900 I'm thinking I'm going to stop there 1058 01:05:37,900 --> 01:05:40,360 because that seems like enough equations for today. 1059 01:05:40,360 --> 01:05:44,270 And then, next time we'll start doing the stress plateau 1060 01:05:44,270 --> 01:05:46,709 and we'll figure out the elastic collapse stress 1061 01:05:46,709 --> 01:05:48,500 from buckling and a plastic collapse stress 1062 01:05:48,500 --> 01:05:50,600 from yielding, and so on, and so on. 1063 01:05:50,600 --> 01:05:54,320 We'll finish doing the modeling next time. 1064 01:05:54,320 --> 01:05:57,300 And we'll probably start doing a little bit of other stuff 1065 01:05:57,300 --> 01:05:59,600 on foams next time.