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 00:00:27,130 --> 00:00:28,130 LORNA GIBSON: All right. 9 00:00:28,130 --> 00:00:29,670 I should probably start. 10 00:00:29,670 --> 00:00:31,850 Last time, we were talking about the honeycombs 11 00:00:31,850 --> 00:00:34,700 and doing some modeling of the mechanical behavior 12 00:00:34,700 --> 00:00:37,440 and we started off talking about the in plane behavior. 13 00:00:37,440 --> 00:00:39,590 We're talking about loading it in this direction 14 00:00:39,590 --> 00:00:41,460 or that direction there. 15 00:00:41,460 --> 00:00:43,810 And we talked about the elastic modulus. 16 00:00:43,810 --> 00:00:45,350 I think I derived a Young's modulus 17 00:00:45,350 --> 00:00:48,630 for the one direction, a Poisson's ratio for loading 18 00:00:48,630 --> 00:00:49,870 in the one direction. 19 00:00:49,870 --> 00:00:52,740 And then we started talking about the stress plateau 20 00:00:52,740 --> 00:00:55,450 and we went over the elastic buckling stress, 21 00:00:55,450 --> 00:00:58,030 for one of these elastomeric honeycombs like this. 22 00:00:58,030 --> 00:01:01,330 And we went through the plastic collapse stress, for, say, 23 00:01:01,330 --> 00:01:03,582 a metal honeycomb that would yield. 24 00:01:03,582 --> 00:01:06,040 And I think I started talking about a brittle honeycomb and 25 00:01:06,040 --> 00:01:07,610 brittle crushing. 26 00:01:07,610 --> 00:01:09,410 The idea with a brittle honeycomb-- 27 00:01:09,410 --> 00:01:13,320 like a ceramic honeycomb-- is it could fail in a brittle manner. 28 00:01:13,320 --> 00:01:16,870 And the failure is going to be controlled by the cell wall 29 00:01:16,870 --> 00:01:17,650 in bending. 30 00:01:17,650 --> 00:01:20,980 And when that bending stress reaches the modulus of rupture, 31 00:01:20,980 --> 00:01:23,120 or the bending strength of the material, 32 00:01:23,120 --> 00:01:24,720 then you get wall fracture. 33 00:01:24,720 --> 00:01:26,860 I think that's where we left it last time, right? 34 00:01:26,860 --> 00:01:29,430 I had written down something about cell wall fracture. 35 00:01:29,430 --> 00:01:32,350 Now, I wanted to do the little derivation. 36 00:01:32,350 --> 00:01:35,120 Here's our little schematic up here. 37 00:01:35,120 --> 00:01:36,990 Here's the honeycomb. 38 00:01:36,990 --> 00:01:41,480 You've loaded it with sigma 1 here to such an extent 39 00:01:41,480 --> 00:01:44,230 that one of these cell walls has reached the modulus of rupture 40 00:01:44,230 --> 00:01:45,200 and has broken. 41 00:01:45,200 --> 00:01:48,460 And this is the little free body diagram that corresponds. 42 00:01:48,460 --> 00:01:50,330 I'm going to go through sigma 1 for loading 43 00:01:50,330 --> 00:01:51,220 in the one direction. 44 00:01:51,220 --> 00:01:54,760 This is the same thing for loading in the two direction. 45 00:01:54,760 --> 00:01:57,840 And the result for that's in the book. 46 00:01:57,840 --> 00:01:58,340 OK. 47 00:01:58,340 --> 00:02:00,410 If I have loading in the one direction, 48 00:02:00,410 --> 00:02:03,270 I can relate that horizontal force p 49 00:02:03,270 --> 00:02:05,098 to the stress in the one direction. 50 00:02:10,490 --> 00:02:14,370 The little p is equal to sigma 1 times h plus sin theta times b. 51 00:02:14,370 --> 00:02:17,310 And remember, b's the depth into the page. 52 00:02:17,310 --> 00:02:22,360 And I'm going to define sigma fs as the modulus of rupture 53 00:02:22,360 --> 00:02:24,430 of the cell wall material. 54 00:02:24,430 --> 00:02:26,920 It's the bending strength of the cell wall material. 55 00:02:35,820 --> 00:02:39,200 And we're going to say that we get fracture of that bent wall 56 00:02:39,200 --> 00:02:43,430 when the applied moment is equal to the fracture moment. 57 00:02:43,430 --> 00:02:46,910 From the plastic collapse stress from last time, 58 00:02:46,910 --> 00:02:51,210 we had the applied moment was equal to p times l 59 00:02:51,210 --> 00:02:54,200 sin theta over 2. 60 00:02:54,200 --> 00:02:56,780 That was just using static equilibrium, 61 00:02:56,780 --> 00:03:00,040 looking at that free body diagram of the beam. 62 00:03:00,040 --> 00:03:03,900 And if I write p, in terms of sigma 1 up here, 63 00:03:03,900 --> 00:03:08,740 I can just write that like this sigma 1 times h plus l 64 00:03:08,740 --> 00:03:11,260 sin theta times b. 65 00:03:11,260 --> 00:03:15,020 And then I've got this other term of l sin theta 66 00:03:15,020 --> 00:03:17,250 and we divide that whole thing by 2. 67 00:03:17,250 --> 00:03:19,270 That's the applied moment. 68 00:03:19,270 --> 00:03:21,270 And we're going to get fracture when 69 00:03:21,270 --> 00:03:22,480 we reach the fracture moment. 70 00:03:29,970 --> 00:03:35,780 I'm going to call that mf-- the moment at fracture. 71 00:03:35,780 --> 00:03:38,690 Last time, we figured out a plastic moment 72 00:03:38,690 --> 00:03:40,470 to form a plastic hinge. 73 00:03:40,470 --> 00:03:43,630 And this is an analogous thing. 74 00:03:43,630 --> 00:03:46,290 But in this case, remember, if we have a beam 75 00:03:46,290 --> 00:03:50,170 and we have the stress profile through the cross section 76 00:03:50,170 --> 00:03:52,860 of the beam, it's going to look something like that. 77 00:03:52,860 --> 00:03:56,590 So for our beam, that's going to be the thickness of the beam 78 00:03:56,590 --> 00:03:57,440 there. 79 00:03:57,440 --> 00:04:00,190 So if it's linear elastic, we get the maximum stress 80 00:04:00,190 --> 00:04:01,320 at the top and the bottom. 81 00:04:01,320 --> 00:04:03,780 And the neutral axis is here in the middle. 82 00:04:03,780 --> 00:04:05,110 There's no stress there. 83 00:04:05,110 --> 00:04:07,740 This is the normal stress distribution here. 84 00:04:07,740 --> 00:04:10,920 And as we increase the stress for a brittle material that's 85 00:04:10,920 --> 00:04:13,467 going to be linear elastic till fracture, 86 00:04:13,467 --> 00:04:15,050 this is going to stay linear like this 87 00:04:15,050 --> 00:04:19,969 until we reach this modulus of rupture stress here. 88 00:04:19,969 --> 00:04:21,510 When we reach that stress, then we're 89 00:04:21,510 --> 00:04:23,960 going to get fracture of the beam. 90 00:04:23,960 --> 00:04:29,890 And we can say that there's some moment associated with that. 91 00:04:29,890 --> 00:04:32,300 I could say that this stress block here 92 00:04:32,300 --> 00:04:34,800 is equivalent to some concentrated force 93 00:04:34,800 --> 00:04:37,900 and this stress block down here is also 94 00:04:37,900 --> 00:04:40,320 equivalent to the-- it's going to the same magnitude, 95 00:04:40,320 --> 00:04:42,260 but the opposite direction force. 96 00:04:42,260 --> 00:04:45,380 And I can get the fracture moment 97 00:04:45,380 --> 00:04:47,980 by figuring out how big those forces are 98 00:04:47,980 --> 00:04:51,400 and multiplying by this moment arm between the two forces. 99 00:04:51,400 --> 00:04:52,500 OK? 100 00:04:52,500 --> 00:04:54,750 The magnitude of those forces is just 101 00:04:54,750 --> 00:04:57,840 going to be the volume, essentially, of this stress 102 00:04:57,840 --> 00:04:59,920 block here. 103 00:04:59,920 --> 00:05:02,450 Imagine there's stresses there. 104 00:05:02,450 --> 00:05:05,330 It's a triangle, so the area of it's 105 00:05:05,330 --> 00:05:11,650 going to be a half times t over 2 times of sigma fs. 106 00:05:11,650 --> 00:05:15,090 And it's going to go b into the page. 107 00:05:15,090 --> 00:05:18,591 So if you think of the force-- if this was the stress-- 108 00:05:18,591 --> 00:05:20,340 if that stress was constant, it would just 109 00:05:20,340 --> 00:05:23,040 be sigma fs times b times t. 110 00:05:23,040 --> 00:05:24,120 But it's not constant. 111 00:05:24,120 --> 00:05:25,510 It's a linear relationship. 112 00:05:25,510 --> 00:05:28,400 So I'm taking the area of that triangle. 113 00:05:28,400 --> 00:05:30,380 That's the force. 114 00:05:30,380 --> 00:05:33,880 And then I want to multiply that times the moment arm. 115 00:05:33,880 --> 00:05:38,130 And the moment arm between those two forces-- each 116 00:05:38,130 --> 00:05:43,140 of these forces acts through the centroid of the area. 117 00:05:43,140 --> 00:05:47,760 The centroid of the area is not in the middle for a triangle, 118 00:05:47,760 --> 00:05:53,470 and that total distance is 2/3 of the thickness, t. 119 00:05:53,470 --> 00:05:54,204 OK? 120 00:05:54,204 --> 00:05:55,620 That's the moment arm that you get 121 00:05:55,620 --> 00:06:00,510 by figuring out where the centroid of these areas are. 122 00:06:00,510 --> 00:06:08,210 I multiply that times 2/3 t and one of the 2's is 123 00:06:08,210 --> 00:06:09,820 going to cancel. 124 00:06:09,820 --> 00:06:17,630 I can rewrite that and sigma fs times b times t squared over 6. 125 00:06:17,630 --> 00:06:18,885 OK? 126 00:06:18,885 --> 00:06:20,260 I think, last time when we talked 127 00:06:20,260 --> 00:06:23,090 about the plastic moment, we did a similar calculation 128 00:06:23,090 --> 00:06:26,580 and it worked out to sigma y bt squared over 4. 129 00:06:26,580 --> 00:06:29,730 So now this is sigma fs, the modulus of rupture, 130 00:06:29,730 --> 00:06:31,994 times bt squred over 6. 131 00:06:31,994 --> 00:06:33,910 The 6 is just slightly different because we've 132 00:06:33,910 --> 00:06:36,890 got a triangle here instead of a square shape like we 133 00:06:36,890 --> 00:06:38,620 had before. 134 00:06:38,620 --> 00:06:41,940 And now I can get the brittle crushing strength 135 00:06:41,940 --> 00:06:44,680 and compression by just equating that applied moment 136 00:06:44,680 --> 00:06:45,980 to this fracture moment. 137 00:06:54,750 --> 00:06:58,030 And if you do that, the result you get 138 00:06:58,030 --> 00:07:02,520 is this plateau stress for brittle crushing 139 00:07:02,520 --> 00:07:04,230 and compression. 140 00:07:04,230 --> 00:07:07,240 In the one direction, it's sigma fs-- 141 00:07:07,240 --> 00:07:11,040 the modulus of rupture of the cell wall-- times t 142 00:07:11,040 --> 00:07:12,680 over l squared. 143 00:07:12,680 --> 00:07:14,595 And then divided by a geometrical factor. 144 00:07:49,270 --> 00:07:51,640 And her regular hexagons, it works out 145 00:07:51,640 --> 00:07:55,303 to 4/9 of the modulus of rupture times t over l squared. 146 00:08:00,500 --> 00:08:01,210 OK? 147 00:08:01,210 --> 00:08:01,816 Are we good? 148 00:08:07,230 --> 00:08:09,990 We've got the in plane compressive properties now. 149 00:08:09,990 --> 00:08:12,310 We've got the elastic moduli and we've 150 00:08:12,310 --> 00:08:16,130 got the three plateau stresses that correspond to the three 151 00:08:16,130 --> 00:08:19,290 mechanisms-- to the elastic buckling failure mechanism, 152 00:08:19,290 --> 00:08:23,580 the plastic yielding mechanism, and then the fracture mechanism 153 00:08:23,580 --> 00:08:24,770 for brittle crushing. 154 00:08:24,770 --> 00:08:27,030 OK? 155 00:08:27,030 --> 00:08:29,130 If you think of the stress-strain curve 156 00:08:29,130 --> 00:08:32,620 of these materials in compression, 157 00:08:32,620 --> 00:08:37,380 the stress strain curves all look something like that. 158 00:08:37,380 --> 00:08:40,030 And now we've figured out equations 159 00:08:40,030 --> 00:08:44,931 that give us the modulus here and our collapse stress there. 160 00:08:44,931 --> 00:08:45,430 OK? 161 00:08:45,430 --> 00:08:49,212 So we can describe that stress-strain curve. 162 00:08:49,212 --> 00:08:50,180 All right. 163 00:08:50,180 --> 00:08:51,710 That's compression. 164 00:08:51,710 --> 00:08:53,987 And the next thing I wanted to talk about is tension. 165 00:08:57,570 --> 00:08:59,580 And if we think about the tensile behavior, 166 00:08:59,580 --> 00:09:02,190 the elastic moduli are going to be just the same. 167 00:09:02,190 --> 00:09:05,530 So the moduli are the same in tension and compression. 168 00:09:05,530 --> 00:09:08,520 And then, if we think about the stress plateau, 169 00:09:08,520 --> 00:09:10,720 we don't really have a stress plateau 170 00:09:10,720 --> 00:09:12,740 for an elastomeric material because there's 171 00:09:12,740 --> 00:09:13,750 no elastic buckling. 172 00:09:13,750 --> 00:09:15,130 If you pull it in tension, you're 173 00:09:15,130 --> 00:09:16,671 not going to get buckling in tension. 174 00:09:16,671 --> 00:09:18,790 You only get buckling if it's in compression. 175 00:09:18,790 --> 00:09:21,510 If you have a material that yields like a metal, 176 00:09:21,510 --> 00:09:25,400 you can get a plastic collapse stress and a plastic plateau. 177 00:09:25,400 --> 00:09:28,210 And that's very similar in tension and compression. 178 00:09:28,210 --> 00:09:30,230 There's a very small geometrical difference, 179 00:09:30,230 --> 00:09:32,272 but you can, essentially, ignore it. 180 00:09:32,272 --> 00:09:34,230 If you're loading the material in compression-- 181 00:09:34,230 --> 00:09:36,931 and imagine this was a metal-- if you load it in compression, 182 00:09:36,931 --> 00:09:39,180 the cell walls are getting a little further apart when 183 00:09:39,180 --> 00:09:39,856 I compress it. 184 00:09:39,856 --> 00:09:41,230 And if you're loading in tension, 185 00:09:41,230 --> 00:09:43,250 like this, the cell walls are getting a little closer 186 00:09:43,250 --> 00:09:43,800 together. 187 00:09:43,800 --> 00:09:45,900 So there's a small geometrical difference. 188 00:09:45,900 --> 00:09:47,460 But if we ignore that, we can say 189 00:09:47,460 --> 00:09:50,580 that the plateau stress for plastic behavior 190 00:09:50,580 --> 00:09:52,750 is about the same in tension and compression. 191 00:09:52,750 --> 00:09:55,040 And so really, the only property that's left 192 00:09:55,040 --> 00:09:57,290 is to look at a brittle honeycomb. 193 00:09:57,290 --> 00:10:00,240 And for a brittle honeycomb, you can have fast fracture 194 00:10:00,240 --> 00:10:03,720 and we can calculate a fracture toughness. 195 00:10:03,720 --> 00:10:07,920 So this next slide describes the fracture toughness calculation 196 00:10:07,920 --> 00:10:09,380 that we're going to do. 197 00:10:09,380 --> 00:10:10,840 Here's our honeycomb. 198 00:10:10,840 --> 00:10:13,010 I'm going to load it in the sigma 1 direction here. 199 00:10:13,010 --> 00:10:14,620 I've turned the honeycomb 90 degrees, 200 00:10:14,620 --> 00:10:16,140 so this is still sigma 1. 201 00:10:16,140 --> 00:10:18,620 And imagine now that we've got a crack here. 202 00:10:18,620 --> 00:10:20,350 And I'm going to consider a situation 203 00:10:20,350 --> 00:10:24,060 where the crack is very large, relative to the cell size. 204 00:10:24,060 --> 00:10:26,210 So it's not a crack in the cell walls. 205 00:10:26,210 --> 00:10:29,176 It's a crack that goes through multiple cells. 206 00:10:29,176 --> 00:10:30,800 I'm going to assume the crack is large, 207 00:10:30,800 --> 00:10:32,660 relative to the cell size. 208 00:10:32,660 --> 00:10:37,470 I'm going to assume that the bending is the main deformation 209 00:10:37,470 --> 00:10:38,170 mode. 210 00:10:38,170 --> 00:10:41,750 And what I'm going to do is look at-- if I have my crack tip 211 00:10:41,750 --> 00:10:44,310 here, I'm going to look at this cell wall 212 00:10:44,310 --> 00:10:46,550 a just ahead of the crack tip. 213 00:10:46,550 --> 00:10:48,489 And I'm going to say, that cell wall is bent. 214 00:10:48,489 --> 00:10:50,030 And I'm going to figure out something 215 00:10:50,030 --> 00:10:52,120 about the stress in that cell wall 216 00:10:52,120 --> 00:10:54,864 and look at when that fails. 217 00:10:54,864 --> 00:10:56,780 And I'm going to assume that the cell wall has 218 00:10:56,780 --> 00:10:58,430 a constant modulus of rupture. 219 00:10:58,430 --> 00:11:00,764 So the cell wall has a constant strength. 220 00:11:00,764 --> 00:11:02,430 You can imagine the cell wall could have 221 00:11:02,430 --> 00:11:04,290 little tiny cracks in it, too. 222 00:11:04,290 --> 00:11:06,430 And if a cell wall has a bigger crack, 223 00:11:06,430 --> 00:11:08,580 it's going to fail at a lower stress. 224 00:11:08,580 --> 00:11:11,550 But let's imagine that the cell walls are all the same strength 225 00:11:11,550 --> 00:11:14,640 and they all have a constant modulus of rupture. 226 00:11:14,640 --> 00:11:16,335 Let me write some of this down. 227 00:11:20,810 --> 00:11:24,386 In tension, the elastic moduli are going to the same 228 00:11:24,386 --> 00:11:25,135 as in compression. 229 00:11:38,010 --> 00:11:42,048 There's no elastic buckling in tension, 230 00:11:42,048 --> 00:11:43,297 so that's not going to happen. 231 00:11:49,030 --> 00:11:53,396 The plastic plateau stress in tension 232 00:11:53,396 --> 00:11:55,520 is going to be very similar to that in compression. 233 00:12:16,864 --> 00:12:19,155 As I mentioned, there's a small geometrical difference, 234 00:12:19,155 --> 00:12:20,590 but we're going to ignore that. 235 00:12:37,368 --> 00:12:39,300 And then, if we had a brittle honeycomb, 236 00:12:39,300 --> 00:12:42,560 like one of those ceramic honeycombs I showed you, 237 00:12:42,560 --> 00:12:43,955 then we can have fast fracture. 238 00:12:48,340 --> 00:12:50,600 What we want to calculate is the fracture toughness. 239 00:13:03,960 --> 00:13:07,690 And I'm going to make a few assumptions here. 240 00:13:07,690 --> 00:13:10,390 I'm going to assume that the crack length is large 241 00:13:10,390 --> 00:13:11,485 compared to the cell size. 242 00:13:24,720 --> 00:13:26,647 And if I do that, I can say that I'm 243 00:13:26,647 --> 00:13:28,230 going to use the continuum assumption. 244 00:13:33,220 --> 00:13:33,720 Hello. 245 00:13:37,900 --> 00:13:39,950 We'll come back to that. 246 00:13:39,950 --> 00:13:43,014 I'm going to say that axial forces can be neglected. 247 00:13:43,014 --> 00:13:44,805 We're just going to look at bending forces. 248 00:13:53,840 --> 00:13:58,360 And I'm also going to assume that the modulus of rupture 249 00:13:58,360 --> 00:14:00,100 is constant for the cell wall. 250 00:14:18,050 --> 00:14:20,760 First, let's just think again about the continuum. 251 00:14:20,760 --> 00:14:28,140 Imagine we just had a solid and we have a plate of the solid 252 00:14:28,140 --> 00:14:31,530 and it's loaded in tension with some remote stress-- 253 00:14:31,530 --> 00:14:34,840 some far away stress-- sigma 1. 254 00:14:34,840 --> 00:14:40,310 And the plate has a crack of length 2c perpendicular 255 00:14:40,310 --> 00:14:42,410 to that normal stress. 256 00:14:42,410 --> 00:14:47,360 And we're going to look at the stress-- local stress 257 00:14:47,360 --> 00:14:53,480 at the crack tip-- some distance r ahead of the crack tip there. 258 00:14:53,480 --> 00:14:56,170 In fracture mechanics, it's been worked out what 259 00:14:56,170 --> 00:14:58,210 that local stress field is. 260 00:14:58,210 --> 00:15:00,250 And it depends on the crack length, 261 00:15:00,250 --> 00:15:02,930 and then how far ahead of the crack tip you are. 262 00:15:02,930 --> 00:15:10,730 So you can say that if you've got a crack length of 2c 263 00:15:10,730 --> 00:15:23,990 in a linear elastic solid, and the crack is 264 00:15:23,990 --> 00:15:32,530 normal to a remote tensile stress-- which 265 00:15:32,530 --> 00:15:38,020 I'm going to call sigma 1-- then that crack 266 00:15:38,020 --> 00:15:43,590 is going to create a local stress field at the crack tip. 267 00:15:56,396 --> 00:15:58,770 And we're going to use this equation for the local stress 268 00:15:58,770 --> 00:15:59,590 field. 269 00:15:59,590 --> 00:16:03,370 The local stress field is equal to the far away field 270 00:16:03,370 --> 00:16:07,510 divided by-- or multiplied by the square root of pi c 271 00:16:07,510 --> 00:16:11,710 and divided by the square root of 2 pi r. 272 00:16:11,710 --> 00:16:14,350 So there's a stress singularity at the crack tip. 273 00:16:14,350 --> 00:16:18,280 And then the local stress decays as you move away 274 00:16:18,280 --> 00:16:19,376 from the crack tip. 275 00:16:27,982 --> 00:16:29,740 AUDIENCE: And what is r? 276 00:16:29,740 --> 00:16:32,020 LORNA GIBSON: r is the distance from the crack tip. 277 00:16:32,020 --> 00:16:34,205 So if that's the tip of my crack there, 278 00:16:34,205 --> 00:16:36,480 then r is my distance out. 279 00:17:00,410 --> 00:17:00,970 OK. 280 00:17:00,970 --> 00:17:03,740 In the honeycomb wall, if we look at the crack here, 281 00:17:03,740 --> 00:17:05,579 and then we look at that cell wall a 282 00:17:05,579 --> 00:17:09,410 that's just ahead of the crack tip, that cell wall is bent. 283 00:17:09,410 --> 00:17:12,880 So in the honeycomb, we're going to be looking at the bent cell 284 00:17:12,880 --> 00:17:13,380 wall. 285 00:17:28,240 --> 00:17:32,800 And that wall is going to fail when the applied moment equals 286 00:17:32,800 --> 00:17:34,086 the fracture moment. 287 00:17:59,120 --> 00:18:05,940 If we look at wall a, we could say 288 00:18:05,940 --> 00:18:11,490 that the applied moment is going to be proportional to p 289 00:18:11,490 --> 00:18:13,390 times l. 290 00:18:13,390 --> 00:18:16,520 Getting ahead of myself there. 291 00:18:16,520 --> 00:18:18,150 I'm going to do this-- because it's 292 00:18:18,150 --> 00:18:21,100 hard to say exactly where the crack tip is because there's 293 00:18:21,100 --> 00:18:21,862 a void there. 294 00:18:21,862 --> 00:18:23,570 I'm going to use that argument here where 295 00:18:23,570 --> 00:18:25,430 I make everything proportional. 296 00:18:25,430 --> 00:18:28,040 The moment's going to be proportional to p times l 297 00:18:28,040 --> 00:18:29,600 on wall a. 298 00:18:29,600 --> 00:18:34,880 And the fracture stress is going to be proportional to sigma fs 299 00:18:34,880 --> 00:18:37,840 times bt squared. 300 00:18:37,840 --> 00:18:40,520 Last time, we said it was sigma fs bt squared over 6. 301 00:18:40,520 --> 00:18:41,420 It's the same thing. 302 00:18:41,420 --> 00:18:43,800 I'm just dropping the 6 out. 303 00:18:43,800 --> 00:18:47,290 And then I can also say that this applied moment, 304 00:18:47,290 --> 00:18:50,600 if it goes as pl-- p is just going 305 00:18:50,600 --> 00:18:53,500 to be my local stress times lb. 306 00:18:53,500 --> 00:18:56,604 And then I multiply times l. 307 00:18:56,604 --> 00:18:58,270 So if you think of just thinking about-- 308 00:18:58,270 --> 00:19:00,610 if you got a load p on this member here, 309 00:19:00,610 --> 00:19:04,140 l, there's going to be some local stress there. 310 00:19:04,140 --> 00:19:06,660 And p is just going to be that local stress times the cell 311 00:19:06,660 --> 00:19:11,700 wall length times the width into the page. 312 00:19:11,700 --> 00:19:14,800 And then, that local stress, sigma l, 313 00:19:14,800 --> 00:19:18,350 I can replace with that equation over there. 314 00:19:18,350 --> 00:19:22,440 So that local stress is going to go as sigma 1 times the root 315 00:19:22,440 --> 00:19:25,750 of c over the root of r. 316 00:19:25,750 --> 00:19:28,620 And I'm going to say the distance ahead of the crack tip 317 00:19:28,620 --> 00:19:30,310 goes as l. 318 00:19:30,310 --> 00:19:33,274 Instead of having r, I'm going to say it's l. 319 00:19:33,274 --> 00:19:34,940 It's not necessarily exactly l, but it's 320 00:19:34,940 --> 00:19:40,310 going to be some fraction of l. 321 00:19:40,310 --> 00:19:42,000 That's my local stress there. 322 00:19:42,000 --> 00:19:47,710 And then I've got an l squared times b. 323 00:19:47,710 --> 00:19:49,970 And if I set that equal to the fracture moment, 324 00:19:49,970 --> 00:19:56,150 that's going to be proportional to sigma fs bt squared. 325 00:19:56,150 --> 00:19:56,935 Are we good here? 326 00:19:59,850 --> 00:20:02,150 You have to think of the crack tip. 327 00:20:02,150 --> 00:20:05,050 And there's some local stress field ahead of the crack tip. 328 00:20:05,050 --> 00:20:07,710 And we're saying that the load p is 329 00:20:07,710 --> 00:20:12,200 equal to that local stress times a cell length times the depth 330 00:20:12,200 --> 00:20:13,850 into the board. 331 00:20:13,850 --> 00:20:16,300 And then multiply it times l to get the moment. 332 00:20:16,300 --> 00:20:18,160 And then I replace that local stress 333 00:20:18,160 --> 00:20:22,010 with this standard equation for the remote stress and the crack 334 00:20:22,010 --> 00:20:24,890 length and the distance ahead of the crank tip. 335 00:20:24,890 --> 00:20:27,470 So here, the b's are going to cancel out. 336 00:20:27,470 --> 00:20:32,980 And now I can solve for a fracture stress in the one 337 00:20:32,980 --> 00:20:34,090 direction. 338 00:20:34,090 --> 00:20:35,700 And that's going to-- well, let me 339 00:20:35,700 --> 00:20:40,700 get proportional-- that's going to be proportional to sigma fs. 340 00:20:40,700 --> 00:20:43,820 Then there's going to be a t over l squared. 341 00:20:43,820 --> 00:20:46,496 And then, this is going to go as the square root of l 342 00:20:46,496 --> 00:20:49,420 over c like that. 343 00:20:49,420 --> 00:20:51,980 And now, if I want to get a fracture toughness, 344 00:20:51,980 --> 00:20:54,330 the fracture toughness is just the strength 345 00:20:54,330 --> 00:20:58,220 times the square root of pi times the half crack length c. 346 00:20:58,220 --> 00:21:05,690 So here, my fracture toughness is just that strength 347 00:21:05,690 --> 00:21:09,100 times the root of pi c. 348 00:21:09,100 --> 00:21:15,710 So I can say that's equal to a constant times sigma fs times t 349 00:21:15,710 --> 00:21:17,720 over l squared. 350 00:21:17,720 --> 00:21:20,165 And now, times the square root of l. 351 00:21:20,165 --> 00:21:22,080 These root of c's have canceled out. 352 00:21:29,240 --> 00:21:31,505 So that's my equation for the fracture toughness. 353 00:21:31,505 --> 00:21:32,880 And one of the interesting things 354 00:21:32,880 --> 00:21:34,338 here is that the fracture toughness 355 00:21:34,338 --> 00:21:35,930 depends on the cell size. 356 00:21:35,930 --> 00:21:38,280 This is the first property that we've 357 00:21:38,280 --> 00:21:40,920 derived an equation for where it depends on the cell size. 358 00:21:54,560 --> 00:21:55,317 OK. 359 00:21:55,317 --> 00:21:57,067 And here, c's just going to be a constant. 360 00:22:02,120 --> 00:22:03,770 All right. 361 00:22:03,770 --> 00:22:06,280 Now we've got a set of equations that describe 362 00:22:06,280 --> 00:22:07,560 the in plane properties. 363 00:22:07,560 --> 00:22:10,020 We've got equations that describe 364 00:22:10,020 --> 00:22:12,480 the linear elastic moduli in the plane. 365 00:22:12,480 --> 00:22:15,440 We've got three equations that describe the compressive stress 366 00:22:15,440 --> 00:22:18,870 for elastic buckling failure, for plastic yielding failure, 367 00:22:18,870 --> 00:22:20,780 and for a brittle crushing failure. 368 00:22:20,780 --> 00:22:23,040 And we've got an equation for the fracture toughness, 369 00:22:23,040 --> 00:22:23,650 as well. 370 00:22:23,650 --> 00:22:25,250 OK? 371 00:22:25,250 --> 00:22:28,210 We've got a description of the in plane behavior 372 00:22:28,210 --> 00:22:29,460 of these hexagonal honeycombs. 373 00:22:41,550 --> 00:22:44,420 The next thing I wanted to do was talk a little bit 374 00:22:44,420 --> 00:22:48,520 about in plane behavior, but for a different cell shape-- 375 00:22:48,520 --> 00:22:49,930 for triangular honeycombs. 376 00:22:49,930 --> 00:22:52,450 Because they deform by a different mechanism. 377 00:22:52,450 --> 00:22:55,650 And they can be used to represent the lattice materials 378 00:22:55,650 --> 00:22:57,980 that we looked at earlier, too. 379 00:22:57,980 --> 00:23:00,120 If we have a triangular honeycomb 380 00:23:00,120 --> 00:23:03,380 with triangular cells, triangulated cells 381 00:23:03,380 --> 00:23:05,370 behave like a truss. 382 00:23:05,370 --> 00:23:08,490 And you can analyze trusses by just saying that the joints are 383 00:23:08,490 --> 00:23:09,640 pin jointed. 384 00:23:09,640 --> 00:23:12,010 There's no moments at the end of the joints 385 00:23:12,010 --> 00:23:13,270 or end of the members. 386 00:23:13,270 --> 00:23:15,527 And the forces are all axial and so 387 00:23:15,527 --> 00:23:17,110 the behavior's a little bit different. 388 00:23:17,110 --> 00:23:21,180 I wanted to show you how these triangular honeycombs work, 389 00:23:21,180 --> 00:23:23,210 too. 390 00:23:23,210 --> 00:23:24,500 I can scoot this up. 391 00:23:49,771 --> 00:23:51,520 Imagine that you've got a honeycomb that's 392 00:23:51,520 --> 00:23:54,870 an array of triangular cells like this. 393 00:23:54,870 --> 00:24:01,560 And say we're applying some bulk stress sigma to it, like that. 394 00:24:01,560 --> 00:24:03,665 And say it's got a depth b into the page. 395 00:24:10,570 --> 00:24:12,660 When we have a triangulated structure like this, 396 00:24:12,660 --> 00:24:14,400 it behaves like a truss. 397 00:24:14,400 --> 00:24:17,080 And we can analyze it as being a pin-jointed structure. 398 00:24:17,080 --> 00:24:18,446 There's no moments at the joint. 399 00:24:47,660 --> 00:24:49,160 And if it's pin-jointed and there's 400 00:24:49,160 --> 00:24:50,750 no moments at the nodes, then we just 401 00:24:50,750 --> 00:24:56,794 get axial forces along the members. 402 00:24:59,640 --> 00:25:01,710 And even if the nodes were fixed-- 403 00:25:01,710 --> 00:25:04,370 as they are in these ceramic honeycombs-- 404 00:25:04,370 --> 00:25:06,310 you can show that if it's triangulated, 405 00:25:06,310 --> 00:25:08,060 even if you accounted for any bending, 406 00:25:08,060 --> 00:25:11,237 it really is a very tiny contribution to the deformation 407 00:25:11,237 --> 00:25:11,820 in the forces. 408 00:25:11,820 --> 00:25:15,130 It's less than a couple of percent. 409 00:25:15,130 --> 00:25:25,470 I'll say even if the ends are fixed-- 410 00:25:25,470 --> 00:25:37,255 I'll just say the bending contributes less than 2% 411 00:25:37,255 --> 00:25:38,630 to the forces in the deformation. 412 00:25:41,140 --> 00:25:45,100 If I have a triangular cell like that, 413 00:25:45,100 --> 00:25:47,630 and say I pick a unit cell like this, 414 00:25:47,630 --> 00:25:52,580 and I say that the bulk stress produces 415 00:25:52,580 --> 00:25:55,630 a load of p on the top and p over 2 416 00:25:55,630 --> 00:25:57,760 at each of the bottom nodes there, 417 00:25:57,760 --> 00:26:00,550 then the force in each member is going to be proportional to p. 418 00:26:15,380 --> 00:26:17,091 And for a given geometry of triangle, 419 00:26:17,091 --> 00:26:19,590 you can figure out exactly what the force distribution would 420 00:26:19,590 --> 00:26:20,677 be in each of the members. 421 00:26:20,677 --> 00:26:23,010 But I'm going to use one of these proportional arguments 422 00:26:23,010 --> 00:26:25,239 again, just to get a general result. 423 00:26:25,239 --> 00:26:27,530 Because I don't really care that much about the details 424 00:26:27,530 --> 00:26:28,196 of the geometry. 425 00:26:56,071 --> 00:26:56,570 OK. 426 00:26:56,570 --> 00:26:59,370 If I have a little set up like this, 427 00:26:59,370 --> 00:27:02,900 I can say that the overall stress 428 00:27:02,900 --> 00:27:08,632 is going to be proportional to p over lb. 429 00:27:08,632 --> 00:27:11,500 And the stress in each member is going 430 00:27:11,500 --> 00:27:14,580 to be proportional to p over l times the thickness-- 431 00:27:14,580 --> 00:27:15,950 or b times the thickness. 432 00:27:15,950 --> 00:27:17,846 This is the overall stress. 433 00:27:17,846 --> 00:27:19,220 The overall strain is going to be 434 00:27:19,220 --> 00:27:22,960 proportional to some deflection of the triangle divided 435 00:27:22,960 --> 00:27:23,920 by the length. 436 00:27:23,920 --> 00:27:31,930 So if I said, say, this length here was a length l. 437 00:27:31,930 --> 00:27:34,270 And then the deflection of each member 438 00:27:34,270 --> 00:27:37,750 is going to be proportional to p times l 439 00:27:37,750 --> 00:27:41,820 over es times the cross sectional area of the member, 440 00:27:41,820 --> 00:27:44,930 and that's just b times t. 441 00:27:44,930 --> 00:27:45,812 OK? 442 00:27:45,812 --> 00:27:47,520 So this is the stress on the whole thing, 443 00:27:47,520 --> 00:27:49,600 the strain on the whole thing, and relating 444 00:27:49,600 --> 00:27:51,390 the delta to the p. 445 00:27:51,390 --> 00:27:54,630 And then, the modulus of the whole honeycomb 446 00:27:54,630 --> 00:27:57,450 is going to go as the stress over the strain. 447 00:27:57,450 --> 00:28:02,816 So that's p over lb divided by delta over l. 448 00:28:06,030 --> 00:28:08,220 These l's here cancel. 449 00:28:08,220 --> 00:28:15,320 And delta here is pl over es bt, and so 450 00:28:15,320 --> 00:28:17,336 the b's cancel and the p's cancel. 451 00:28:21,150 --> 00:28:24,070 And the modulus I get for the honeycomb is just 452 00:28:24,070 --> 00:28:28,150 some constant related to the cell geometry times the modulus 453 00:28:28,150 --> 00:28:31,260 of the solid times t over l. 454 00:28:31,260 --> 00:28:33,330 And if you did an exact calculation 455 00:28:33,330 --> 00:28:50,910 for equilateral triangles, you'd find that that constant's 1.15. 456 00:28:50,910 --> 00:28:52,400 The interesting thing to note here 457 00:28:52,400 --> 00:28:55,480 is that the modulus for these triangular honeycombs 458 00:28:55,480 --> 00:28:58,460 goes as t over l, not as t over l cubed. 459 00:28:58,460 --> 00:29:02,410 For the hexagonal honeycomb, it went as t over l cubed. 460 00:29:02,410 --> 00:29:04,110 And here, because the deformations 461 00:29:04,110 --> 00:29:06,905 are axial-- not bending-- it's much stiffer. 462 00:29:06,905 --> 00:29:08,280 And it's much stiffer to have one 463 00:29:08,280 --> 00:29:11,460 of these triangulated structures. 464 00:29:11,460 --> 00:29:21,270 I'll just say, here, that the modulus goes 465 00:29:21,270 --> 00:29:24,170 as t over l cubed for the hexagonal 466 00:29:24,170 --> 00:29:33,230 honeycombs due to the bending. 467 00:29:41,567 --> 00:29:44,150 One of the reasons that people are interested in those lattice 468 00:29:44,150 --> 00:29:46,200 materials is that they, too, have 469 00:29:46,200 --> 00:29:48,000 moduli that go as t over l. 470 00:29:48,000 --> 00:29:51,860 That basically go with the relative density, 471 00:29:51,860 --> 00:29:53,930 rather than with the relative density cubed. 472 00:29:53,930 --> 00:29:59,000 So they're much stiffer than, say, a hexagonal honeycomb. 473 00:29:59,000 --> 00:30:00,440 OK? 474 00:30:00,440 --> 00:30:04,440 Are we good with the triangulated honeycombs? 475 00:30:04,440 --> 00:30:05,292 Yes? 476 00:30:05,292 --> 00:30:06,472 AUDIENCE: What is c? 477 00:30:06,472 --> 00:30:08,680 LORNA GIBSON: c's just a constant related to the cell 478 00:30:08,680 --> 00:30:09,179 geometry. 479 00:30:24,230 --> 00:30:26,850 For equilateral triangles, it's 1.15. 480 00:30:26,850 --> 00:30:28,500 You could work it out, but it just 481 00:30:28,500 --> 00:30:31,930 makes the whole thing a little more complicated to do that. 482 00:30:31,930 --> 00:30:32,430 OK. 483 00:30:32,430 --> 00:30:35,140 That's the in-plane behavior. 484 00:30:35,140 --> 00:30:38,140 And next, I wanted to talk about the out-of-plane behavior. 485 00:30:38,140 --> 00:30:42,180 Remember, we said the hexagonal honeycombs are orthotropic 486 00:30:42,180 --> 00:30:45,430 and the orthotropic materials have nine elastic constants. 487 00:30:45,430 --> 00:30:47,390 And we've figured out four so far. 488 00:30:47,390 --> 00:30:50,620 We've figured out the four in-plane elastic constants. 489 00:30:50,620 --> 00:30:52,740 There's five out-of-plane elastic constants 490 00:30:52,740 --> 00:30:56,690 to describe the elastic behavior completely. 491 00:30:56,690 --> 00:30:59,890 And so we want to talk about these other elastic constants. 492 00:30:59,890 --> 00:31:02,630 The honeycombs are also-- I should just 493 00:31:02,630 --> 00:31:03,760 back up a little bit. 494 00:31:03,760 --> 00:31:05,991 The honeycombs are used in sandwich panels. 495 00:31:05,991 --> 00:31:07,740 And when they're used in sandwich panels-- 496 00:31:07,740 --> 00:31:10,260 I brought a little panel in with carbon fiber faces 497 00:31:10,260 --> 00:31:12,040 and a nomex core. 498 00:31:12,040 --> 00:31:14,730 If you bend that panel like that, 499 00:31:14,730 --> 00:31:17,470 you're going to get shear stresses in the core. 500 00:31:17,470 --> 00:31:19,220 And the shear stresses are going to be 501 00:31:19,220 --> 00:31:22,600 going this way and this way on, and that way, that way on. 502 00:31:22,600 --> 00:31:26,160 And so those shear stresses are out-of-plane. 503 00:31:26,160 --> 00:31:30,580 They're in the x1, x3, or x2, x3 planes. 504 00:31:30,580 --> 00:31:32,810 And so you need the out-of-plane properties 505 00:31:32,810 --> 00:31:36,800 for the shear properties in the sandwich panels. 506 00:31:36,800 --> 00:31:38,380 Honeycombs are also sometimes used 507 00:31:38,380 --> 00:31:41,070 as energy absorption devices. 508 00:31:41,070 --> 00:31:43,734 Not these rubber ones, but imagine there was a metal one. 509 00:31:43,734 --> 00:31:45,900 And when they're used for energy absorption devices, 510 00:31:45,900 --> 00:31:47,720 they're typically loaded this way on. 511 00:31:47,720 --> 00:31:49,804 Again, that's the out-of-plane direction 512 00:31:49,804 --> 00:31:51,512 and you need the out-of-plane properties. 513 00:31:54,110 --> 00:31:55,980 And for the out-of-plane properties, 514 00:31:55,980 --> 00:31:57,600 the cell walls don't tend to bend. 515 00:31:57,600 --> 00:32:00,450 Instead, they just extend or contract. 516 00:32:00,450 --> 00:32:04,362 And you get stiffer and stronger properties. 517 00:32:04,362 --> 00:32:06,320 Let me just write something down and then we'll 518 00:32:06,320 --> 00:32:08,028 start to derive some of those properties. 519 00:33:23,440 --> 00:33:30,950 The cell walls contract or expand instead of bending, 520 00:33:30,950 --> 00:33:33,290 and that gives stiffer and stronger properties. 521 00:33:55,084 --> 00:33:55,583 OK. 522 00:34:43,311 --> 00:34:43,810 OK. 523 00:34:43,810 --> 00:34:48,740 There's five elastic constants in the out-of-plane directions. 524 00:35:00,419 --> 00:35:01,960 We'll start with the Young's modulus. 525 00:35:11,370 --> 00:35:13,680 And if I take my honeycomb and I load it 526 00:35:13,680 --> 00:35:17,270 in the x3 direction-- just taking this thing here and just 527 00:35:17,270 --> 00:35:21,290 loading it like that-- the cell walls just axially contract 528 00:35:21,290 --> 00:35:24,360 and the stiffness just depends on how much cell wall I've got. 529 00:35:24,360 --> 00:35:26,510 So the modulus in the three direction 530 00:35:26,510 --> 00:35:28,500 is just equal to the area fraction 531 00:35:28,500 --> 00:35:30,177 times the modulus of the solid. 532 00:35:30,177 --> 00:35:32,010 That's just the same as the volume fraction, 533 00:35:32,010 --> 00:35:33,050 or the relative density. 534 00:35:33,050 --> 00:35:37,340 So it's quite straightforward. 535 00:35:37,340 --> 00:35:40,728 The cell walls contract or extend axially. 536 00:36:17,450 --> 00:36:21,650 e3 is just es times the relative density. 537 00:36:21,650 --> 00:36:25,920 And that's just es times t over l. 538 00:36:25,920 --> 00:36:29,710 And then there's a geometrical factor here. 539 00:36:29,710 --> 00:36:33,410 h over l plus 2 over 2. 540 00:36:33,410 --> 00:36:38,950 h over la plus sin theta times cos theta. 541 00:36:48,700 --> 00:36:50,920 Again, a little bit like those triangular honeycombs. 542 00:36:50,920 --> 00:36:53,450 The thing to notice here is that in the three direction, 543 00:36:53,450 --> 00:36:56,070 the modulus goes linearly with t over l, 544 00:36:56,070 --> 00:36:57,954 whereas in the in-plane directions, 545 00:36:57,954 --> 00:36:59,120 it goes with t over l cubed. 546 00:36:59,120 --> 00:37:02,130 So there's a huge anisotropy in the honeycombs 547 00:37:02,130 --> 00:37:03,540 because of this difference. 548 00:37:03,540 --> 00:37:06,790 Imagine a honeycomb might be 10% dense. 549 00:37:06,790 --> 00:37:10,360 t over l might be something like a tenth-- 0.1. 550 00:37:10,360 --> 00:37:13,515 So e star 3 is going to 0.1 of es, 551 00:37:13,515 --> 00:37:16,080 roughly, and in the other direction, 552 00:37:16,080 --> 00:37:17,410 it's going to be 1/1000th. 553 00:37:17,410 --> 00:37:19,614 So there's a huge anisotropy because of this. 554 00:37:45,740 --> 00:37:48,413 Let me just-- square honeycombs. 555 00:37:52,150 --> 00:37:55,180 This just shows looking at the out-of-plane directions 556 00:37:55,180 --> 00:37:57,385 and the different stresses and properties 557 00:37:57,385 --> 00:37:59,799 that we're going to look at here. 558 00:37:59,799 --> 00:38:02,090 The next one we're going to look at is Poisson's ratio. 559 00:38:10,710 --> 00:38:14,010 And first, we're going to look at loading in the x3 direction. 560 00:38:14,010 --> 00:38:15,770 And if we load it in the x3 direction, 561 00:38:15,770 --> 00:38:19,760 the cell wall's just strain by whatever the Poisson's ratio 562 00:38:19,760 --> 00:38:22,490 is for the solid times the strain in the three direction 563 00:38:22,490 --> 00:38:24,480 in the other two directions. 564 00:38:24,480 --> 00:38:39,050 We'll say for loading in the x3 direction, the cell wall's 565 00:38:39,050 --> 00:38:43,710 strain by nu of the solid times whatever 566 00:38:43,710 --> 00:38:46,777 the strain is in the three direction in the other two 567 00:38:46,777 --> 00:38:47,276 directions. 568 00:38:54,359 --> 00:38:56,400 If we load it in the x3 directions and everything 569 00:38:56,400 --> 00:39:00,080 contracts by that much in the other two directions, that just 570 00:39:00,080 --> 00:39:06,160 means that the Poisson's ratios-- nu 3 1 and nu 3 2 571 00:39:06,160 --> 00:39:07,555 are going to be the same. 572 00:39:07,555 --> 00:39:09,680 And they're just going to be equal to the Poisson's 573 00:39:09,680 --> 00:39:11,420 ratio of the solid. 574 00:39:11,420 --> 00:39:14,515 So if each wall is going to contract by that amount, 575 00:39:14,515 --> 00:39:16,640 the whole thing's going to contract by that amount. 576 00:39:16,640 --> 00:39:21,850 And that's going to give you that Poisson's ratio. 577 00:39:21,850 --> 00:39:25,900 Let me just say, here, also-- and remember 578 00:39:25,900 --> 00:39:32,680 that I'm defining nu ij as minus epsilon j over epsilon i. 579 00:39:32,680 --> 00:39:36,450 We're loading in the three direction here. 580 00:39:36,450 --> 00:39:39,890 And then you can get the other two Poisson's ratios 581 00:39:39,890 --> 00:39:42,530 using those reciprocal relationships. 582 00:39:42,530 --> 00:39:49,240 So nu 1 3 and nu 2 3 can be found 583 00:39:49,240 --> 00:39:50,600 from the reciprocal relations. 584 00:40:03,300 --> 00:40:05,130 And remember, those relationships 585 00:40:05,130 --> 00:40:07,750 come from saying that the compliance 586 00:40:07,750 --> 00:40:10,091 tensor, or the stiffness tensor, is symmetric. 587 00:40:13,510 --> 00:40:18,370 We can write, for instance, that nu 1 3 over e1 588 00:40:18,370 --> 00:40:22,536 is equal to nu 3 1 over e3. 589 00:40:25,950 --> 00:40:28,230 So I can write that like that. 590 00:40:28,230 --> 00:40:33,250 And then I can say nu 1 3-- that is going to equal 591 00:40:33,250 --> 00:40:42,050 to nu 3 1 times e1 over e3. 592 00:40:42,050 --> 00:40:45,501 And we just saw that nu 3 1 was equal to nu s. 593 00:40:45,501 --> 00:40:50,140 And we see, from before, the e1 is equal to some constant. 594 00:40:50,140 --> 00:40:56,270 Let me just call it c1 times es times t over l cubed. 595 00:40:56,270 --> 00:41:01,040 And e3 is going to be some other constant times es 596 00:41:01,040 --> 00:41:03,060 time t over l. 597 00:41:03,060 --> 00:41:05,310 The es's are going to go. 598 00:41:05,310 --> 00:41:07,970 And if t over l is small-- even if it's say, 599 00:41:07,970 --> 00:41:10,740 a 10th-- and this is going as t over l cubed 600 00:41:10,740 --> 00:41:13,000 and that's going as t over l, then 601 00:41:13,000 --> 00:41:15,640 I can say this thing is about equal to 0. 602 00:41:15,640 --> 00:41:16,599 It's going to be small. 603 00:41:16,599 --> 00:41:18,140 It's not to be exactly [INAUDIBLE] 0, 604 00:41:18,140 --> 00:41:20,560 but it's going to be small so we're going to say it's 0. 605 00:41:23,512 --> 00:41:29,810 So I'll just say for small t over l. 606 00:41:29,810 --> 00:41:33,870 And then, similarly, nu 2 3 is going to be close to 0, 607 00:41:33,870 --> 00:41:34,370 as well. 608 00:41:44,680 --> 00:41:46,176 So there's the Poisson's ratios. 609 00:42:17,910 --> 00:42:22,182 We've got the Young's modulus, the Poisson's ratios, 610 00:42:22,182 --> 00:42:23,890 and next we want to get the Shear moduli. 611 00:42:27,820 --> 00:42:30,790 And the shear moduli is little more complicated. 612 00:42:30,790 --> 00:42:32,780 The cell walls are loaded in shear 613 00:42:32,780 --> 00:42:35,470 but the neighboring cell walls constrain them 614 00:42:35,470 --> 00:42:37,281 and they produce some non-uniform strain. 615 00:42:37,281 --> 00:42:39,030 I'm talking about shearing it this way on. 616 00:42:39,030 --> 00:42:41,430 You can see on this figure here, we're 617 00:42:41,430 --> 00:42:48,330 talking about shearing it, like tau 2 3 or tau 1 3, this way. 618 00:42:48,330 --> 00:42:50,210 And so each wall is going to shear, 619 00:42:50,210 --> 00:42:51,980 but the walls are attached to each other 620 00:42:51,980 --> 00:42:54,210 so they can't just do it independently. 621 00:42:54,210 --> 00:42:58,110 They have to be constrained by each other. 622 00:42:58,110 --> 00:43:01,177 And the exact solution is a little bit complicated. 623 00:43:01,177 --> 00:43:03,260 And I'm just going to give you an estimate of what 624 00:43:03,260 --> 00:43:04,890 that modulus is. 625 00:43:04,890 --> 00:43:08,330 And we're going to see that it depends linearly on t over l, 626 00:43:08,330 --> 00:43:09,834 as well. 627 00:43:09,834 --> 00:43:11,875 I'll just say the cell walls are loaded in shear. 628 00:44:04,500 --> 00:44:12,020 An estimate is g star 1 3 is equal to g of the solid times 629 00:44:12,020 --> 00:44:15,130 t over l times a geometric function. 630 00:44:15,130 --> 00:44:20,520 It's cos theta over h over l plus sin theta. 631 00:44:20,520 --> 00:44:22,810 And for regular hexagonal honeycombs, 632 00:44:22,810 --> 00:44:27,932 it's 1 over root 3 times gs times t over l. 633 00:44:43,820 --> 00:44:45,870 Again, just note the linear dependence 634 00:44:45,870 --> 00:44:47,890 of the modulus on t over l. 635 00:44:57,710 --> 00:45:00,010 And in the book, there's a method 636 00:45:00,010 --> 00:45:01,800 using upper and lower bounds that 637 00:45:01,800 --> 00:45:04,954 gives an estimate for g 2 3. 638 00:45:04,954 --> 00:45:06,120 I'm not going to go into it. 639 00:45:06,120 --> 00:45:07,880 I just want you to notice that the shear 640 00:45:07,880 --> 00:45:11,097 moduli go as t over l, just like the Young's modulus does. 641 00:45:11,097 --> 00:45:12,930 And Sardar, who's sitting in the back there, 642 00:45:12,930 --> 00:45:17,480 has done even more involved calculations and analysis 643 00:45:17,480 --> 00:45:20,420 of the shear moduli of the honeycombs in this direction. 644 00:45:20,420 --> 00:45:24,081 So I'm not going to go into all the gory details on that. 645 00:45:24,081 --> 00:45:24,580 OK. 646 00:45:24,580 --> 00:45:26,030 That gives us the moduli now. 647 00:45:26,030 --> 00:45:28,680 So now we've got all nine elastic moduli. 648 00:45:28,680 --> 00:45:30,928 OK? 649 00:45:30,928 --> 00:45:35,310 And the next thing to do is, then, 650 00:45:35,310 --> 00:45:36,970 to figure out the compressive strength. 651 00:45:36,970 --> 00:45:38,400 So we're going to look at compression again, 652 00:45:38,400 --> 00:45:40,010 and then we'll look at tension. 653 00:45:40,010 --> 00:45:41,720 If we look at compressive strengths, 654 00:45:41,720 --> 00:45:44,910 again, we've got different modes of failure. 655 00:45:44,910 --> 00:45:47,720 And if I have an elastomeric honeycomb like this one 656 00:45:47,720 --> 00:45:50,480 here-- if these cell walls were a little longer, 657 00:45:50,480 --> 00:45:52,040 I might be able to actually do it. 658 00:45:52,040 --> 00:45:54,250 If you compress this enough, you produce 659 00:45:54,250 --> 00:45:56,130 buckling in the cell walls. 660 00:45:56,130 --> 00:46:00,580 And this is a schematic of this buckling pattern here. 661 00:46:00,580 --> 00:46:02,580 And you can see there's a diamond pattern where 662 00:46:02,580 --> 00:46:07,300 it alternates up and down in the different cell walls. 663 00:46:07,300 --> 00:46:09,780 We're going to do some approximate calculations, 664 00:46:09,780 --> 00:46:14,590 but you can see the idea of how the material behaves 665 00:46:14,590 --> 00:46:17,500 in this direction, just from these approximate calculations. 666 00:46:41,480 --> 00:46:41,980 OK. 667 00:46:41,980 --> 00:46:47,660 Say we have our honeycomb like this, 668 00:46:47,660 --> 00:46:51,050 and here's the prism axis this way. 669 00:46:51,050 --> 00:46:52,610 And now, we're going to load it up 670 00:46:52,610 --> 00:46:55,880 with some stress in the three direction. 671 00:46:55,880 --> 00:47:01,060 I'm going to call this sigma star elastic 3 when it buckles. 672 00:47:05,460 --> 00:47:08,850 And what we're going to do is just look at a single plate. 673 00:47:08,850 --> 00:47:10,410 And look at the buckling of a plate. 674 00:47:16,630 --> 00:47:19,760 We're going to analyze it just looking at a single plate 675 00:47:19,760 --> 00:47:22,740 and then adding up how many plates we have per unit cell. 676 00:47:22,740 --> 00:47:24,450 It's actually more complicated than this 677 00:47:24,450 --> 00:47:26,250 because, obviously, the plates are attached together 678 00:47:26,250 --> 00:47:28,540 and there's some constraint by attaching the plates. 679 00:47:28,540 --> 00:47:30,330 But we're not going to worry about that. 680 00:47:39,490 --> 00:47:41,150 If you have a column-- just a, say, 681 00:47:41,150 --> 00:47:42,820 circular cross-section column-- and you 682 00:47:42,820 --> 00:47:46,100 apply a compressive load to it, it buckles at the Euler load. 683 00:47:46,100 --> 00:47:48,680 And similarly, there's an Euler load for plates. 684 00:47:48,680 --> 00:47:51,070 And that equation is usually written 685 00:47:51,070 --> 00:47:54,910 as a p critical is equal to some end constraint factor. 686 00:47:54,910 --> 00:47:58,290 For plates, it's usually called k instead of n. 687 00:47:58,290 --> 00:47:59,835 So this is an end constraint factor. 688 00:47:59,835 --> 00:48:02,545 It depends on the modulus of the plate. 689 00:48:02,545 --> 00:48:05,190 It goes as t cubed. 690 00:48:05,190 --> 00:48:09,420 Then, there's a factor of 1 minus mu of the solid squared 691 00:48:09,420 --> 00:48:11,960 and the length of the plate. 692 00:48:11,960 --> 00:48:16,060 Say this plate here-- actually the width 693 00:48:16,060 --> 00:48:20,500 of the plate there is h and the length here is b. 694 00:48:20,500 --> 00:48:25,730 And this thickness here is t like that. 695 00:48:25,730 --> 00:48:27,560 Here, k is an end constraint factor. 696 00:48:35,140 --> 00:48:38,694 And it's going to depend on the stiffness of the adjacent cell 697 00:48:38,694 --> 00:48:39,194 walls. 698 00:48:52,670 --> 00:49:01,350 If I had a honeycomb, and say it was-- these walls 699 00:49:01,350 --> 00:49:03,890 here-- the adjacent walls-- were thicker, 700 00:49:03,890 --> 00:49:06,470 then you can imagine those thicker walls-- it'd be harder 701 00:49:06,470 --> 00:49:07,800 to get them to deform. 702 00:49:07,800 --> 00:49:10,110 And the end constraint for the plate 703 00:49:10,110 --> 00:49:12,390 is going to depend on those thicker walls. 704 00:49:12,390 --> 00:49:15,370 So that the end constraint, k, depends 705 00:49:15,370 --> 00:49:19,345 on these-- say I'm looking at this wall here of width h 706 00:49:19,345 --> 00:49:21,560 here, then how stiff these other two walls 707 00:49:21,560 --> 00:49:26,860 are is going to affect that end constraint factor. 708 00:49:26,860 --> 00:49:29,460 What we're going to do is just do something very approximate. 709 00:49:29,460 --> 00:49:32,480 We're going to say if these vertical edges here-- if this 710 00:49:32,480 --> 00:49:35,700 edge here and that one there-- if they were simply supported-- 711 00:49:35,700 --> 00:49:39,760 if they're just pinned to the next column, 712 00:49:39,760 --> 00:49:42,770 the next member-- then k has some value. 713 00:49:42,770 --> 00:49:45,120 And if they're fixed, it has some other value. 714 00:49:45,120 --> 00:49:47,030 And we're going to pick a value in between. 715 00:49:47,030 --> 00:49:49,030 So we're going to do something very approximate. 716 00:50:20,700 --> 00:50:27,520 I'll say if those vertical edges are simply 717 00:50:27,520 --> 00:50:41,770 supported-- that means they're free to rotate-- then 718 00:50:41,770 --> 00:50:44,820 k is equal to 2.0. 719 00:50:44,820 --> 00:50:51,040 And this is if b is bigger than three times the length. 720 00:50:51,040 --> 00:50:55,140 So this is h here, or we could say l. 721 00:50:55,140 --> 00:50:57,150 Either way. 722 00:50:57,150 --> 00:50:59,050 It's really the-- it's the length when 723 00:50:59,050 --> 00:51:00,591 we look at the honeycomb this way on, 724 00:51:00,591 --> 00:51:05,470 but it's the width in that picture there. 725 00:51:05,470 --> 00:51:15,130 And if the vertical edges are clamped, or fixed, 726 00:51:15,130 --> 00:51:19,130 then k is equal to 6.2. 727 00:51:19,130 --> 00:51:23,517 These are values you can look up in tables of plate buckling. 728 00:51:23,517 --> 00:51:25,100 And we're just going to approximate it 729 00:51:25,100 --> 00:51:26,560 by saying k is equal to 4. 730 00:51:32,260 --> 00:51:36,850 We're just picking a value that's in between those two. 731 00:51:36,850 --> 00:51:39,150 And then, the p total is going to be 732 00:51:39,150 --> 00:51:43,110 the sum of the p criticals for the columns 733 00:51:43,110 --> 00:51:44,660 that make up a unit cell. 734 00:51:52,460 --> 00:51:56,170 For the unit cell, I have one wall of length h 735 00:51:56,170 --> 00:51:57,645 and two of length l. 736 00:52:16,010 --> 00:52:17,510 And if you just take that total load 737 00:52:17,510 --> 00:52:20,080 and divide by the area of the cell, 738 00:52:20,080 --> 00:52:24,810 you get that this compressive strength for elastic buckling 739 00:52:24,810 --> 00:52:28,310 is approximately equal to es over 1 minus nu 740 00:52:28,310 --> 00:52:33,980 s squared times t over l cubed. 741 00:52:33,980 --> 00:52:36,055 And then there's a geometrical factor here. 742 00:52:58,900 --> 00:53:04,060 And if you had regular hexagonal cells, 743 00:53:04,060 --> 00:53:08,730 this buckling stress works out to 5.2 times es times 744 00:53:08,730 --> 00:53:10,211 t over l cubed. 745 00:53:13,900 --> 00:53:17,690 If you remember, for the loading in the two direction-- 746 00:53:17,690 --> 00:53:21,250 in the in-plane direction-- it has the same form. 747 00:53:21,250 --> 00:53:25,210 And goes as es times t over l cubed, but it's much smaller. 748 00:53:25,210 --> 00:53:27,020 This number here was, I think, 0.2. 749 00:53:27,020 --> 00:53:28,456 It was much smaller. 750 00:53:46,670 --> 00:53:49,580 So it has the same form, but it's a lot bigger. 751 00:53:49,580 --> 00:53:51,090 OK? 752 00:53:51,090 --> 00:53:52,590 Are we good with that? 753 00:53:52,590 --> 00:53:55,080 The idea is we just use the standard equations 754 00:53:55,080 --> 00:53:56,010 for plate buckling. 755 00:53:56,010 --> 00:53:58,510 We make some estimate of what that end constraint factor is. 756 00:53:58,510 --> 00:54:02,100 And we just have an approximate calculation here. 757 00:54:26,510 --> 00:54:27,224 OK. 758 00:54:27,224 --> 00:54:28,390 That's the elastic buckling. 759 00:54:28,390 --> 00:54:31,210 If I had a metal honeycomb, then it 760 00:54:31,210 --> 00:54:33,310 might not fail by elastic buckling like that. 761 00:54:33,310 --> 00:54:35,400 Instead, we'd probably get yielding. 762 00:54:35,400 --> 00:54:37,260 If it was dense enough, we could just 763 00:54:37,260 --> 00:54:40,160 get axial yielding that-- if you just loaded it, 764 00:54:40,160 --> 00:54:41,320 you'd have axial forces. 765 00:54:41,320 --> 00:54:43,740 And at some point, you'd reach the yield stress. 766 00:54:43,740 --> 00:54:46,460 And so you can get failure by just uniaxial yield. 767 00:54:46,460 --> 00:54:48,060 That's one option. 768 00:55:03,705 --> 00:55:05,080 And if you get that, then it just 769 00:55:05,080 --> 00:55:08,427 depends on how much solid you've got again. 770 00:55:08,427 --> 00:55:10,510 So it's just the yield strength of the solid times 771 00:55:10,510 --> 00:55:12,130 the relative density. 772 00:55:12,130 --> 00:55:15,590 But usually, the honeycomb is thinner walled than that. 773 00:55:15,590 --> 00:55:18,240 And usually, you get plastic buckling proceeding that. 774 00:55:43,790 --> 00:55:46,590 In plastic buckling, you can think of it 775 00:55:46,590 --> 00:55:49,120 as-- say if you have a tube-- this is just shown 776 00:55:49,120 --> 00:55:50,930 for an individual tube here. 777 00:55:50,930 --> 00:55:53,470 You can see how the tube folds up. 778 00:55:53,470 --> 00:55:56,345 And you can get that same kind of thing with the honeycomb. 779 00:55:56,345 --> 00:55:57,220 Here's a single tube. 780 00:55:57,220 --> 00:56:00,840 It's been loaded along the prism axis of the tube. 781 00:56:00,840 --> 00:56:02,700 And you can see, you get these folds, 782 00:56:02,700 --> 00:56:05,610 and the more you load it, the more number of folds you get. 783 00:56:05,610 --> 00:56:09,070 And the more the folds concertina up. 784 00:56:09,070 --> 00:56:12,070 To do an exact analysis for the honeycombs, 785 00:56:12,070 --> 00:56:17,510 you would have to take into account not just one tube, 786 00:56:17,510 --> 00:56:20,040 but the constraint of the neighboring tubes again. 787 00:56:20,040 --> 00:56:22,630 And again, that gets to be a complicated, messy thing. 788 00:56:22,630 --> 00:56:25,030 So again, we're going to do a more approximate thing. 789 00:56:25,030 --> 00:56:28,770 What we're going to do is just say that we have members 790 00:56:28,770 --> 00:56:30,410 that are folding up like that. 791 00:56:30,410 --> 00:56:32,457 So the same geometry. 792 00:56:32,457 --> 00:56:34,540 But we're just going to look at a single cell wall 793 00:56:34,540 --> 00:56:36,350 and see what the single cell wall does. 794 00:56:36,350 --> 00:56:39,340 And someone else has done the more exact calculation. 795 00:56:39,340 --> 00:56:41,560 We'll just compare our approximate calculation 796 00:56:41,560 --> 00:56:44,171 to the exact one. 797 00:56:44,171 --> 00:56:44,670 OK. 798 00:56:48,924 --> 00:56:51,048 We're going to consider an approximate calculation. 799 00:57:21,600 --> 00:57:25,090 What we're going do is look at our isolated cell wall. 800 00:57:25,090 --> 00:57:26,720 And if you look at the figure here, 801 00:57:26,720 --> 00:57:29,207 the wall is going to be vertical, initially. 802 00:57:29,207 --> 00:57:31,290 And as we load it, eventually it's going to buckle 803 00:57:31,290 --> 00:57:33,414 and we're going to form one of those plastic hinges 804 00:57:33,414 --> 00:57:34,470 in the middle here. 805 00:57:34,470 --> 00:57:36,060 And then, the thing is then going 806 00:57:36,060 --> 00:57:39,460 to rotate about that plastic hinge and just fold up. 807 00:57:39,460 --> 00:57:42,230 So at the bottom here, it's completely folded up. 808 00:57:42,230 --> 00:57:43,010 OK? 809 00:57:43,010 --> 00:57:45,330 And we're going to do a little work calculations. 810 00:57:45,330 --> 00:57:47,500 We're going to look at the internal work done 811 00:57:47,500 --> 00:57:49,709 and we're going to look at the external work done. 812 00:57:49,709 --> 00:57:51,250 The external work is just going to be 813 00:57:51,250 --> 00:57:55,320 this load p times that deflection delta 814 00:57:55,320 --> 00:57:57,510 that the p moves through. 815 00:57:57,510 --> 00:58:01,370 And if we say this is half of a wavelength-- 816 00:58:01,370 --> 00:58:04,530 if you think of this thing going through multiple wavelengths, 817 00:58:04,530 --> 00:58:06,920 just consider when it folds up like that, 818 00:58:06,920 --> 00:58:08,400 that's a half of a wavelength. 819 00:58:08,400 --> 00:58:11,550 It would go two of those to get a full wavelength. 820 00:58:11,550 --> 00:58:13,410 That's lambda over 2. 821 00:58:13,410 --> 00:58:18,140 And so to go from this stage to that stage over here, 822 00:58:18,140 --> 00:58:21,220 the external work done is going to be approximately p 823 00:58:21,220 --> 00:58:22,960 times lambda over 2. 824 00:58:22,960 --> 00:58:25,920 Say that it's thin and that 2t is small compared to lambda. 825 00:58:25,920 --> 00:58:28,890 So it's going to be about p times lambda over 2. 826 00:58:28,890 --> 00:58:31,890 And then, we're also going to look at the work done 827 00:58:31,890 --> 00:58:34,380 by the plastic moment. 828 00:58:34,380 --> 00:58:36,400 And when we form the plastic hinge here, 829 00:58:36,400 --> 00:58:38,140 there's a plastic moment. 830 00:58:38,140 --> 00:58:41,800 And that moment is going to rotate through an angle of pi. 831 00:58:41,800 --> 00:58:45,390 So we start out straight here, we end up folded up like that, 832 00:58:45,390 --> 00:58:46,960 and we've gone from straight to that. 833 00:58:46,960 --> 00:58:49,990 We had to go through 180 degrees to get there. 834 00:58:49,990 --> 00:58:52,340 So it goes through an angle of pi. 835 00:58:52,340 --> 00:58:54,730 And if you have a moment going through a rotation, 836 00:58:54,730 --> 00:58:56,900 the work done is the moment times the rotation. 837 00:58:59,700 --> 00:59:03,560 We're going to equate those two works done. 838 00:59:03,560 --> 00:59:13,050 We're going to look at the rotation of the cell wall 839 00:59:13,050 --> 00:59:15,410 by an angle of pi at the plastic hinge. 840 00:59:30,880 --> 00:59:33,270 Our plastic moment-- it's going to be the yield 841 00:59:33,270 --> 00:59:36,220 strength of the solid again times t 842 00:59:36,220 --> 00:59:38,340 squared over 4, the same as when we 843 00:59:38,340 --> 00:59:41,050 were talking about the plastic moment before 844 00:59:41,050 --> 00:59:42,610 for the other loading direction. 845 00:59:42,610 --> 00:59:45,320 But now, instead of multiplying this times b, 846 00:59:45,320 --> 00:59:49,210 we're multiplying it times 2l plus h. 847 00:59:49,210 --> 00:59:50,750 That's the length of the cell wall 848 00:59:50,750 --> 00:59:52,930 that's associated with one cell. 849 00:59:52,930 --> 00:59:54,780 And now, it's not b because now we've 850 00:59:54,780 --> 00:59:56,240 turned the thing the other way on. 851 00:59:56,240 --> 00:59:57,800 We're loading it the other way on. 852 00:59:57,800 --> 01:00:00,460 And this plastic hinge-- if I think 853 01:00:00,460 --> 01:00:02,260 of-- if this was b before. 854 01:00:02,260 --> 01:00:08,080 And now that b is l plus 2h-- or 2l plus h, rather. 855 01:00:08,080 --> 01:00:12,150 That's the dimension of the-- let me draw a little hexagon so 856 01:00:12,150 --> 01:00:12,980 maybe you can see. 857 01:00:23,360 --> 01:00:23,860 OK. 858 01:00:23,860 --> 01:00:27,821 Now we're forming a plastic hinge halfway down the board. 859 01:00:27,821 --> 01:00:30,070 Imagine that this has some length b that way and we're 860 01:00:30,070 --> 01:00:31,510 halfway down the board. 861 01:00:31,510 --> 01:00:33,920 And now, the plastic hinge has to form all the way 862 01:00:33,920 --> 01:00:37,360 around these members here for one cell. 863 01:00:37,360 --> 01:00:41,590 Or you could think about it as this guy plus these guys 864 01:00:41,590 --> 01:00:42,359 is one cell. 865 01:00:42,359 --> 01:00:44,400 You can think about the unit cell different ways, 866 01:00:44,400 --> 01:00:48,130 but it's one h plus two l's. 867 01:00:48,130 --> 01:00:48,790 OK? 868 01:00:48,790 --> 01:00:49,675 Are we OK with that? 869 01:00:52,420 --> 01:00:55,280 OK. 870 01:00:55,280 --> 01:01:05,780 Then the internal plastic work is that plastic moment 871 01:01:05,780 --> 01:01:09,870 times the rotation pho-- or pi, rather. 872 01:01:09,870 --> 01:01:10,370 Sorry. 873 01:01:34,330 --> 01:01:35,830 Are we OK with this? 874 01:01:35,830 --> 01:01:38,750 That the work done is m times our angle? 875 01:01:38,750 --> 01:01:43,120 Imagine-- let me get rid of my honeycomb here. 876 01:01:43,120 --> 01:01:45,120 Imagine you have a point here and you 877 01:01:45,120 --> 01:01:46,610 have some force over here. 878 01:01:46,610 --> 01:01:48,110 Let's call that f. 879 01:01:48,110 --> 01:01:50,970 And say, the force is at distance r from f. 880 01:01:50,970 --> 01:01:54,890 And say that it moves through some distance. 881 01:01:54,890 --> 01:01:59,920 The moment here would be r times f. 882 01:01:59,920 --> 01:02:02,800 And if that rotates, say, through some angle-- 883 01:02:02,800 --> 01:02:07,110 let's call it alpha-- and here is f here, then 884 01:02:07,110 --> 01:02:09,210 this distance here that the force moves through 885 01:02:09,210 --> 01:02:11,300 is just r times alpha. 886 01:02:11,300 --> 01:02:16,590 So the work done is going to be r times alpha times f, 887 01:02:16,590 --> 01:02:18,664 or just the moment times alpha. 888 01:02:18,664 --> 01:02:19,480 OK? 889 01:02:19,480 --> 01:02:22,821 So that's all that we're doing. 890 01:02:22,821 --> 01:02:23,320 OK. 891 01:02:23,320 --> 01:02:25,380 That's the internal plastic work. 892 01:02:25,380 --> 01:02:27,622 And now we have to look at the external work done. 893 01:02:34,100 --> 01:02:39,150 And that's equal to p times lambda over 2. 894 01:02:39,150 --> 01:02:43,500 Here, lambda is the half wavelength of the buckling. 895 01:02:51,280 --> 01:02:54,190 I'm going to say for these tubular kinds of things, 896 01:02:54,190 --> 01:02:56,470 it's in the order of l. 897 01:02:56,470 --> 01:03:01,181 So if you look at that last slide here-- oops. 898 01:03:01,181 --> 01:03:01,680 Rats. 899 01:03:01,680 --> 01:03:02,736 How'd that happen? 900 01:03:06,050 --> 01:03:07,928 Let me scoot back down here. 901 01:03:11,481 --> 01:03:11,980 There. 902 01:03:11,980 --> 01:03:15,740 If we look at that guy again, the magnitude 903 01:03:15,740 --> 01:03:20,610 of the buckling wavelength is on the order of l. 904 01:03:20,610 --> 01:03:24,360 And here, below p, can be related to the stress 905 01:03:24,360 --> 01:03:25,400 in the three direction. 906 01:03:25,400 --> 01:03:27,760 We'll just multiply it times the area of the unit cell. 907 01:03:34,900 --> 01:03:38,790 And so if I equate the internal work and the external work, 908 01:03:38,790 --> 01:03:43,430 I can say p times lambda over 2 is 909 01:03:43,430 --> 01:03:45,507 equal to pi times my plastic moment. 910 01:03:48,260 --> 01:03:53,070 And then, for p, I can write sigma 3 h plus l sin 911 01:03:53,070 --> 01:03:57,040 theta times 2l cos theta. 912 01:04:00,610 --> 01:04:05,910 And then, lambda is l divided by 2 is equal to pi. 913 01:04:05,910 --> 01:04:08,078 And then I've got my plastic moment over there. 914 01:04:18,190 --> 01:04:22,470 And then if I solve for sigma 3, that's my compressor strength. 915 01:04:35,250 --> 01:04:41,560 I've got pi by 4, the strength of the solid, sigma ys, times 916 01:04:41,560 --> 01:04:44,290 t over l squared. 917 01:04:44,290 --> 01:04:51,380 Then h over l plus 2 divided by h over l plus sin theta times 918 01:04:51,380 --> 01:04:51,950 cos theta. 919 01:04:56,470 --> 01:05:03,250 And for the regular hexagons, this 920 01:05:03,250 --> 01:05:14,950 works out to about 2 sigma ys times to over l squared. 921 01:05:14,950 --> 01:05:23,670 And the exact calculation for regular honeycombs 922 01:05:23,670 --> 01:05:30,340 is equal to 5.6 times sigma ys times t over l 923 01:05:30,340 --> 01:05:32,310 to the 5/3 power. 924 01:05:32,310 --> 01:05:37,150 This power here-- 5/3-- is a little less than 2. 925 01:05:37,150 --> 01:05:39,070 And that's because the additional constraint 926 01:05:39,070 --> 01:05:41,020 of the neighboring cell walls. 927 01:05:41,020 --> 01:05:44,230 But the main thing we're interested in, 928 01:05:44,230 --> 01:05:46,770 in these sorts of calculations, is the power dependence 929 01:05:46,770 --> 01:05:49,722 on the density and this simple calculation. 930 01:05:49,722 --> 01:05:51,763 Obviously, it's not exact, but it gets you close. 931 01:05:55,896 --> 01:05:56,396 OK. 932 01:06:36,305 --> 01:06:38,596 I'm just going to wait for people to catch up a little. 933 01:06:46,761 --> 01:06:47,260 OK. 934 01:06:47,260 --> 01:06:48,843 The next property I'm going to look at 935 01:06:48,843 --> 01:06:50,710 is out-of-plane brittle fractures. 936 01:06:50,710 --> 01:06:54,570 Say we loaded in tension, and if we had no cracks in the walls, 937 01:06:54,570 --> 01:06:57,350 we'd just see uniaxial tension and the strength would just 938 01:06:57,350 --> 01:06:59,870 be the strength of the solid times the relative density 939 01:06:59,870 --> 01:07:02,950 times the amount of solid. 940 01:07:02,950 --> 01:07:16,915 We'll just say if defect free, the walls see uniaxial tension. 941 01:07:20,230 --> 01:07:24,350 And then the fracture stress in the three direction 942 01:07:24,350 --> 01:07:28,100 is just equal to the relative density times the fracture 943 01:07:28,100 --> 01:07:29,063 strength of the solid. 944 01:07:37,660 --> 01:07:41,510 If the cell walls are cracked, and if the crack length 945 01:07:41,510 --> 01:07:45,980 is very much bigger than the cell length, 946 01:07:45,980 --> 01:07:48,250 then the crack propagates normal to x3. 947 01:07:57,120 --> 01:08:06,410 Then we can say the toughness gc-- or the critical strain 948 01:08:06,410 --> 01:08:09,980 energy release rate-- is just equal to the volume 949 01:08:09,980 --> 01:08:15,430 fraction of solid times gc for the solid. 950 01:08:15,430 --> 01:08:25,899 And then the fracture toughness, k1c, 951 01:08:25,899 --> 01:08:30,142 is equal to the square root of the Young's modulus times gc. 952 01:08:33,240 --> 01:08:37,020 And that's just equal to the relative density 953 01:08:37,020 --> 01:08:39,260 times the modulus of the solid. 954 01:08:39,260 --> 01:08:45,840 And then the relative density times the toughness 955 01:08:45,840 --> 01:08:47,740 of the solid. 956 01:08:47,740 --> 01:08:51,460 So it's just equal to the relative density 957 01:08:51,460 --> 01:08:53,290 times the fracture toughness of the solid. 958 01:08:55,957 --> 01:08:57,290 It's just straightforward there. 959 01:09:29,510 --> 01:09:33,460 Then we've got one last out-of-plane property. 960 01:09:33,460 --> 01:09:35,734 And that's brittle crushing and compression. 961 01:09:45,520 --> 01:09:48,319 And if we have some compressive strength of the cell wall-- 962 01:09:48,319 --> 01:09:53,120 say I call it cs-- then it's just the relative density 963 01:09:53,120 --> 01:09:54,063 times that strength. 964 01:10:15,640 --> 01:10:19,200 And for brittle materials, that crushing strength 965 01:10:19,200 --> 01:10:23,858 is typically around 12 times the modulus of rupture, or fracture 966 01:10:23,858 --> 01:10:24,358 strength. 967 01:10:36,520 --> 01:10:38,410 We could say that's about equal to 12 968 01:10:38,410 --> 01:10:45,172 times the relative density times sigma fs, a fracture strength. 969 01:10:47,940 --> 01:10:48,440 OK. 970 01:10:48,440 --> 01:10:52,730 That's the modeling of the honeycombs. 971 01:10:52,730 --> 01:10:55,620 I know there's been a lot of equations and derivations, 972 01:10:55,620 --> 01:10:57,757 but that's the basis of a lot of the things 973 01:10:57,757 --> 01:10:59,590 we're going to do in the rest of the course. 974 01:10:59,590 --> 01:11:02,920 The modeling we're going to do on the foams is based on this 975 01:11:02,920 --> 01:11:05,989 and the mathematics is just easier 976 01:11:05,989 --> 01:11:08,280 because we're going to use these dimensional arguments. 977 01:11:08,280 --> 01:11:09,655 We're not going to figure out all 978 01:11:09,655 --> 01:11:11,940 these geometrical parameters. 979 01:11:11,940 --> 01:11:13,440 Before we get to the foams, I wanted 980 01:11:13,440 --> 01:11:15,902 to talk a little bit about honeycombs in nature. 981 01:11:15,902 --> 01:11:17,610 And we've only got a couple minutes left, 982 01:11:17,610 --> 01:11:19,220 so I won't really get that far. 983 01:11:19,220 --> 01:11:23,540 But I wanted to talk a little bit about honeycomb materials 984 01:11:23,540 --> 01:11:24,140 in nature. 985 01:11:24,140 --> 01:11:26,056 And the two examples we're going to talk about 986 01:11:26,056 --> 01:11:28,454 are wood and cork. 987 01:11:28,454 --> 01:11:30,870 I'm going to talk a little bit about the structure of wood 988 01:11:30,870 --> 01:11:31,990 next time. 989 01:11:31,990 --> 01:11:34,710 Then, we'll see how we can apply these models to understanding 990 01:11:34,710 --> 01:11:36,100 how wood behaves. 991 01:11:36,100 --> 01:11:38,200 And we'll see how you can use these models 992 01:11:38,200 --> 01:11:41,130 to predict the density dependence of wood properties 993 01:11:41,130 --> 01:11:43,930 and also the anisotropy in wood properties. 994 01:11:43,930 --> 01:11:47,750 And I guess we'll probably, maybe, start it Wednesday 995 01:11:47,750 --> 01:11:49,010 next week. 996 01:11:49,010 --> 01:11:50,720 We'll talk about cork, as well. 997 01:11:50,720 --> 01:11:52,360 Those of you who took 3032 know that I 998 01:11:52,360 --> 01:11:56,000 like cork because of Robert Hooke and his drawing of cork. 999 01:11:56,000 --> 01:11:58,750 And I made a new video that I'm going to show you. 1000 01:11:58,750 --> 01:12:01,022 Remember in 3032, I showed you the video 1001 01:12:01,022 --> 01:12:03,480 from the Bodleian Library, where they had the first edition 1002 01:12:03,480 --> 01:12:05,860 of Hooke's Micrographia. 1003 01:12:05,860 --> 01:12:08,170 Well, it turns out Harvard has a first edition. 1004 01:12:08,170 --> 01:12:10,461 Harvard has three first editions. 1005 01:12:10,461 --> 01:12:10,960 Yeah. 1006 01:12:10,960 --> 01:12:11,690 Exactly. 1007 01:12:11,690 --> 01:12:14,300 MIT has zero first editions. 1008 01:12:14,300 --> 01:12:16,480 Gee, why does that surprise me? 1009 01:12:16,480 --> 01:12:18,480 And I have a friend who's a librarian at Harvard 1010 01:12:18,480 --> 01:12:20,063 and she arranged for me to go and make 1011 01:12:20,063 --> 01:12:22,380 a little video with the first edition of Micrographia. 1012 01:12:22,380 --> 01:12:25,155 So I can-- I don't if we'll play the whole thing, 1013 01:12:25,155 --> 01:12:27,030 but I'll show you the first little bit of it. 1014 01:12:27,030 --> 01:12:29,620 And you can watch it at your leisure. 1015 01:12:29,620 --> 01:12:30,470 And Sardar came. 1016 01:12:30,470 --> 01:12:32,687 You came and saw it with me. 1017 01:12:32,687 --> 01:12:34,770 You came and saw the first edition with me, right? 1018 01:12:34,770 --> 01:12:35,150 AUDIENCE: Yes. 1019 01:12:35,150 --> 01:12:35,610 LORNA GIBSON: Yeah. 1020 01:12:35,610 --> 01:12:36,180 Yeah. 1021 01:12:36,180 --> 01:12:40,030 It's very beautiful and you'll see some of the nice drawings. 1022 01:12:40,030 --> 01:12:42,010 And I talk about the cellular structure 1023 01:12:42,010 --> 01:12:43,340 of some of the drawings. 1024 01:12:43,340 --> 01:12:45,360 So we'll talk about wood and cork next time. 1025 01:12:45,360 --> 01:12:47,610 But I think I'm going to stop there because that seems 1026 01:12:47,610 --> 01:12:50,940 like enough equations for now.