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 --> 00:00:10,680 continue to offer high-quality educational resources for free. 5 00:00:10,680 --> 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 00:00:27,100 --> 00:00:28,890 LORNA GIBSON: OK, so it's five after. 9 00:00:28,890 --> 00:00:31,820 We should probably start. 10 00:00:31,820 --> 00:00:33,999 So last time we were talking about honeycombs, 11 00:00:33,999 --> 00:00:35,790 and I just wanted to quickly kind of review 12 00:00:35,790 --> 00:00:37,623 what we had talked about, and then today I'm 13 00:00:37,623 --> 00:00:39,610 going to start deriving equations 14 00:00:39,610 --> 00:00:42,850 for the mechanical properties of the honeycombs, OK? 15 00:00:42,850 --> 00:00:45,290 So this is a slide of our honeycomb setup here. 16 00:00:45,290 --> 00:00:48,170 These are the hexagonal cells we're going to look at. 17 00:00:48,170 --> 00:00:50,000 We talked about the stress-strain behavior. 18 00:00:50,000 --> 00:00:53,120 The curves on the left-hand side are for compression 19 00:00:53,120 --> 00:00:55,752 and the ones on the right-hand side are for tension. 20 00:00:55,752 --> 00:00:57,460 And so what we're going to be doing today 21 00:00:57,460 --> 00:01:00,500 is we're going to start out by calculating a Young's modulus, 22 00:01:00,500 --> 00:01:01,950 this slope here. 23 00:01:01,950 --> 00:01:05,830 We're going to calculate the stress plateaus for failure 24 00:01:05,830 --> 00:01:08,850 by elastic buckling in elastomeric honeycombs, 25 00:01:08,850 --> 00:01:12,120 by failure from plastic yielding in, say, a metal honeycomb, 26 00:01:12,120 --> 00:01:14,350 and by failure by a brittle crushing 27 00:01:14,350 --> 00:01:15,984 in, say, a ceramic honeycomb. 28 00:01:15,984 --> 00:01:18,150 And if we have time, we'll get to the tension stuff. 29 00:01:18,150 --> 00:01:20,320 I don't know if we'll get to that today or next time. 30 00:01:20,320 --> 00:01:21,778 So we're going to start calculating 31 00:01:21,778 --> 00:01:22,900 those properties today. 32 00:01:22,900 --> 00:01:24,650 And these were the deformation mechanisms. 33 00:01:24,650 --> 00:01:27,150 Remember, we said the linear elastic behavior was related 34 00:01:27,150 --> 00:01:29,890 to bending of the cell walls, and then the plateau 35 00:01:29,890 --> 00:01:32,530 was related to buckling if it was an elastomer. 36 00:01:32,530 --> 00:01:34,460 And the plateau was related to yielding 37 00:01:34,460 --> 00:01:37,889 if it was, say, a metal that had a yield point. 38 00:01:37,889 --> 00:01:39,430 And then this was sort of an overview 39 00:01:39,430 --> 00:01:40,846 of the stress-strain curve showing 40 00:01:40,846 --> 00:01:42,674 those different regions, OK? 41 00:01:42,674 --> 00:01:44,590 So what I'm going to talk about today to start 42 00:01:44,590 --> 00:01:46,744 is the linear elastic behavior. 43 00:01:46,744 --> 00:01:49,160 And we're going to be starting with the in-plane behavior. 44 00:01:49,160 --> 00:01:52,730 So in-plane means in the plane of the hexagonal cells. 45 00:01:52,730 --> 00:01:55,860 And then next time we'll do the out-of-plane behavior, this way 46 00:01:55,860 --> 00:01:56,376 on. 47 00:01:56,376 --> 00:01:58,000 So if I had to form my little honeycomb 48 00:01:58,000 --> 00:02:00,130 like this, what initially happens 49 00:02:00,130 --> 00:02:02,530 is the inclined cell walls bend. 50 00:02:02,530 --> 00:02:05,050 So if you can see over here, we've kind of 51 00:02:05,050 --> 00:02:06,970 exaggerated it on this sketch. 52 00:02:06,970 --> 00:02:08,490 So this wall here is bent. 53 00:02:08,490 --> 00:02:11,220 This one here just kind of moves along, goes for the ride. 54 00:02:11,220 --> 00:02:12,410 And this guy here is bent. 55 00:02:12,410 --> 00:02:14,050 So this is for loading in the what 56 00:02:14,050 --> 00:02:16,590 we're calling the x1 direction, sigma 1. 57 00:02:16,590 --> 00:02:18,030 And the same kind of thing happens 58 00:02:18,030 --> 00:02:19,530 when we load in the other direction, 59 00:02:19,530 --> 00:02:22,510 in the sigma 2 direction, these guys still bend. 60 00:02:22,510 --> 00:02:24,590 Now the honeycomb gets wider that way. 61 00:02:24,590 --> 00:02:26,690 It gets shorter this way, wider that way. 62 00:02:26,690 --> 00:02:29,090 And we can calculate the Young's modulus 63 00:02:29,090 --> 00:02:33,310 if we can relate the load on the beam in the moments 64 00:02:33,310 --> 00:02:34,950 to this deflection here, right? 65 00:02:34,950 --> 00:02:38,610 So the Young's modulus is going to be related to the stiffness 66 00:02:38,610 --> 00:02:41,367 and the stiffness is going to be related to how much deformation 67 00:02:41,367 --> 00:02:43,950 you get for a certain amount of load that you put on the beam. 68 00:02:43,950 --> 00:02:46,730 So I'm going to calculate the modulus for the x1 direction, 69 00:02:46,730 --> 00:02:48,590 the thing on the left there. 70 00:02:48,590 --> 00:02:50,822 And you can do the same thing for the x2 direction 71 00:02:50,822 --> 00:02:52,280 on the right, but I won't calculate 72 00:02:52,280 --> 00:02:54,880 that because it's exactly the same kind of process. 73 00:02:54,880 --> 00:02:58,199 OK, so let me start here. 74 00:02:58,199 --> 00:02:58,740 Get my chalk. 75 00:03:01,710 --> 00:03:03,863 So I'm going to draw a one-unit cell here. 76 00:03:07,770 --> 00:03:12,660 So here's my unit cell there, like that. 77 00:03:12,660 --> 00:03:17,220 And this member here is of length h. 78 00:03:17,220 --> 00:03:21,360 That member there is of length l. 79 00:03:21,360 --> 00:03:24,270 That angle there is theta. 80 00:03:24,270 --> 00:03:26,560 I'm going to say all the walls have equal thickness, 81 00:03:26,560 --> 00:03:28,630 and I'm going to call it t. 82 00:03:28,630 --> 00:03:32,000 And I'm going to define an x1 and x2 axis like this. 83 00:03:32,000 --> 00:03:37,130 So the horizontal is x1 and the vertical axis is x2. 84 00:03:37,130 --> 00:03:41,580 And I'm going to say that I apply a sort of global stress 85 00:03:41,580 --> 00:03:44,380 to it, sigma 1. 86 00:03:44,380 --> 00:03:49,395 So there's a stress in the one direction there, sigma 1, OK? 87 00:03:49,395 --> 00:03:51,270 And I'm going to say my honeycomb has a depth 88 00:03:51,270 --> 00:03:58,530 b into the page, but the depth s-- 89 00:03:58,530 --> 00:04:00,622 is because the honeycomb's prismatic, 90 00:04:00,622 --> 00:04:03,080 the b's are always going to cancel out of all the equations 91 00:04:03,080 --> 00:04:05,010 that we're going to get, because everything's 92 00:04:05,010 --> 00:04:07,330 uniform in that direction. 93 00:04:07,330 --> 00:04:10,000 And we can think about a unit cell here, 94 00:04:10,000 --> 00:04:15,170 and in the x1 direction, we could 95 00:04:15,170 --> 00:04:24,700 say the length of our unit cell is 2l cos theta. 96 00:04:24,700 --> 00:04:28,820 So in the x1 direction, that's our unit cell there, 97 00:04:28,820 --> 00:04:34,296 and that's 2l cos of theta. 98 00:04:34,296 --> 00:04:37,020 And in the x2 direction, you might 99 00:04:37,020 --> 00:04:38,950 think that you go from this vertex 100 00:04:38,950 --> 00:04:41,150 up here down to that vertex there, 101 00:04:41,150 --> 00:04:43,950 but if you did that, then on the next layer of cells, 102 00:04:43,950 --> 00:04:47,440 you wouldn't have the same distance. 103 00:04:47,440 --> 00:04:51,880 So the unit length in the x2 direction 104 00:04:51,880 --> 00:04:54,074 is actually from here to here, and then you 105 00:04:54,074 --> 00:04:56,240 can see the next cell, you would get the same thing. 106 00:04:56,240 --> 00:04:58,440 You get this bit here from the inclined member, 107 00:04:58,440 --> 00:05:01,400 and then you would get h down here from the next member. 108 00:05:01,400 --> 00:05:08,240 So this bit here is equal to h plus l times sine of theta. 109 00:05:08,240 --> 00:05:11,142 So I can say in the x2 direction, 110 00:05:11,142 --> 00:05:18,870 the length of the unit cell is h plus l sine theta, OK? 111 00:05:18,870 --> 00:05:21,230 So that's kind of the setup. 112 00:05:21,230 --> 00:05:23,874 And then what we want to look at is that inclined member 113 00:05:23,874 --> 00:05:25,290 that bends, we want to look at how 114 00:05:25,290 --> 00:05:28,080 this guy bends under the load. 115 00:05:28,080 --> 00:05:32,290 And if we can relate the forces on it to the deflection, 116 00:05:32,290 --> 00:05:34,830 and we need the component of the deflection in the one 117 00:05:34,830 --> 00:05:37,940 direction, then we're going to be able to get the modulus. 118 00:05:37,940 --> 00:05:41,020 So I'm going to draw that inclined member again 119 00:05:41,020 --> 00:05:45,210 over here, and it's going to see some loads 120 00:05:45,210 --> 00:05:48,240 that I'm going to call p, and that's going 121 00:05:48,240 --> 00:05:51,620 to cause this thing to bend. 122 00:05:51,620 --> 00:05:54,840 So I've kind of exaggerated it there, but there's the bending. 123 00:05:54,840 --> 00:06:00,540 And there's some end deflection there, delta. 124 00:06:00,540 --> 00:06:05,910 And there's moments at either end of the beam, or either end 125 00:06:05,910 --> 00:06:07,590 of that member, as well. 126 00:06:07,590 --> 00:06:10,420 And this member here has a length. 127 00:06:10,420 --> 00:06:12,040 That length is l. 128 00:06:12,040 --> 00:06:15,570 OK, are we good? 129 00:06:15,570 --> 00:06:18,030 So it's just kind of the setup. 130 00:06:18,030 --> 00:06:19,900 And I'm going to draw the deflection 131 00:06:19,900 --> 00:06:21,730 delta bigger over here. 132 00:06:21,730 --> 00:06:23,190 So say that's delta. 133 00:06:23,190 --> 00:06:26,360 That's the same parallel as this guy here. 134 00:06:26,360 --> 00:06:30,170 What I'm going to want is the deflection in the x1 direction, 135 00:06:30,170 --> 00:06:32,900 and when I come to calculate the Poisson's ratio, 136 00:06:32,900 --> 00:06:35,820 I'm going to want the deflection in the x2 direction. 137 00:06:35,820 --> 00:06:40,360 And if this angle here is theta, between the horizontal 138 00:06:40,360 --> 00:06:42,820 and the inclined member, then this angle up here 139 00:06:42,820 --> 00:06:47,780 is also theta, and so this bit here is delta sine theta. 140 00:06:47,780 --> 00:06:50,350 And this bit here is delta cos theta. 141 00:06:55,130 --> 00:06:56,130 Ba-doop-ba-doop-ba-doop. 142 00:07:00,460 --> 00:07:02,230 So the Young's modulus is going to be 143 00:07:02,230 --> 00:07:04,210 the stress in the one direction divided 144 00:07:04,210 --> 00:07:06,000 by the strain in the one direction. 145 00:07:06,000 --> 00:07:08,050 So I need to get the stress and the strain 146 00:07:08,050 --> 00:07:09,890 in the one direction. 147 00:07:09,890 --> 00:07:14,020 So here the stress in the one direction. 148 00:07:14,020 --> 00:07:18,020 If I'm applying my load, like this, sigma 1, 149 00:07:18,020 --> 00:07:19,920 the stress in the one direction is 150 00:07:19,920 --> 00:07:23,130 going to be this load p-- so this load, 151 00:07:23,130 --> 00:07:26,530 say, p on this member here, divided by this length 152 00:07:26,530 --> 00:07:28,300 here, the unit cell length, and then 153 00:07:28,300 --> 00:07:32,830 divided by b into the board, the width end of the board. 154 00:07:32,830 --> 00:07:40,600 So sigma 1 is going to be p divided by h plus l sine theta 155 00:07:40,600 --> 00:07:41,337 times b. 156 00:07:44,660 --> 00:07:47,580 And epsilon 1 is going to be the strain in the one direction, 157 00:07:47,580 --> 00:07:50,120 is going to be the deformation in the one direction divided 158 00:07:50,120 --> 00:07:52,380 by the unit cell length in the one direction. 159 00:07:52,380 --> 00:07:59,630 So that's going to be delta sine theta divided by l cos theta. 160 00:07:59,630 --> 00:08:03,130 So here, even though I've said this unit cell is 2l cos theta, 161 00:08:03,130 --> 00:08:05,414 there'd be two of the members here 162 00:08:05,414 --> 00:08:07,330 that would be twice that deflection in the one 163 00:08:07,330 --> 00:08:09,240 direction. 164 00:08:09,240 --> 00:08:12,480 So it's, for one beam, it's delta sine theta 165 00:08:12,480 --> 00:08:14,520 over l cos theta. 166 00:08:14,520 --> 00:08:16,710 Are we OK so far? 167 00:08:16,710 --> 00:08:18,890 So far so good? 168 00:08:18,890 --> 00:08:20,450 So for the hexagons, because we're 169 00:08:20,450 --> 00:08:25,620 going to figure out the equations more or less exactly, 170 00:08:25,620 --> 00:08:28,221 we're going to keep track of all the geometrical factors. 171 00:08:28,221 --> 00:08:29,720 When we come to the foams, we're not 172 00:08:29,720 --> 00:08:31,997 going to keep track of all the geometrical factors. 173 00:08:31,997 --> 00:08:34,580 So one of the things that makes us look a little kind of hairy 174 00:08:34,580 --> 00:08:36,038 is just the fact that we're keeping 175 00:08:36,038 --> 00:08:38,630 track of all these sines and cosines and all the dimensions 176 00:08:38,630 --> 00:08:39,802 and things. 177 00:08:39,802 --> 00:08:42,696 All right, whoop. 178 00:08:48,160 --> 00:08:52,050 So I need to be able to relate my load p to my deformation 179 00:08:52,050 --> 00:08:55,820 delta to get a stiffness out of this, to get a modulus, OK? 180 00:08:55,820 --> 00:08:57,270 So the way I do that is, remember 181 00:08:57,270 --> 00:08:59,360 in 3032, we did those bending moment diagrams 182 00:08:59,360 --> 00:09:01,280 and we did the deflection of the beams? 183 00:09:01,280 --> 00:09:03,210 This is where this comes in handy. 184 00:09:03,210 --> 00:09:05,990 So I'm going to draw my beam a little bit differently now. 185 00:09:05,990 --> 00:09:08,050 I'm going to turn it on its side. 186 00:09:08,050 --> 00:09:11,190 So this is still my length l, but I'm 187 00:09:11,190 --> 00:09:13,410 going to turn on my side just so that you 188 00:09:13,410 --> 00:09:15,390 can see it the same kind of way we did 189 00:09:15,390 --> 00:09:16,580 the bending moment diagrams. 190 00:09:20,160 --> 00:09:24,620 So this is still my length l across here. 191 00:09:24,620 --> 00:09:32,100 And there's end moments, M and M here. 192 00:09:32,100 --> 00:09:35,060 And P sin theta is just the perpendicular component 193 00:09:35,060 --> 00:09:35,940 of the load p. 194 00:09:35,940 --> 00:09:38,260 So p sin theta is just the component perpendicular 195 00:09:38,260 --> 00:09:39,470 to my beam. 196 00:09:46,970 --> 00:09:48,760 So I could draw a shear diagram here 197 00:09:48,760 --> 00:09:51,070 and I could draw a bending moment diagram here. 198 00:09:51,070 --> 00:09:54,830 And, if you remember, the shear diagram, 199 00:09:54,830 --> 00:09:57,920 if I have no concentrated load along here, 200 00:09:57,920 --> 00:10:02,200 and I have no distributed load along here, if this is zero, 201 00:10:02,200 --> 00:10:06,760 down here, it's just going to go up my P sin theta, 202 00:10:06,760 --> 00:10:12,590 and then be horizontal, and then come down by P sin theta, OK? 203 00:10:12,590 --> 00:10:14,300 So that's the shear diagram. 204 00:10:14,300 --> 00:10:20,614 And then the bending moment diagram, 205 00:10:20,614 --> 00:10:21,780 I'm going to draw down here. 206 00:10:25,010 --> 00:10:27,850 So I've got some moment at the end here, 207 00:10:27,850 --> 00:10:31,912 and this would tend to bend like that. 208 00:10:31,912 --> 00:10:33,370 So this would be a negative moment. 209 00:10:33,370 --> 00:10:35,869 Remember, bending moments were negative 210 00:10:35,869 --> 00:10:38,160 if there was tension on the top, and they were positive 211 00:10:38,160 --> 00:10:39,618 if there was tension on the bottom. 212 00:10:39,618 --> 00:10:41,730 So over here we'd have tension on the top, 213 00:10:41,730 --> 00:10:44,100 so that would give us a negative bending moment. 214 00:10:44,100 --> 00:10:47,470 And then, if you also remember, the moment 215 00:10:47,470 --> 00:10:52,100 at a particular point is equal to the integral from, say, A 216 00:10:52,100 --> 00:10:54,760 to B-- well maybe I should write this another way. 217 00:10:54,760 --> 00:10:59,460 And B minus Ma is the integral of the shear 218 00:10:59,460 --> 00:11:02,829 diagram between the two points. 219 00:11:02,829 --> 00:11:03,495 A little sloppy. 220 00:11:06,810 --> 00:11:08,020 OK. 221 00:11:08,020 --> 00:11:11,610 So if I know how I have some moment here minus M, 222 00:11:11,610 --> 00:11:14,680 if I integrate this shear diagram up, 223 00:11:14,680 --> 00:11:17,630 then this is just going to be linear here, 224 00:11:17,630 --> 00:11:20,667 and then I'm going to be at plus M over there. 225 00:11:30,250 --> 00:11:33,760 So if you look at this shear and bending moment diagram, 226 00:11:33,760 --> 00:11:35,790 it's really just the same as the shear 227 00:11:35,790 --> 00:11:38,060 and bending moment diagram for two cantilevers that 228 00:11:38,060 --> 00:11:39,720 are attached to each other. 229 00:11:39,720 --> 00:11:44,730 So let me just draw over here what the cantilever looks like. 230 00:11:44,730 --> 00:11:45,670 Let's see. 231 00:11:45,670 --> 00:11:49,250 So imagine I just had a cantilever like this, 232 00:11:49,250 --> 00:11:52,680 and I have some force F on it like that. 233 00:11:52,680 --> 00:11:56,290 And I call this distance here capital L, 234 00:11:56,290 --> 00:11:59,940 and I'm going to call that deflection capital delta, 235 00:11:59,940 --> 00:12:01,500 like that. 236 00:12:01,500 --> 00:12:04,220 If I drew the shear diagram for that, 237 00:12:04,220 --> 00:12:07,710 there'd be a reaction here, F, there would be a moment here, 238 00:12:07,710 --> 00:12:10,030 FL. 239 00:12:10,030 --> 00:12:12,820 Doot, doot, yup. 240 00:12:12,820 --> 00:12:18,870 So this would look-- whoops-- it's a little too long. 241 00:12:18,870 --> 00:12:21,730 Shear diagram here would look like this. 242 00:12:21,730 --> 00:12:23,280 That would be zero. 243 00:12:23,280 --> 00:12:25,700 This would be FL. 244 00:12:25,700 --> 00:12:28,160 And the moment diagram would look like this. 245 00:12:28,160 --> 00:12:29,650 Whoops. 246 00:12:29,650 --> 00:12:33,070 A little too long again. 247 00:12:33,070 --> 00:12:35,092 And that would be minus FL. 248 00:12:35,092 --> 00:12:36,050 And that would be zero. 249 00:12:40,190 --> 00:12:44,100 So do you see how the shear and the bending moment diagram 250 00:12:44,100 --> 00:12:47,710 here are really just like two cantilevers, OK? 251 00:12:47,710 --> 00:12:54,600 So I know that the deflection for a cantilever, delta, 252 00:12:54,600 --> 00:13:01,560 is equal to F capital L cubed over 3EI. 253 00:13:01,560 --> 00:13:04,800 It's kind of a standard result. And so I 254 00:13:04,800 --> 00:13:07,710 can take this and apply that to this beam here. 255 00:13:07,710 --> 00:13:10,330 So instead of working everything out from first principles, 256 00:13:10,330 --> 00:13:12,820 I'm just going to say that my beam here 257 00:13:12,820 --> 00:13:16,840 is like two cantilevers, and instead of F, I've 258 00:13:16,840 --> 00:13:18,770 got P sin theta. 259 00:13:18,770 --> 00:13:21,840 And instead of capital L here as the length, 260 00:13:21,840 --> 00:13:25,889 I've got l/2 because l/2 would be the length of one 261 00:13:25,889 --> 00:13:26,680 of the cantilevers. 262 00:13:29,576 --> 00:13:38,700 OK, so for the honeycomb, I've got 263 00:13:38,700 --> 00:13:49,770 two cantilevers of length l/2. 264 00:13:49,770 --> 00:13:54,440 So delta for the inclined member on the honeycomb 265 00:13:54,440 --> 00:13:58,330 is going to be 2-- because I've got 266 00:13:58,330 --> 00:14:02,710 two cantilevers-- the force, instead of having F, 267 00:14:02,710 --> 00:14:04,930 I'm going to have P sin theta. 268 00:14:04,930 --> 00:14:07,280 And instead of having capital L, I'm 269 00:14:07,280 --> 00:14:11,770 going to have l/2, all cubed. 270 00:14:11,770 --> 00:14:17,990 So this is like F capital L cubed over 3. 271 00:14:17,990 --> 00:14:20,400 And here, the modulus that I want 272 00:14:20,400 --> 00:14:23,040 is the modulus of the solid cell wall material, 273 00:14:23,040 --> 00:14:29,140 so I'm going to call that ES, and over the moment of inertia. 274 00:14:29,140 --> 00:14:30,940 So you see how I've done it? 275 00:14:30,940 --> 00:14:32,370 Is that OK? 276 00:14:32,370 --> 00:14:35,600 So then I can just kind of simplify this thing here. 277 00:14:35,600 --> 00:14:40,580 I've got P sin theta l cubed. 278 00:14:40,580 --> 00:14:43,600 1/2 cubed is going to be 1/8. 279 00:14:43,600 --> 00:14:48,335 So this is going to be 2, if that's 1/8, times 3 is 24. 280 00:14:48,335 --> 00:14:49,785 So 2/24 is 12. 281 00:14:52,712 --> 00:14:56,220 So I've got delta for my honeycomb member 282 00:14:56,220 --> 00:15:00,620 is P sin theta l cubed over 12 EsI. 283 00:15:00,620 --> 00:15:08,080 And here I is the moment of inertia of that inclined 284 00:15:08,080 --> 00:15:12,540 member of the honeycomb. 285 00:15:12,540 --> 00:15:15,070 And that's BT cubed over 12, OK? 286 00:15:15,070 --> 00:15:19,125 So B is the depth into the board, and T is the thickness. 287 00:15:19,125 --> 00:15:20,990 We'll cube that and divide by 12. 288 00:15:20,990 --> 00:15:22,115 It's a rectangular section. 289 00:15:22,115 --> 00:15:22,818 Yeah? 290 00:15:22,818 --> 00:15:24,424 AUDIENCE: What was ES again? 291 00:15:24,424 --> 00:15:26,590 LORNA GIBSON: ES is the Young's modulus of the solid 292 00:15:26,590 --> 00:15:27,890 that it's made from. 293 00:15:27,890 --> 00:15:30,790 So clearly, if my honeycomb is made up 294 00:15:30,790 --> 00:15:33,900 of these members, whatever material 295 00:15:33,900 --> 00:15:36,370 the members are made of is going to affect the stiffness 296 00:15:36,370 --> 00:15:38,410 of the whole thing. 297 00:15:38,410 --> 00:15:40,410 Are we good? 298 00:15:40,410 --> 00:15:42,630 Because once we have this part, then we just 299 00:15:42,630 --> 00:15:45,260 combine these equations for the stress 300 00:15:45,260 --> 00:15:47,510 and the strain in the one direction. 301 00:15:47,510 --> 00:15:51,010 And we have this equation relating delta and P, 302 00:15:51,010 --> 00:15:54,500 and we're going to be able to get our Young's modulus, OK? 303 00:15:54,500 --> 00:15:55,910 We're happy? 304 00:15:55,910 --> 00:15:56,540 OK. 305 00:15:56,540 --> 00:15:57,390 All right. 306 00:16:07,206 --> 00:16:09,830 So I'm going to call the Young's modulus in the one direction E 307 00:16:09,830 --> 00:16:10,640 star 1. 308 00:16:10,640 --> 00:16:14,240 So everything with a star refers to a cellular solid property, 309 00:16:14,240 --> 00:16:16,550 and 1 because it's in one direction. 310 00:16:16,550 --> 00:16:21,030 So that's going to be sigma 1 over epsilon 1. 311 00:16:21,030 --> 00:16:22,870 So if I go back up there, I can say 312 00:16:22,870 --> 00:16:31,550 sigma 1 is equal to P divided by h plus l sin theta b. 313 00:16:31,550 --> 00:16:38,880 And epsilon 1 is equal to delta sine theta in the denominator 314 00:16:38,880 --> 00:16:40,030 over l cos theta. 315 00:16:45,620 --> 00:16:47,400 And now instead of having delta here, 316 00:16:47,400 --> 00:16:50,389 I can substitute this thing here in for delta. 317 00:16:50,389 --> 00:16:52,430 And then I'm going to able to cancel the P's out. 318 00:17:03,650 --> 00:17:14,430 So delta was equal to P sine theta l cubed over 12 Es, 319 00:17:14,430 --> 00:17:17,160 and there was an I, a moment of inertia, 320 00:17:17,160 --> 00:17:20,941 and I was equal to bt cubed over 12. 321 00:17:20,941 --> 00:17:23,690 And let's see here. 322 00:17:23,690 --> 00:17:25,329 So that's delta. 323 00:17:25,329 --> 00:17:26,793 And there's another sine theta here 324 00:17:26,793 --> 00:17:28,876 so I'm just going to square that sine theta there. 325 00:17:33,570 --> 00:17:36,100 So now the P's cancel out. 326 00:17:36,100 --> 00:17:38,195 The b's are going to cancel out. 327 00:17:38,195 --> 00:17:41,400 The 12s are going to cancel out. 328 00:17:41,400 --> 00:17:44,770 And I'm going to rearrange this a little bit. 329 00:17:44,770 --> 00:17:47,640 So I'm going to write Young's modulus 330 00:17:47,640 --> 00:17:50,530 of the solid out in the front. 331 00:17:50,530 --> 00:17:53,660 Then I've got a term here of t cubed 332 00:17:53,660 --> 00:17:56,879 and I'm going to multiply that by 1/l squared, 333 00:17:56,879 --> 00:17:58,670 and then everything else-- well, let's see. 334 00:17:58,670 --> 00:18:00,620 We can take this l cubed here. 335 00:18:00,620 --> 00:18:02,856 I can take that. 336 00:18:02,856 --> 00:18:04,230 Put it underneath that, so that's 337 00:18:04,230 --> 00:18:09,000 going to give me t/l cubed. 338 00:18:09,000 --> 00:18:11,249 And then I've got an h plus l sine theta here, 339 00:18:11,249 --> 00:18:12,790 and I've got an l there, so I'm going 340 00:18:12,790 --> 00:18:17,730 to take that to be h/l plus sine theta. 341 00:18:24,850 --> 00:18:25,600 Boop-boop-da-doop. 342 00:18:30,000 --> 00:18:32,760 So I've got this term with [? h/l's ?] in the thetas. 343 00:18:32,760 --> 00:18:35,570 There's a cos theta from the numerator here. 344 00:18:35,570 --> 00:18:38,554 This term here turns into h/l plus sine theta, 345 00:18:38,554 --> 00:18:40,720 and then I've got my sine squared thetas down there. 346 00:18:43,860 --> 00:18:46,950 And that's my result for the Young's modulus 347 00:18:46,950 --> 00:18:47,865 in the one direction. 348 00:18:51,000 --> 00:18:51,500 OK? 349 00:18:54,452 --> 00:18:56,330 Let's make sure that seems right. 350 00:18:56,330 --> 00:18:57,630 It seems good. 351 00:18:57,630 --> 00:18:58,700 OK. 352 00:18:58,700 --> 00:19:00,350 So one of the things to notice here 353 00:19:00,350 --> 00:19:04,260 is there's three types of parameters that are important. 354 00:19:04,260 --> 00:19:07,230 So one is the solid properties. 355 00:19:07,230 --> 00:19:11,192 So the Young's modulus of the solid comes into this. 356 00:19:11,192 --> 00:19:12,650 So the stiffness of the whole thing 357 00:19:12,650 --> 00:19:14,990 depends on the stiffness of whatever it's made from. 358 00:19:14,990 --> 00:19:18,040 There's this factor of t/l cubed-- 359 00:19:18,040 --> 00:19:21,020 that's directly related to the relative density or the volume 360 00:19:21,020 --> 00:19:23,330 fraction of solids. 361 00:19:23,330 --> 00:19:27,090 So what this is saying is the relative density goes as t/l, 362 00:19:27,090 --> 00:19:30,230 so the Young's modulus depends on the cube 363 00:19:30,230 --> 00:19:31,230 of the relative density. 364 00:19:31,230 --> 00:19:33,520 So it's very sensitive to the relative density. 365 00:19:33,520 --> 00:19:36,190 And then this factor here really is just 366 00:19:36,190 --> 00:19:39,069 a factor that depends on the cell geometry. 367 00:19:39,069 --> 00:19:41,610 Remember when we talked about the structure of the honeycomb, 368 00:19:41,610 --> 00:19:44,770 we said we could define the cell geometry by the ratio of h/l 369 00:19:44,770 --> 00:19:50,160 and theta, OK? 370 00:19:50,160 --> 00:19:54,260 And since we often deal with regular hexagonal honeycombs, 371 00:19:54,260 --> 00:19:56,030 I'm just going to write down what 372 00:19:56,030 --> 00:19:59,890 this works out to be for regular hexagonal honeycombs. 373 00:20:17,630 --> 00:20:20,640 So for a regular hexagonal honeycomb, h/l is 1. 374 00:20:20,640 --> 00:20:22,940 All the members have the same length. 375 00:20:22,940 --> 00:20:25,300 And theta's 3, and the modulus works out 376 00:20:25,300 --> 00:20:30,510 to 4 over root 3 times Es times t/l cubed, OK? 377 00:20:30,510 --> 00:20:32,080 So do you see how we do these things? 378 00:20:32,080 --> 00:20:35,460 So all the other properties work in a similar kind of way. 379 00:20:35,460 --> 00:20:38,050 You have to say something about what the sort of bulk stress 380 00:20:38,050 --> 00:20:39,770 is on the whole thing and relate that 381 00:20:39,770 --> 00:20:41,880 to the loads on the members. 382 00:20:41,880 --> 00:20:44,559 You have to say something about how the loads are related 383 00:20:44,559 --> 00:20:46,600 to deflections, or when we look at the strengths, 384 00:20:46,600 --> 00:20:47,860 we're going to look at moments and how 385 00:20:47,860 --> 00:20:49,480 the moments are related to failure 386 00:20:49,480 --> 00:20:51,170 moments of one sort or another. 387 00:20:51,170 --> 00:20:55,010 But it's all just like a little structural analysis, OK? 388 00:20:55,010 --> 00:20:56,046 Are we good? 389 00:20:56,046 --> 00:20:56,935 You good, Teddy? 390 00:20:56,935 --> 00:20:58,810 I thought you were going to put your hand up? 391 00:20:58,810 --> 00:20:59,690 No? 392 00:20:59,690 --> 00:21:00,210 You're OK? 393 00:21:00,210 --> 00:21:00,710 OK. 394 00:21:03,420 --> 00:21:06,850 OK, so the next property we're going to look at 395 00:21:06,850 --> 00:21:08,056 is Poisson's ratio. 396 00:21:14,179 --> 00:21:16,720 And I'm going to look at it for loading in the one direction. 397 00:21:36,250 --> 00:21:41,029 So Poisson's 1 2, say we load uniaxially 398 00:21:41,029 --> 00:21:42,570 in the one direction, we want to know 399 00:21:42,570 --> 00:21:44,700 what the strain is in the two direction, 400 00:21:44,700 --> 00:21:47,840 it's minus epsilon 2 over epsilon 1. 401 00:21:47,840 --> 00:21:51,610 And again, if I look at my inclined member, 402 00:21:51,610 --> 00:21:56,450 and I say that member's going to bend something like that, 403 00:21:56,450 --> 00:22:01,226 and that's my deflection delta there, and, say, 404 00:22:01,226 --> 00:22:04,650 got the same x1 and x2 axes. 405 00:22:04,650 --> 00:22:07,112 And again, if I look at delta here, 406 00:22:07,112 --> 00:22:08,980 it's the same little sketch I had before. 407 00:22:08,980 --> 00:22:12,540 That's delta sine theta. 408 00:22:12,540 --> 00:22:14,952 And this is delta cos theta. 409 00:22:14,952 --> 00:22:16,660 I'm going to need those components to get 410 00:22:16,660 --> 00:22:20,920 the two strains in the different directions. 411 00:22:20,920 --> 00:22:27,144 So epsilon 1 is going to be delta sine theta over l cos 412 00:22:27,144 --> 00:22:28,380 theta. 413 00:22:28,380 --> 00:22:32,660 And if I'm compressing it, that would get shorter. 414 00:22:32,660 --> 00:22:39,660 And we get-- and epsilon 2 is going 415 00:22:39,660 --> 00:22:49,520 to be delta cos theta divided by h plus l sine theta. 416 00:22:49,520 --> 00:22:50,797 And that would get longer. 417 00:22:58,000 --> 00:22:59,650 So these two have opposite signs, 418 00:22:59,650 --> 00:23:01,845 and so the minus sign is going to disappear here. 419 00:23:06,249 --> 00:23:07,040 Doodle-doodle-doot. 420 00:23:10,340 --> 00:23:11,859 So then I can get my Poisson's ratio 421 00:23:11,859 --> 00:23:13,650 by just taking the ratio of those two guys. 422 00:23:16,810 --> 00:23:18,560 So I could put a minus sign there 423 00:23:18,560 --> 00:23:22,390 and say that's the opposite sign to epsilon 2. 424 00:23:22,390 --> 00:23:29,820 Then this would be delta cos theta divided by h plus l sine 425 00:23:29,820 --> 00:23:30,990 theta. 426 00:23:30,990 --> 00:23:41,320 And epsilon 1 would be delta sine theta over l cos theta. 427 00:23:41,320 --> 00:23:43,550 And the thing that's convenient here 428 00:23:43,550 --> 00:23:45,810 is that the two deltas just cancel out. 429 00:23:45,810 --> 00:23:48,939 So the Poisson's ratio is the ratio of two strains. 430 00:23:48,939 --> 00:23:51,480 Each one of the strains is going to be proportional to delta, 431 00:23:51,480 --> 00:23:54,170 and so the two deltas are just going to cancel out. 432 00:23:54,170 --> 00:23:57,080 And so I can rewrite this thing here 433 00:23:57,080 --> 00:24:04,810 as cos squared theta divided by h/l plus sine 434 00:24:04,810 --> 00:24:10,980 theta times sine theta. 435 00:24:10,980 --> 00:24:12,480 And so one of the interesting things 436 00:24:12,480 --> 00:24:14,370 to notice that the Poisson's ratio only 437 00:24:14,370 --> 00:24:16,070 depends on the cell geometry. 438 00:24:16,070 --> 00:24:19,380 It doesn't depend on what solid the material is made from. 439 00:24:19,380 --> 00:24:21,550 It doesn't depend on the relative density. 440 00:24:21,550 --> 00:24:26,015 It only depends on the cell geometry. 441 00:24:49,387 --> 00:24:49,887 Oops. 442 00:25:25,790 --> 00:25:27,270 OK, and then we can also work out 443 00:25:27,270 --> 00:25:29,925 what the value is for a regular hexagonal cell. 444 00:25:36,610 --> 00:25:41,790 And if we plug-in h is equal to l and theta's equal to 30, 445 00:25:41,790 --> 00:25:43,670 you get that it's equal to 1. 446 00:25:43,670 --> 00:25:47,140 So one is kind of an unusual number for a Poisson's ratio. 447 00:25:47,140 --> 00:25:50,320 When we think of most materials, it's around 0.3, 448 00:25:50,320 --> 00:25:52,444 so it's kind of unusual that it's that large. 449 00:25:52,444 --> 00:25:53,860 The other thing that's interesting 450 00:25:53,860 --> 00:25:55,950 is that it can be negative. 451 00:26:06,950 --> 00:26:10,675 So if theta is less than 0, then you can get a negative value. 452 00:26:10,675 --> 00:26:12,880 If the cos squared is going to be a positive value, 453 00:26:12,880 --> 00:26:15,500 but you've got a sine theta down here, then that's going 454 00:26:15,500 --> 00:26:17,610 to give you a negative value. 455 00:26:17,610 --> 00:26:20,000 So you can get negative values. 456 00:26:20,000 --> 00:26:23,280 So let me just plug in an example. 457 00:26:23,280 --> 00:26:27,020 So say h/l is equal to 2, and theta 458 00:26:27,020 --> 00:26:36,170 is equal to minus 30 degrees, then this turns out to be 3/4. 459 00:26:36,170 --> 00:26:41,850 So cos of 30 is root 3/2, so square of that is 3/4. 460 00:26:41,850 --> 00:26:45,510 h/l is 2, sine theta is 1/2, but it's minus 1/2. 461 00:26:45,510 --> 00:26:49,950 So 2 minus 1/2 is 1 and 1/2. 462 00:26:49,950 --> 00:26:53,810 And then the sine theta is minus 1/2. 463 00:26:53,810 --> 00:26:55,380 And so it works out to be minus 1 464 00:26:55,380 --> 00:26:59,040 for that particular combination. 465 00:26:59,040 --> 00:27:00,800 And I brought my little honeycomb 466 00:27:00,800 --> 00:27:03,480 that has a negative Poisson's ratio in. 467 00:27:03,480 --> 00:27:05,079 So this guy here-- let's see, I don't 468 00:27:05,079 --> 00:27:06,370 think there's an overhead here. 469 00:27:06,370 --> 00:27:06,960 No overhead? 470 00:27:06,960 --> 00:27:07,660 Guess not. 471 00:27:07,660 --> 00:27:08,740 I'll just pass it around. 472 00:27:08,740 --> 00:27:12,240 So if you take it, put your hands on the flat side 473 00:27:12,240 --> 00:27:14,449 and load it like this, and don't smoosh it like that. 474 00:27:14,449 --> 00:27:16,740 Just load it a little bit, because you [? want to be ?] 475 00:27:16,740 --> 00:27:17,400 linear elastic. 476 00:27:17,400 --> 00:27:18,550 If you load it just a little bit, 477 00:27:18,550 --> 00:27:20,258 you can see that as you push it this way, 478 00:27:20,258 --> 00:27:22,150 it contracts in sideways that way. 479 00:27:22,150 --> 00:27:22,904 So don't smash it. 480 00:27:22,904 --> 00:27:25,070 Just load it a little bit and you can kind of see it 481 00:27:25,070 --> 00:27:25,980 with your hands. 482 00:27:25,980 --> 00:27:27,780 And if you put on a piece of lined paper, 483 00:27:27,780 --> 00:27:30,250 it's easier to see it. 484 00:27:30,250 --> 00:27:33,510 OK, so that's kind of interesting. 485 00:27:33,510 --> 00:27:36,216 So are we good with getting the Young's modulus in the one 486 00:27:36,216 --> 00:27:37,590 direction and the Poisson's ratio 487 00:27:37,590 --> 00:27:39,230 for loading in one direction? 488 00:27:39,230 --> 00:27:41,040 OK, so you can do the same sort of thing 489 00:27:41,040 --> 00:27:43,090 to get the Young's modulus in the two 490 00:27:43,090 --> 00:27:44,530 direction and the Poisson's ratio 491 00:27:44,530 --> 00:27:45,620 for loading in the two direction. 492 00:27:45,620 --> 00:27:47,286 And you get slightly different formulas, 493 00:27:47,286 --> 00:27:49,280 but it's the same idea. 494 00:27:49,280 --> 00:27:51,220 And you can also get a shear modulus this way, 495 00:27:51,220 --> 00:27:52,604 and in-plane shear modulus. 496 00:27:52,604 --> 00:27:55,020 It's a little bit-- the geometry of it's a little bit more 497 00:27:55,020 --> 00:27:55,950 complicated. 498 00:27:55,950 --> 00:27:58,590 So all of those things are derived in the book, 499 00:27:58,590 --> 00:27:59,830 in the cellular solids book. 500 00:27:59,830 --> 00:28:02,500 So if you wanted to figure those out, look at that, 501 00:28:02,500 --> 00:28:03,920 you could look at the book. 502 00:28:03,920 --> 00:28:06,149 So let me just comment on that. 503 00:28:36,570 --> 00:28:42,760 All right, so those are the in-plane linear elastic moduli, 504 00:28:42,760 --> 00:28:47,320 and remember we said that four of them 505 00:28:47,320 --> 00:28:51,410 describe the in-plane properties for an anisotropic honeycomb. 506 00:28:51,410 --> 00:28:53,830 And you can use that reciprocal relationship 507 00:28:53,830 --> 00:28:57,985 to relate the two Young's moduli and the two Poisson's ratios. 508 00:28:57,985 --> 00:29:00,110 All right, so the next thing I wanted to talk about 509 00:29:00,110 --> 00:29:01,580 was the compressive strength. 510 00:29:01,580 --> 00:29:05,000 So let me just back up here a second. 511 00:29:05,000 --> 00:29:06,860 So if we go back to here, remember 512 00:29:06,860 --> 00:29:09,780 we had for an elastomeric honeycomb, 513 00:29:09,780 --> 00:29:12,480 this stress plateau was related to elastic buckling. 514 00:29:12,480 --> 00:29:15,430 So we're going to look at that buckling stress first. 515 00:29:15,430 --> 00:29:17,700 And this plateau here is related to yielding. 516 00:29:17,700 --> 00:29:20,480 And then we'll look at the yielding stress next. 517 00:29:20,480 --> 00:29:22,690 And then this plateau here is related 518 00:29:22,690 --> 00:29:27,600 to a brittle sort of crushing, and we'll do that one third. 519 00:29:27,600 --> 00:29:30,400 So we're going to go through each of those next. 520 00:29:30,400 --> 00:29:33,612 And this is kind of a schematic for the elastic buckling. 521 00:29:33,612 --> 00:29:35,320 So when you look at the elastic buckling, 522 00:29:35,320 --> 00:29:37,370 one of the things to note is that when 523 00:29:37,370 --> 00:29:38,850 you load the honeycomb this way on, 524 00:29:38,850 --> 00:29:40,350 if you load it in the one direction, 525 00:29:40,350 --> 00:29:41,330 you don't get buckling. 526 00:29:41,330 --> 00:29:43,115 It just sort of continues to-- whoops, 527 00:29:43,115 --> 00:29:44,742 if I can keep it in plane. 528 00:29:44,742 --> 00:29:46,450 It all just kind of folds up, so you just 529 00:29:46,450 --> 00:29:48,240 get larger and larger bending deflections. 530 00:29:48,240 --> 00:29:49,510 You don't really get buckling. 531 00:29:49,510 --> 00:29:52,620 But when you load it this way on, these vertical members 532 00:29:52,620 --> 00:29:55,280 here, the ones of length h, they're going to buckle. 533 00:29:55,280 --> 00:29:59,320 So, see if I do that, my honeycomb looks 534 00:29:59,320 --> 00:30:04,170 like those cells up on the schematic there, OK? 535 00:30:04,170 --> 00:30:05,594 So, whoops. 536 00:30:13,788 --> 00:30:16,571 So we're going to look at the compressive stress or strength 537 00:30:16,571 --> 00:30:17,070 next. 538 00:30:27,040 --> 00:30:28,871 That's sometimes called the plateau stress. 539 00:30:43,540 --> 00:30:46,180 So we can get cell collapse by elastic buckling, 540 00:30:46,180 --> 00:30:49,800 if, for instance, the honeycomb is made of a polymer. 541 00:30:49,800 --> 00:30:52,790 And then the stress-strain curve looks something like that. 542 00:30:52,790 --> 00:30:54,550 And what happens is you get buckling 543 00:30:54,550 --> 00:31:28,270 of those vertical struts throughout the honeycomb 544 00:31:28,270 --> 00:31:35,130 And then you could also get a stress plateau 545 00:31:35,130 --> 00:31:36,290 by plastic yielding. 546 00:31:41,094 --> 00:31:43,010 And what happens when you get plastic yielding 547 00:31:43,010 --> 00:31:45,100 is you get localization of the deformation. 548 00:31:45,100 --> 00:31:48,390 So one band of cells will begin to yield initially, and then 549 00:31:48,390 --> 00:31:51,780 as the deformation proceeds, that deformation ban will 550 00:31:51,780 --> 00:31:53,690 propagate and get bigger and bigger, 551 00:31:53,690 --> 00:31:55,310 and you get a wider and wider band 552 00:31:55,310 --> 00:31:57,430 of cells yielding and failing. 553 00:32:04,100 --> 00:32:06,300 So you get localization of yield, 554 00:32:06,300 --> 00:32:12,870 and then as deformation progresses, 555 00:32:12,870 --> 00:32:16,010 the deformation band widens throughout the material. 556 00:32:27,310 --> 00:32:31,110 So if I go back and look-- if I look at this one 557 00:32:31,110 --> 00:32:35,510 here, when you look at this middle picture here, 558 00:32:35,510 --> 00:32:38,430 you can see how one band of cells has started to collapse 559 00:32:38,430 --> 00:32:39,880 and started to fail. 560 00:32:39,880 --> 00:32:42,820 And as you continue to compress that in the one direction, 561 00:32:42,820 --> 00:32:45,382 this way on, then more and more neighboring cells 562 00:32:45,382 --> 00:32:47,090 are going to collapse and the whole thing 563 00:32:47,090 --> 00:32:50,136 will get wider until the whole thing has collapsed. 564 00:32:50,136 --> 00:32:51,510 And that's kind of characteristic 565 00:32:51,510 --> 00:32:53,040 of the plastic failure. 566 00:32:57,040 --> 00:33:00,530 And then the third possibility is brittle crushing. 567 00:33:16,510 --> 00:33:19,580 And then you get these kind of serrated plateau. 568 00:33:19,580 --> 00:33:22,540 And the peaks and valleys correspond to fractures 569 00:33:22,540 --> 00:33:23,920 of individual cell walls. 570 00:34:31,731 --> 00:34:33,230 OK, so we're going to start off with 571 00:34:33,230 --> 00:34:36,120 the elastic buckling failure. 572 00:34:56,980 --> 00:35:00,160 And I'm going to call these plateau stresses sigma star, 573 00:35:00,160 --> 00:35:02,130 for the sort of compressive strength. 574 00:35:02,130 --> 00:35:04,640 And el means it's by elastic buckling. 575 00:35:04,640 --> 00:35:07,260 And as I mentioned, you don't get it in the one direction. 576 00:35:07,260 --> 00:35:08,270 The cells just fold up. 577 00:35:08,270 --> 00:35:10,311 You only get it for loading in the two direction, 578 00:35:10,311 --> 00:35:11,930 so it's going to be sigma star el 2. 579 00:35:16,820 --> 00:35:19,680 Oops, need a different piece of chalk. 580 00:36:35,910 --> 00:36:38,270 So you get this elastic buckling for loading 581 00:36:38,270 --> 00:36:42,700 in the x2 direction, and the cell walls of length h buckle. 582 00:36:42,700 --> 00:36:44,950 And you don't get it for loading in the one direction, 583 00:36:44,950 --> 00:36:46,710 the cells just fold up. 584 00:36:46,710 --> 00:36:50,040 So again, let me draw a little kind of unit cell here. 585 00:37:01,930 --> 00:37:07,370 And here is our stress sigma 2, like that. 586 00:37:07,370 --> 00:37:10,650 And here's our little wall of length h 587 00:37:10,650 --> 00:37:12,050 that's going to buckle. 588 00:37:12,050 --> 00:37:14,480 So if I load it up, initially it'll be linear elastic. 589 00:37:14,480 --> 00:37:16,120 And then eventually, at some stress, 590 00:37:16,120 --> 00:37:18,970 it will get large enough that this wall here will buckle. 591 00:37:18,970 --> 00:37:22,500 And we can relate to that plateau 592 00:37:22,500 --> 00:37:24,230 stress, or that compressive stress, 593 00:37:24,230 --> 00:37:25,750 to some Euler buckling load. 594 00:37:30,350 --> 00:37:33,390 So you remember, if we have a pin-ended column, so just 595 00:37:33,390 --> 00:37:36,030 a single column, pins on either end, 596 00:37:36,030 --> 00:37:37,870 the Euler buckling load says you get 597 00:37:37,870 --> 00:37:42,820 buckling when the critical load is equal to some end constraint 598 00:37:42,820 --> 00:37:44,350 factor, n squared. 599 00:37:44,350 --> 00:37:50,330 So n squared pi squared E, and here it's E of the solid, 600 00:37:50,330 --> 00:37:54,410 I over the length of the column, and in this case, 601 00:37:54,410 --> 00:37:57,132 the column length is h-- so h squared. 602 00:37:57,132 --> 00:38:00,310 OK, so that's just the Euler formula. 603 00:38:00,310 --> 00:38:02,480 And here, n is an end constraint factor. 604 00:38:10,164 --> 00:38:15,552 And if you remember for a pin column, 605 00:38:15,552 --> 00:38:19,020 so if our column is pinned at both ends like that, 606 00:38:19,020 --> 00:38:24,150 and just buckles out like that, then n is equal to 1. 607 00:38:24,150 --> 00:38:30,970 And if the column is fixed at both ends, 608 00:38:30,970 --> 00:38:35,530 something like that, then the column looks like that 609 00:38:35,530 --> 00:38:39,700 and then it's equal to 2, OK? 610 00:38:39,700 --> 00:38:43,680 So if I know what the end condition is, I know what n is 611 00:38:43,680 --> 00:38:46,700 and I can use my Euler formula here. 612 00:38:46,700 --> 00:38:50,575 So the trick to this is that it's not so obvious what n is. 613 00:38:50,575 --> 00:38:51,504 Yes? 614 00:38:51,504 --> 00:38:54,468 AUDIENCE: So, when you're loading in the x2 direction 615 00:38:54,468 --> 00:38:56,410 here, the first thing you're going to get 616 00:38:56,410 --> 00:38:58,089 is the incline members deforming? 617 00:38:58,089 --> 00:38:58,880 LORNA GIBSON: Yeah. 618 00:38:58,880 --> 00:39:00,440 AUDIENCE: And then at some point, 619 00:39:00,440 --> 00:39:02,682 you hit a P critical that will cause 620 00:39:02,682 --> 00:39:04,250 the vertical members to buckle? 621 00:39:04,250 --> 00:39:04,730 LORNA GIBSON: Exactly. 622 00:39:04,730 --> 00:39:04,995 AUDIENCE: OK. 623 00:39:04,995 --> 00:39:05,911 LORNA GIBSON: Exactly. 624 00:39:05,911 --> 00:39:08,480 That's exactly right. 625 00:39:08,480 --> 00:39:10,492 Hello. 626 00:39:10,492 --> 00:39:12,200 So the trick here is that we don't really 627 00:39:12,200 --> 00:39:13,940 know what this n is, initially. 628 00:39:13,940 --> 00:39:15,420 They're not really pinned, pinned; 629 00:39:15,420 --> 00:39:16,720 they're not fixed, fixed. 630 00:39:16,720 --> 00:39:20,220 And if you think about the setup with the honeycomb here, 631 00:39:20,220 --> 00:39:22,800 the constraint on that vertical member 632 00:39:22,800 --> 00:39:25,610 depends on how stiff the adjacent members are. 633 00:39:25,610 --> 00:39:27,284 So you can kind of imagine, if I'm 634 00:39:27,284 --> 00:39:28,950 looking at one of these vertical members 635 00:39:28,950 --> 00:39:32,520 here, if these two adjacent inclined members were 636 00:39:32,520 --> 00:39:36,240 big honking thick things, it would be more constrained. 637 00:39:36,240 --> 00:39:39,200 And if they were little thin, kind of teeny little membranes, 638 00:39:39,200 --> 00:39:40,800 it would be less constrained. 639 00:39:40,800 --> 00:39:43,990 And you can think of it in terms of a rotational stiffness, 640 00:39:43,990 --> 00:39:47,130 that when the honeycomb buckles, you kind of 641 00:39:47,130 --> 00:39:50,930 see the member h goes from being horizontal to sort 642 00:39:50,930 --> 00:39:53,190 of it buckles over like this. 643 00:39:53,190 --> 00:39:55,340 But that whole end joint, see the end joint 644 00:39:55,340 --> 00:39:57,880 at the top here or the end joint at the bottom, that 645 00:39:57,880 --> 00:39:59,850 whole joint rotates a little bit. 646 00:39:59,850 --> 00:40:03,150 And so there's some rotational stiffness of that joint. 647 00:40:03,150 --> 00:40:06,450 And that rotational stiffness depends on how stiff 648 00:40:06,450 --> 00:40:10,020 the member h is and how stiff those inclined members are. 649 00:40:10,020 --> 00:40:14,790 So there's a thing called the elastic line analysis that you 650 00:40:14,790 --> 00:40:16,510 can use to calculate what n is. 651 00:40:16,510 --> 00:40:18,520 And basically what that does is it matches 652 00:40:18,520 --> 00:40:21,490 the rotational stiffness of the column h 653 00:40:21,490 --> 00:40:24,000 with the rotational stiffness of those inclined members. 654 00:40:24,000 --> 00:40:25,240 So we're not going to get into that. 655 00:40:25,240 --> 00:40:26,660 I'm just going to tell you what the answer is. 656 00:40:26,660 --> 00:40:28,230 But if you want to go through it, 657 00:40:28,230 --> 00:40:29,950 it's in an appendix in the book. 658 00:40:29,950 --> 00:40:31,720 So you can look at it, if you want. 659 00:40:35,920 --> 00:40:46,260 So here I'm just going to say that the constraint 660 00:40:46,260 --> 00:40:51,200 n depends on the stiffness of the adjacent inclined members. 661 00:41:16,532 --> 00:41:18,200 And we can find that by something called 662 00:41:18,200 --> 00:41:19,600 the elastic line analysis. 663 00:41:26,700 --> 00:41:29,020 And if you have the book, you can look in the appendix 664 00:41:29,020 --> 00:41:32,360 and see how that works. 665 00:41:32,360 --> 00:41:33,930 But essentially what it does is it 666 00:41:33,930 --> 00:41:38,100 matches the rotational stiffness of the column h 667 00:41:38,100 --> 00:41:40,810 with the rotational stiffness of the inclined members. 668 00:42:24,130 --> 00:42:28,860 So what you find is that n depends on the ratio of h/l. 669 00:42:28,860 --> 00:42:32,710 And I'm just going to give you a table with a few values. 670 00:42:32,710 --> 00:42:39,210 So for h/l equal to 1, then n is equal to 0.686. 671 00:42:39,210 --> 00:42:44,890 For h equal to 1.5, it's equal to 0.76. 672 00:42:44,890 --> 00:42:52,810 And for h/l equal to 2, it's equal to 0.806. 673 00:42:52,810 --> 00:42:57,460 OK, so now if we have values for n, 674 00:42:57,460 --> 00:43:01,689 we can just substitute in to get the critical buckling load. 675 00:43:01,689 --> 00:43:03,230 And if I take that load and divide it 676 00:43:03,230 --> 00:43:05,260 by the area of the unit cell, I'm 677 00:43:05,260 --> 00:43:06,650 going to get my buckling stress. 678 00:43:06,650 --> 00:43:08,890 So it's pretty straightforward from this. 679 00:43:49,360 --> 00:43:54,180 So my buckling stress is going to be that critical load 680 00:43:54,180 --> 00:44:00,250 divided by my unit cell area. 681 00:44:00,250 --> 00:44:03,260 So it's divided by the unit cell length in the x1 direction 682 00:44:03,260 --> 00:44:06,477 to l cos theta times the depth b into the page. 683 00:44:25,610 --> 00:44:29,860 So it's equal to n squared pi squared Es times I. 684 00:44:29,860 --> 00:44:32,510 And I is bt cubed over 12. 685 00:44:32,510 --> 00:44:35,910 Divided by the length of the column, h squared, 686 00:44:35,910 --> 00:44:39,769 and then divided by the area of the unit cell, 2l cos theta b, 687 00:44:39,769 --> 00:44:40,268 OK? 688 00:44:43,020 --> 00:44:45,510 And I can rearrange that somewhat 689 00:44:45,510 --> 00:44:48,750 to put it in terms of dimensionless groups. 690 00:44:54,320 --> 00:44:55,990 So if I pull all the constants out, 691 00:44:55,990 --> 00:45:00,720 it's n squared pi squared over 24 times the modulus 692 00:45:00,720 --> 00:45:08,020 of the solid, t/l cubed in the numerator divided 693 00:45:08,020 --> 00:45:14,231 by h/l squared times cos theta in the denominator. 694 00:45:23,870 --> 00:45:26,080 So again, you can see that the buckling stress, 695 00:45:26,080 --> 00:45:29,900 the compressive sort of elastic collapse stress, 696 00:45:29,900 --> 00:45:31,540 depends on the solid property. 697 00:45:31,540 --> 00:45:34,300 So here is the modulus of the cell wall in here. 698 00:45:34,300 --> 00:45:38,860 Depends on the relative density through t/l cubed. 699 00:45:38,860 --> 00:45:43,230 And then it depends on the cell geometry through h/l cos theta, 700 00:45:43,230 --> 00:45:46,176 and n depends on h/l as well, OK? 701 00:45:48,756 --> 00:45:50,130 And then we can do the same thing 702 00:45:50,130 --> 00:45:52,588 where we figure out what it is for regular hexagonal cells. 703 00:46:07,200 --> 00:46:12,470 And it's 0.22 Es times t/l cubed. 704 00:46:12,470 --> 00:46:18,530 And then we can also notice that since E in the 2 direction, 705 00:46:18,530 --> 00:46:21,421 for a regular hexagonal cell, E is the same in the 2 706 00:46:21,421 --> 00:46:22,670 direction and the 1 direction. 707 00:46:22,670 --> 00:46:24,140 It's isotropic. 708 00:46:24,140 --> 00:46:31,280 So E2 is also equal to 4 over root 3 Es times t/l cubed. 709 00:46:31,280 --> 00:46:41,890 That's equal to-- whoops-- it's equals to 2.31 Es t/l cubed. 710 00:46:41,890 --> 00:46:45,869 And we can say that the strain at which that buckling happens 711 00:46:45,869 --> 00:46:47,035 is just equal to a constant. 712 00:46:55,680 --> 00:46:59,110 And for regular hexagonal honeycombs, 713 00:46:59,110 --> 00:47:00,650 it works out to a strain of 10%. 714 00:47:17,880 --> 00:47:18,785 Are we good? 715 00:47:18,785 --> 00:47:19,910 So we have a buckling load. 716 00:47:19,910 --> 00:47:21,580 We divide by the area. 717 00:47:21,580 --> 00:47:24,070 The only complicated thing is finding n. 718 00:47:24,070 --> 00:47:28,500 And you can find it by this elastic line analysis thing. 719 00:47:28,500 --> 00:47:30,620 So each of these calculations is like 720 00:47:30,620 --> 00:47:33,000 a little structural analysis, only 721 00:47:33,000 --> 00:47:35,875 on a little teeny weeny scale of the cells. 722 00:47:35,875 --> 00:47:38,000 So you see where my background in civil engineering 723 00:47:38,000 --> 00:47:40,320 comes in handy. 724 00:47:40,320 --> 00:47:41,270 Yup. 725 00:47:41,270 --> 00:47:41,770 OK. 726 00:47:47,770 --> 00:47:50,980 So the honeycombs involve the most sort 727 00:47:50,980 --> 00:47:53,380 of complicated equations. 728 00:47:53,380 --> 00:47:55,110 When we come to do the foams, we're 729 00:47:55,110 --> 00:47:57,230 going to use a dimensional analysis 730 00:47:57,230 --> 00:48:00,430 and all the equations are going to be much simpler. 731 00:48:00,430 --> 00:48:02,430 So this is the most kind of tedious part 732 00:48:02,430 --> 00:48:05,200 of the whole thing. 733 00:48:05,200 --> 00:48:07,100 So the next property I want to look at 734 00:48:07,100 --> 00:48:10,300 is the plastic collapse stress. 735 00:48:10,300 --> 00:48:13,430 Say we had a metal honeycomb and we 736 00:48:13,430 --> 00:48:18,470 wanted to calculate the stress plateau for a metal honeycomb. 737 00:48:18,470 --> 00:48:20,930 So we have this little schematic here, 738 00:48:20,930 --> 00:48:24,500 and say we load it in the one direction again. 739 00:48:24,500 --> 00:48:27,100 So we're loading it here. 740 00:48:27,100 --> 00:48:30,090 And we've got some load P, like that. 741 00:48:30,090 --> 00:48:33,830 And if we have our honeycomb, we load it this way on, initially, 742 00:48:33,830 --> 00:48:35,567 the cell walls bend. 743 00:48:35,567 --> 00:48:37,150 And you have linear elasticity and you 744 00:48:37,150 --> 00:48:38,305 have some Young's modulus. 745 00:48:38,305 --> 00:48:41,402 But if you have a metal, if you continue to deform it 746 00:48:41,402 --> 00:48:43,600 and you continue to load it more and more, 747 00:48:43,600 --> 00:48:46,101 eventually you're going to hit the yield stress and the cell 748 00:48:46,101 --> 00:48:46,600 wall. 749 00:48:46,600 --> 00:48:48,240 So the stresses in the cell wall are 750 00:48:48,240 --> 00:48:49,770 going to hit the yield stress. 751 00:48:49,770 --> 00:48:52,089 And initially, the stresses are just 752 00:48:52,089 --> 00:48:53,880 going to be-- remember, if you have a beam, 753 00:48:53,880 --> 00:48:56,701 the stresses are maximum at the top and the bottom of the beam. 754 00:48:56,701 --> 00:48:58,450 So initially you're going to hit the yield 755 00:48:58,450 --> 00:49:00,707 stress at the top and the bottom of the beam first. 756 00:49:00,707 --> 00:49:02,290 But as you continue to load it, you're 757 00:49:02,290 --> 00:49:04,730 going to end up yielding the cross-section 758 00:49:04,730 --> 00:49:06,470 through the entire section. 759 00:49:06,470 --> 00:49:08,770 So the entire section is going to be yielded. 760 00:49:08,770 --> 00:49:10,970 And once the entire section yields, 761 00:49:10,970 --> 00:49:13,440 it forms what's called a plastic hinge. 762 00:49:13,440 --> 00:49:16,830 Once the whole thing's yielded, then you can add more force 763 00:49:16,830 --> 00:49:18,280 and the thing just rotates. 764 00:49:18,280 --> 00:49:20,801 And because it rotates, it's called a plastic hinge. 765 00:49:20,801 --> 00:49:22,300 You know, if you take a coat hanger, 766 00:49:22,300 --> 00:49:24,220 and you bend it back and forth and bend it back and forth. 767 00:49:24,220 --> 00:49:27,320 If you bend it enough, you form a plastic hinge because it just 768 00:49:27,320 --> 00:49:31,640 can bend easily. 769 00:49:31,640 --> 00:49:36,250 So these little schematics here, if you look at the, 770 00:49:36,250 --> 00:49:37,900 say, one of these inclined members, 771 00:49:37,900 --> 00:49:39,575 the moments are maximum at the end. 772 00:49:39,575 --> 00:49:42,387 So Remember when we had the linear elastic deformation 773 00:49:42,387 --> 00:49:44,470 and I looked at the little bending moment diagram? 774 00:49:44,470 --> 00:49:46,590 The moments are maximum at the ends, 775 00:49:46,590 --> 00:49:49,180 and you're going to form those plastic hinges initially 776 00:49:49,180 --> 00:49:50,330 at the ends. 777 00:49:50,330 --> 00:49:54,100 And so these little ellipsey things here, all the ends, 778 00:49:54,100 --> 00:49:57,510 those kind of show where the plastic hinges are. 779 00:49:57,510 --> 00:49:59,010 So those plastic hinges are forming. 780 00:49:59,010 --> 00:50:01,250 So here's for loading in the x1 direction, 781 00:50:01,250 --> 00:50:05,440 and here's for loading in the x2 direction, there. 782 00:50:05,440 --> 00:50:07,200 So the thing we want to calculate 783 00:50:07,200 --> 00:50:10,860 is what stress does it take to form those plastic hinges 784 00:50:10,860 --> 00:50:14,510 and get this kind of plastic plateau stress? 785 00:50:40,090 --> 00:50:44,930 OK, so we can say we get failure by yielding in the cell walls. 786 00:50:56,967 --> 00:50:59,300 And I'm going to say the yield strength of the cell wall 787 00:50:59,300 --> 00:51:00,020 is sigma ys. 788 00:51:09,820 --> 00:51:12,830 So sigma y for yield and s for the solid. 789 00:51:12,830 --> 00:51:16,170 And the plastic hinge forms when the cross-section has fully 790 00:51:16,170 --> 00:51:16,670 yielded. 791 00:51:53,930 --> 00:51:56,610 So let's look at the stress distribution 792 00:51:56,610 --> 00:51:59,940 through the cross-section when its first linear elastic. 793 00:51:59,940 --> 00:52:05,100 So say that's the thickness t of the member. 794 00:52:05,100 --> 00:52:08,030 And if the beam was linear elastic, 795 00:52:08,030 --> 00:52:11,820 the stress would just [? vary ?] linearly, like that, right? 796 00:52:11,820 --> 00:52:15,190 And this would be the neutral axis, here, 797 00:52:15,190 --> 00:52:16,818 where there is no normal stress. 798 00:52:20,870 --> 00:52:23,320 So that's what happens if it's linear elastic, 799 00:52:23,320 --> 00:52:26,820 and I'm hoping you remember something vaguely like that. 800 00:52:26,820 --> 00:52:27,412 Sounds good? 801 00:52:30,180 --> 00:52:31,890 But as we increase the load on it, 802 00:52:31,890 --> 00:52:34,090 and we increase the sort of external stress, 803 00:52:34,090 --> 00:52:37,290 this stress in the member is going to get bigger and bigger, 804 00:52:37,290 --> 00:52:39,860 and eventually, that's going to reach the yield stress, OK? 805 00:52:39,860 --> 00:52:42,620 And once that reaches the yield stress, 806 00:52:42,620 --> 00:52:44,560 if we continue to load it, what happens 807 00:52:44,560 --> 00:52:47,220 is the yielding propagates down through the thickness 808 00:52:47,220 --> 00:52:48,170 of the thing here. 809 00:52:48,170 --> 00:52:51,270 So we get yielding through the whole cross-section. 810 00:52:51,270 --> 00:52:54,590 So let me scoot over here. 811 00:52:54,590 --> 00:52:55,570 AUDIENCE: Professor? 812 00:52:55,570 --> 00:52:56,320 LORNA GIBSON: Yup? 813 00:52:56,320 --> 00:53:00,070 AUDIENCE: When it starts to yield, does this curve change? 814 00:53:00,070 --> 00:53:00,820 LORNA GIBSON: Yes. 815 00:53:00,820 --> 00:53:01,770 I'm going to draw it for you. 816 00:53:01,770 --> 00:53:02,670 AUDIENCE: Oh, OK. 817 00:53:02,670 --> 00:53:05,040 LORNA GIBSON: That's the next step. 818 00:53:05,040 --> 00:53:06,367 That would be the next thing. 819 00:53:29,520 --> 00:53:33,190 OK, so once the stress at the outer fiber 820 00:53:33,190 --> 00:53:59,450 is the yield strength of the solid, 821 00:53:59,450 --> 00:54:03,524 then the yielding begins and it progresses through the section 822 00:54:03,524 --> 00:54:04,440 as the load increases. 823 00:54:35,900 --> 00:54:38,180 So the stress distribution starts to look something 824 00:54:38,180 --> 00:54:39,589 like this once it yields. 825 00:54:43,990 --> 00:54:47,770 OK, so that's sigma y of the solid. 826 00:54:47,770 --> 00:54:50,150 Actually, let me rub that out because then I 827 00:54:50,150 --> 00:54:53,110 can show you something else. 828 00:54:53,110 --> 00:54:56,680 So in 3D, this would be through the thickness of the beam. 829 00:54:56,680 --> 00:54:58,715 That would be the thickness of the beam there. 830 00:55:01,685 --> 00:55:05,090 And boop, boop. 831 00:55:05,090 --> 00:55:06,960 It would look something like that. 832 00:55:06,960 --> 00:55:08,210 OK? 833 00:55:08,210 --> 00:55:10,305 And then this is still our neutral axis here. 834 00:55:14,670 --> 00:55:17,640 And then eventually, as you load it more and more, 835 00:55:17,640 --> 00:55:19,626 the whole cross-section is going to yield. 836 00:55:19,626 --> 00:55:20,126 Whoops. 837 00:55:53,580 --> 00:55:56,180 And I'm assuming that the material is elastic, perfectly 838 00:55:56,180 --> 00:55:56,680 plastic. 839 00:56:12,990 --> 00:56:15,660 So the stress-strain curve from the solid I'm 840 00:56:15,660 --> 00:56:17,098 idealizing as-- whoops. 841 00:56:20,770 --> 00:56:23,150 That's not quite right. 842 00:56:23,150 --> 00:56:25,610 I'm idealizing as that, OK? 843 00:56:28,270 --> 00:56:30,050 So when you get to this point here, 844 00:56:30,050 --> 00:56:31,860 the entire cross-section has yielded, 845 00:56:31,860 --> 00:56:34,050 and that means you form the plastic hinge. 846 00:56:56,507 --> 00:56:58,090 The idea here is that the section then 847 00:56:58,090 --> 00:57:01,968 just rotates like a pin. 848 00:57:09,610 --> 00:57:10,110 All right. 849 00:57:28,990 --> 00:57:31,430 So we can figure out the plateau stress 850 00:57:31,430 --> 00:57:33,980 that corresponds to this by looking 851 00:57:33,980 --> 00:57:37,630 at the moment that's associated with the plastic hinge 852 00:57:37,630 --> 00:57:38,220 formation. 853 00:57:38,220 --> 00:57:41,040 So there's some internal moment associated with that. 854 00:57:41,040 --> 00:57:43,840 And then equating that to the applied moment 855 00:57:43,840 --> 00:57:45,770 from the applied stress. 856 00:57:45,770 --> 00:57:47,080 So doodle-loodle-oot. 857 00:57:49,830 --> 00:57:52,690 Let me see me, maybe back up here. 858 00:57:52,690 --> 00:57:54,940 So there's some-- if I have the stress distribution 859 00:57:54,940 --> 00:57:59,160 here, I could say this whole kind of stress block 860 00:57:59,160 --> 00:58:01,330 is equivalent to some force acting out 861 00:58:01,330 --> 00:58:04,350 like that and some force acting out like that. 862 00:58:04,350 --> 00:58:10,360 It would be sigma ys times b comes t/2 would be f. 863 00:58:10,360 --> 00:58:14,250 And I can say there's some plastic moment. 864 00:58:14,250 --> 00:58:17,240 If I think of the force here and the force there, 865 00:58:17,240 --> 00:58:19,865 they act as a couple and they have some moment, 866 00:58:19,865 --> 00:58:22,490 and that's called the plastic moment. 867 00:58:22,490 --> 00:58:24,020 So that's like an internal moment 868 00:58:24,020 --> 00:58:25,660 when the plastic hinge forms. 869 00:58:29,610 --> 00:58:38,079 So I'll say the internal moment at the formation 870 00:58:38,079 --> 00:58:38,870 of a plastic hinge. 871 00:58:46,580 --> 00:58:49,570 I'm going to call that Mp, for plastic moment. 872 00:58:53,050 --> 00:58:56,670 And we can work out Mp by looking at that stress 873 00:58:56,670 --> 00:58:59,600 distribution when the entire cross section has yielded. 874 00:58:59,600 --> 00:59:06,990 The force F is going to be sigma ys times b comes t/2. 875 00:59:06,990 --> 00:59:10,120 It's the stress times that area. 876 00:59:10,120 --> 00:59:15,440 And then the moment arm between the two forces is also t/2. 877 00:59:15,440 --> 00:59:20,340 And so that plastic moment is just sigma ys 878 00:59:20,340 --> 00:59:24,360 bt squared over 4, OK? 879 00:59:24,360 --> 00:59:26,740 Are we good? 880 00:59:26,740 --> 00:59:27,648 Sonya? 881 00:59:27,648 --> 00:59:30,460 AUDIENCE: What's the second [INAUDIBLE]? 882 00:59:30,460 --> 00:59:32,210 LORNA GIBSON: OK, so this is the force. 883 00:59:32,210 --> 00:59:35,760 This thing here is the force F. And I 884 00:59:35,760 --> 00:59:38,280 have to-- if I'm getting a moment, 885 00:59:38,280 --> 00:59:43,160 I'm saying that that force, if I doot-doot-doot-- 886 00:59:43,160 --> 00:59:47,520 the distance between those two forces there is t/2. 887 00:59:47,520 --> 00:59:50,769 So each force acts through the middle of the block, 888 00:59:50,769 --> 00:59:52,602 and so the distance between [? it is ?] t/2. 889 01:00:00,260 --> 01:00:04,810 And I'm going to equate that moment to the applied moment 890 01:00:04,810 --> 01:00:06,803 from the sort of applied stress. 891 01:00:15,450 --> 01:00:18,405 And then if I go back to my inclined member-- whoops, 892 01:00:18,405 --> 01:00:19,720 let's see. 893 01:00:19,720 --> 01:00:21,900 Let me get a little more inclined. 894 01:00:21,900 --> 01:00:24,268 That's my inclined member, there, of length l. 895 01:00:28,950 --> 01:00:33,800 I've got modes p that are applied 896 01:00:33,800 --> 01:00:35,880 at the end from sigma 1. 897 01:00:35,880 --> 01:00:39,170 And I've got moments that are induced at the ends. 898 01:00:39,170 --> 01:00:42,720 And that angle there would be theta. 899 01:00:42,720 --> 01:00:45,940 This length here is l, like that. 900 01:00:45,940 --> 01:00:47,680 And if I just use static equilibrium 901 01:00:47,680 --> 01:00:51,130 on that, I can say that I've got 2 times 902 01:00:51,130 --> 01:00:53,800 the moment, so I've got one at each end-- they're 903 01:00:53,800 --> 01:00:57,160 both the same sign-- minus P. And then 904 01:00:57,160 --> 01:00:59,590 the distance between these two P's, say I take moments 905 01:00:59,590 --> 01:01:03,400 about here, I've got M applied plus M applied, 906 01:01:03,400 --> 01:01:05,686 I've got minus P times l sine theta. 907 01:01:09,280 --> 01:01:10,474 That's equal to 0. 908 01:01:13,140 --> 01:01:18,728 So the applied moment there is just Pl sine theta over 2. 909 01:01:23,200 --> 01:01:25,370 So now what I'm going to do is I'm 910 01:01:25,370 --> 01:01:27,090 going to equate this applied moment 911 01:01:27,090 --> 01:01:29,190 with this plastic moment, and I'm 912 01:01:29,190 --> 01:01:32,780 going to relate P to my applied stress sigma 1. 913 01:01:32,780 --> 01:01:35,840 And then I'm going to get a strength in terms of the yield 914 01:01:35,840 --> 01:01:38,980 strength of the solid, there's going to be a t/l factor 915 01:01:38,980 --> 01:01:42,400 and there's going to be some geometrical factor. 916 01:01:42,400 --> 01:01:44,875 So that's just the last step. 917 01:01:44,875 --> 01:01:45,750 Boop-ba-doop-ba-doop. 918 01:02:19,000 --> 01:02:21,060 So we get plastic collapse of the honeycomb. 919 01:02:35,810 --> 01:02:38,902 And the stress I'm going to call sigma star plastic with a 1, 920 01:02:38,902 --> 01:02:40,860 because I'm going to look at the one direction. 921 01:02:44,450 --> 01:02:49,000 And that happens when that internal plastic moment 922 01:02:49,000 --> 01:02:50,363 equals the applied moment. 923 01:02:55,190 --> 01:02:55,920 So let's see. 924 01:02:55,920 --> 01:02:58,230 I've got that. 925 01:02:58,230 --> 01:03:01,350 Let me also write down over here, 926 01:03:01,350 --> 01:03:05,900 I've also got this sigma 1 is equal to P over 927 01:03:05,900 --> 01:03:13,670 h plus l sine theta times b. 928 01:03:13,670 --> 01:03:18,090 So here I can write P in terms of sigma 1, in this thing. 929 01:03:18,090 --> 01:03:20,960 And then write that, get the applied moment 930 01:03:20,960 --> 01:03:23,326 in terms of that, and then equate it to that. 931 01:03:48,930 --> 01:03:51,900 So this term on the left-hand side 932 01:03:51,900 --> 01:03:55,340 corresponds to this expression for the applied moment 933 01:03:55,340 --> 01:03:56,520 where I've plugged in. 934 01:03:56,520 --> 01:03:59,585 For P, I've plugged in sigma 1 times h plus l sine 935 01:03:59,585 --> 01:04:01,610 theta times b. 936 01:04:01,610 --> 01:04:06,210 And that's my plastic moment on the right-hand side. 937 01:04:06,210 --> 01:04:08,960 So if I just rearrange this, I can then 938 01:04:08,960 --> 01:04:11,110 solve for this plastic collapse stress. 939 01:04:13,930 --> 01:04:18,480 So it's equal to the yield strength of the solid times 940 01:04:18,480 --> 01:04:23,330 t/l squared, and then times another geometrical factor. 941 01:04:23,330 --> 01:04:28,850 2 times h/l plus sine theta times sine theta. 942 01:04:31,345 --> 01:04:31,970 Doop-doop-doop. 943 01:04:41,584 --> 01:04:43,000 So the same kind of thing, there's 944 01:04:43,000 --> 01:04:46,890 a solid property, a t/l, a relative density term, 945 01:04:46,890 --> 01:04:49,350 in then a cell geometry term. 946 01:04:49,350 --> 01:04:53,380 And we can calculate with this for regular hexagonal cells. 947 01:05:09,850 --> 01:05:11,930 And we can do a similar kind of calculation 948 01:05:11,930 --> 01:05:13,560 for loading in the other direction. 949 01:05:29,830 --> 01:05:34,554 And you can get a shear strength if you want to do that, too. 950 01:05:34,554 --> 01:05:36,595 AUDIENCE: If you're going in the other direction, 951 01:05:36,595 --> 01:05:40,626 only the E or the M apply changes, right? 952 01:05:40,626 --> 01:05:41,750 [? Or ?] like that section. 953 01:05:41,750 --> 01:05:43,010 LORNA GIBSON: Yeah, this thing here is the same. 954 01:05:43,010 --> 01:05:43,380 AUDIENCE: That stays. 955 01:05:43,380 --> 01:05:44,213 LORNA GIBSON: Right. 956 01:05:44,213 --> 01:05:46,500 And this is-- there's a different geometry to it. 957 01:05:46,500 --> 01:05:48,918 Because now you're loading it this way on. 958 01:06:11,870 --> 01:06:15,630 OK, so we've calculated an elastic buckling plateau 959 01:06:15,630 --> 01:06:20,070 stress and a sort of plastic collapse plateau stress. 960 01:06:20,070 --> 01:06:23,140 And if you have thin enough walled, say, even aluminum 961 01:06:23,140 --> 01:06:25,440 honeycombs, then the elastic buckling 962 01:06:25,440 --> 01:06:29,110 could precede the plastic collapse. 963 01:06:29,110 --> 01:06:31,790 And so I'm just going to work out what the criterion would 964 01:06:31,790 --> 01:06:32,872 be for that to happen. 965 01:07:12,070 --> 01:07:16,018 So the two stresses can be equated. 966 01:07:19,300 --> 01:07:22,193 And then that's going to give us some criterion. 967 01:07:33,370 --> 01:07:35,550 So the two are equal, I'm just going to write down 968 01:07:35,550 --> 01:07:37,400 the equations that we had. 969 01:07:37,400 --> 01:07:39,455 So the buckling stress was n squared 970 01:07:39,455 --> 01:07:46,120 pi squared over 24 times E of the solid times t/l 971 01:07:46,120 --> 01:07:54,350 cubed divided by h/l squared times cos of theta. 972 01:07:54,350 --> 01:08:00,260 And the plastic collapse stress for the 2 direction 973 01:08:00,260 --> 01:08:05,850 was sigma ys times t/l squared divided by 2 cos squared theta. 974 01:08:11,510 --> 01:08:14,140 So I can write this-- because this has a t/l cubed term, 975 01:08:14,140 --> 01:08:16,510 and that has a t/l squared term, I 976 01:08:16,510 --> 01:08:18,902 can write this in terms of a t/l critical. 977 01:08:22,520 --> 01:08:25,090 So if I leave it t/l here and I put everything else 978 01:08:25,090 --> 01:08:34,300 on the other side, I've got 12 over n squared pi squared, 979 01:08:34,300 --> 01:08:45,362 then h/l squared over cos theta times sigma ys over Es. 980 01:08:48,670 --> 01:08:50,399 So if t/l is less than that, I'm going 981 01:08:50,399 --> 01:08:51,659 to get elastic buckling first. 982 01:08:51,659 --> 01:08:52,920 And if it's more than that, I'm going 983 01:08:52,920 --> 01:08:54,260 to get plastic yielding first. 984 01:09:00,960 --> 01:09:02,710 And we can work out an exact number 985 01:09:02,710 --> 01:09:04,220 for regular hexagonal honeycombs, 986 01:09:04,220 --> 01:09:05,220 so I'm going to do that. 987 01:09:13,140 --> 01:09:15,010 So if I have a particular geometry, 988 01:09:15,010 --> 01:09:16,740 I can figure out what n is. 989 01:09:16,740 --> 01:09:20,930 So for regular hexagonal honeycombs, 990 01:09:20,930 --> 01:09:25,260 t/l critical just works out to 3 times 991 01:09:25,260 --> 01:09:28,710 the yield strength of the solid over the Young's modulus 992 01:09:28,710 --> 01:09:29,315 of the solid. 993 01:09:34,060 --> 01:09:36,090 So if we know that ratio of the yield 994 01:09:36,090 --> 01:09:38,340 strength of the modulus of the solid, 995 01:09:38,340 --> 01:09:43,390 we can get some idea of what that critical t/l would be. 996 01:09:43,390 --> 01:09:46,529 So we'll do that next. 997 01:09:46,529 --> 01:09:50,693 AUDIENCE: And you said if t/l is less than that critical, 998 01:09:50,693 --> 01:09:52,527 then you're going to get the yielding first. 999 01:09:52,527 --> 01:09:54,984 LORNA GIBSON: No, if it's less, you get the buckling first. 1000 01:09:54,984 --> 01:09:57,127 If it's really skinny, it tends to buckle first. 1001 01:10:27,510 --> 01:10:31,480 So, for example, for metals, the yield strength over the modulus 1002 01:10:31,480 --> 01:10:35,350 is roughly 0.002, like the 0.2% yield strength. 1003 01:10:35,350 --> 01:10:38,510 And so that means that t/l, the sort 1004 01:10:38,510 --> 01:10:44,350 of transition or the critical value is at 0.6%. 1005 01:10:44,350 --> 01:10:46,924 So most metal honeycombs are denser than that. 1006 01:10:46,924 --> 01:10:48,090 That's a pretty low density. 1007 01:11:13,730 --> 01:11:19,010 But if we look at polymers, you can get polymers 1008 01:11:19,010 --> 01:11:22,590 with a yield strength relative to the modulus 1009 01:11:22,590 --> 01:11:30,860 of about 3% to 5%, and then that critical t/l is 1010 01:11:30,860 --> 01:11:32,220 equal to about 10%, 15%. 1011 01:11:35,790 --> 01:11:38,390 So low-density polymers with yield points 1012 01:11:38,390 --> 01:11:39,870 may buckle before they yield. 1013 01:12:18,640 --> 01:12:22,540 So we have one more of these compressive plateau stresses, 1014 01:12:22,540 --> 01:12:24,500 and that's for the brittle honeycomb. 1015 01:12:24,500 --> 01:12:26,500 So I don't think I'm going to finish this today, 1016 01:12:26,500 --> 01:12:29,139 but let me set it up and then we'll finish it next time. 1017 01:13:23,254 --> 01:13:25,670 So the idea here is that if you have a ceramic honeycomb-- 1018 01:13:25,670 --> 01:13:28,210 remember I showed you some of those ceramic honeycombs-- 1019 01:13:28,210 --> 01:13:31,400 that if you compress them, they can fail by a brittle crushing 1020 01:13:31,400 --> 01:13:32,760 mode. 1021 01:13:32,760 --> 01:13:35,990 So ceramic honeycombs can fail in a brittle manner. 1022 01:13:54,620 --> 01:13:57,950 And again, initially there would be some cell wall bending, 1023 01:13:57,950 --> 01:14:00,570 but at some point, you're going to reach the bending strength 1024 01:14:00,570 --> 01:14:02,470 of the material. 1025 01:14:02,470 --> 01:14:04,950 And bending strengths are called modulus of rupture. 1026 01:14:04,950 --> 01:14:07,940 So you reach the modulus of rupture of the cell wall. 1027 01:14:47,757 --> 01:14:49,590 So I'm not going to write the equations down 1028 01:14:49,590 --> 01:14:51,506 today because we're not going to get very far, 1029 01:14:51,506 --> 01:14:52,782 so I'll do that next time. 1030 01:14:52,782 --> 01:14:54,740 But we're going to set this up exactly the same 1031 01:14:54,740 --> 01:14:57,260 as we did for the last one, for the plastic yielding. 1032 01:14:57,260 --> 01:14:59,540 But instead of getting that sort of blocky, 1033 01:14:59,540 --> 01:15:03,007 fully yielded cross-section stress distribution, 1034 01:15:03,007 --> 01:15:05,090 we're just going to have the linear elastic stress 1035 01:15:05,090 --> 01:15:07,150 distribution, and when the maximum stress 1036 01:15:07,150 --> 01:15:09,400 reaches that modulus [? rupture, ?] the thing's 1037 01:15:09,400 --> 01:15:10,360 going to fail. 1038 01:15:10,360 --> 01:15:12,020 So the form of the equations is going 1039 01:15:12,020 --> 01:15:15,060 to be very similar to what we had for the plastic collapse 1040 01:15:15,060 --> 01:15:17,800 stress, but there's a slightly different geometrical factor-- 1041 01:15:17,800 --> 01:15:18,616 that's all. 1042 01:15:18,616 --> 01:15:19,740 So we'll do that next time. 1043 01:15:19,740 --> 01:15:21,198 And then next time we're also going 1044 01:15:21,198 --> 01:15:23,450 to talk about the tensile behavior 1045 01:15:23,450 --> 01:15:24,860 of honeycombs in-plane. 1046 01:15:24,860 --> 01:15:26,516 We'll work out a fracture toughness 1047 01:15:26,516 --> 01:15:28,640 and then we'll start talking about the out-of-plane 1048 01:15:28,640 --> 01:15:29,960 properties, as well. 1049 01:15:29,960 --> 01:15:34,200 So on Wednesday, we'll do the out-of-plane properties, OK? 1050 01:15:34,200 --> 01:15:36,530 So hopefully we'll finish the out-of-plane properties 1051 01:15:36,530 --> 01:15:37,465 Wednesday. 1052 01:15:37,465 --> 01:15:39,090 And then next week, I was going to talk 1053 01:15:39,090 --> 01:15:41,130 about some natural materials that 1054 01:15:41,130 --> 01:15:44,690 have honeycomb-like structures, so things like wood and cork, 1055 01:15:44,690 --> 01:15:45,860 OK? 1056 01:15:45,860 --> 01:15:48,640 All right, so this is the kind of most equationy lecture 1057 01:15:48,640 --> 01:15:50,540 in the whole course.