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 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 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 00:00:27,440 --> 00:00:30,190 LORNA GIBSON: All right, so I guess I should start. 9 00:00:30,190 --> 00:00:32,430 So I think last time we were talking about cell 10 00:00:32,430 --> 00:00:34,520 structure and cell geometry. 11 00:00:34,520 --> 00:00:38,180 And I got as far as putting this image here up. 12 00:00:38,180 --> 00:00:40,520 And I talked a little bit about how this works. 13 00:00:40,520 --> 00:00:42,430 And I was going to go over it again and then 14 00:00:42,430 --> 00:00:44,290 write the notes down today. 15 00:00:44,290 --> 00:00:45,410 So we'll start from there. 16 00:00:45,410 --> 00:00:47,743 So we're going to do a little bit more on cell structure 17 00:00:47,743 --> 00:00:48,390 today. 18 00:00:48,390 --> 00:00:50,810 We're going to talk about some topological laws 19 00:00:50,810 --> 00:00:54,160 for cellular materials for polyhedral cells. 20 00:00:54,160 --> 00:00:56,940 And then we'll start talking about modeling honeycomb 21 00:00:56,940 --> 00:00:58,830 materials, and talk about how we look 22 00:00:58,830 --> 00:01:01,550 at the mechanical properties of honeycombs. 23 00:01:01,550 --> 00:01:03,830 So I think where we left off last time was 24 00:01:03,830 --> 00:01:06,310 we started talking about mean intercept length. 25 00:01:06,310 --> 00:01:10,210 So the idea is that with-- hello, hello-- 26 00:01:10,210 --> 00:01:13,260 if you have a honeycomb, it's fairly easy to define the cell 27 00:01:13,260 --> 00:01:16,890 shape in terms of the ratio of the cell edge lengths, 28 00:01:16,890 --> 00:01:19,490 h over l, and the angle that the inclined cell 29 00:01:19,490 --> 00:01:21,360 wall is to the vertical one. 30 00:01:21,360 --> 00:01:23,470 But for a foam it's a little more difficult. 31 00:01:23,470 --> 00:01:25,720 And what people do is they measure this mean intercept 32 00:01:25,720 --> 00:01:26,220 length. 33 00:01:26,220 --> 00:01:28,870 So last time I talked about it briefly. 34 00:01:28,870 --> 00:01:31,910 So the idea is you take an image of the structure you're 35 00:01:31,910 --> 00:01:33,310 interested in. 36 00:01:33,310 --> 00:01:38,060 You sort of draw out the outline of a planar surface. 37 00:01:38,060 --> 00:01:40,400 And then you put on that a test circle. 38 00:01:40,400 --> 00:01:43,770 And you superimpose on the test circle equidistant 39 00:01:43,770 --> 00:01:44,710 parallel lines. 40 00:01:44,710 --> 00:01:47,610 So say we start off at theta equals 0. 41 00:01:47,610 --> 00:01:49,840 We then count how many times the cell walls 42 00:01:49,840 --> 00:01:51,020 intercept those lines. 43 00:01:51,020 --> 00:01:54,490 And we get a number of cells per unit length. 44 00:01:54,490 --> 00:01:57,890 And that gives us an intercept length for that orientation. 45 00:01:57,890 --> 00:01:59,680 Then we rotate it around a little bit, 46 00:01:59,680 --> 00:02:01,900 say 5 degrees or something. 47 00:02:01,900 --> 00:02:04,740 And we measure a new intercept length for that orientation. 48 00:02:04,740 --> 00:02:07,590 And we keep doing it all the way around, 180 degrees. 49 00:02:07,590 --> 00:02:09,780 And then you get-- then what you do 50 00:02:09,780 --> 00:02:13,170 is you plot those points-- if I can find my little pointer-- we 51 00:02:13,170 --> 00:02:14,020 plot those points. 52 00:02:14,020 --> 00:02:17,350 And it will form an ellipse. 53 00:02:17,350 --> 00:02:20,160 And the major and minor axes of the ellipse 54 00:02:20,160 --> 00:02:23,290 correspond to the principal dimensions of the cells 55 00:02:23,290 --> 00:02:24,480 on that plane. 56 00:02:24,480 --> 00:02:27,000 And then you can do the same for perpendicular planes 57 00:02:27,000 --> 00:02:28,140 and form an ellipsoid. 58 00:02:28,140 --> 00:02:30,920 And you can get the three principal dimensions. 59 00:02:30,920 --> 00:02:33,630 You can also get the orientation of the cells 60 00:02:33,630 --> 00:02:35,890 by getting the orientation of this ellipse, 61 00:02:35,890 --> 00:02:38,750 or the ellipsoid in three dimensions. 62 00:02:38,750 --> 00:02:40,900 So let me just write down some notes that kind of 63 00:02:40,900 --> 00:02:42,870 summarize all of that. 64 00:02:42,870 --> 00:02:46,070 So I'm going to say that for foams, we characterize the cell 65 00:02:46,070 --> 00:02:49,940 shape and orientation by using these mean intercepts-- mean 66 00:02:49,940 --> 00:02:50,862 intercept lengths. 67 00:03:16,050 --> 00:03:22,595 So we consider a circular test area on a plane section. 68 00:03:31,590 --> 00:03:33,305 So you want to use a circle and not say, 69 00:03:33,305 --> 00:03:34,305 a square or a rectangle. 70 00:03:34,305 --> 00:03:35,520 Because if you had a square, then 71 00:03:35,520 --> 00:03:37,311 you'd have different total lengths of line, 72 00:03:37,311 --> 00:03:39,360 depending on what orientation you took it. 73 00:03:39,360 --> 00:03:42,510 So you want to use a circular test section. 74 00:03:42,510 --> 00:03:45,420 And then you draw equidistant parallel lines. 75 00:03:55,520 --> 00:03:59,780 So for example you might start at theta equals to 0. 76 00:03:59,780 --> 00:04:02,480 And then you count the number of intercepts of the cell walls 77 00:04:02,480 --> 00:04:03,260 with the lines. 78 00:04:21,260 --> 00:04:24,600 So if we say that Nc is the number of cells 79 00:04:24,600 --> 00:04:39,260 per unit length of line, then we can get the intercept length 80 00:04:39,260 --> 00:04:41,490 for that orientation. 81 00:04:41,490 --> 00:04:46,950 Say for theta equals to 0, it works out to 1.5/Nc. 82 00:04:46,950 --> 00:04:49,957 So it's not just 1/Nc, because you may not 83 00:04:49,957 --> 00:04:51,540 be cutting the cell-- you know, you're 84 00:04:51,540 --> 00:04:53,620 cutting the cell-- say you've got a three dimensional 85 00:04:53,620 --> 00:04:54,360 cell like this. 86 00:04:54,360 --> 00:04:56,520 You're cutting it at different places along here. 87 00:04:56,520 --> 00:04:58,980 So people have worked out the stereology of that. 88 00:04:58,980 --> 00:05:04,710 And there's a constant that's 1.5 that fits into there. 89 00:05:04,710 --> 00:05:06,920 So then you just repeat that process 90 00:05:06,920 --> 00:05:08,606 for different increments of theta. 91 00:05:11,870 --> 00:05:25,430 So you increment theta by some amount, something 92 00:05:25,430 --> 00:05:27,076 like say, 5 degrees. 93 00:05:27,076 --> 00:05:28,700 And then you repeat that whole process. 94 00:05:33,570 --> 00:05:39,920 And then you plot a polar diagram 95 00:05:39,920 --> 00:05:41,980 of the intercept lengths versus theta. 96 00:05:55,660 --> 00:05:58,680 And you then fit an ellipse to the points. 97 00:06:05,580 --> 00:06:07,750 And if you did it in 3D, you'd have an ellipsoid. 98 00:06:11,730 --> 00:06:15,100 And then the principal axes of the ellipsoid 99 00:06:15,100 --> 00:06:18,193 are the principal dimensions of the cells. 100 00:06:37,560 --> 00:06:39,220 And the orientation of the ellipsoid 101 00:06:39,220 --> 00:06:41,020 is the orientation of the cells. 102 00:06:44,961 --> 00:06:45,460 Hello. 103 00:06:59,440 --> 00:07:01,903 And you can write the equation of the ellipsoid. 104 00:07:07,690 --> 00:07:12,400 So it would be something like Ax1 squared plus B 105 00:07:12,400 --> 00:07:25,950 x2 squared plus Cx3 squared plus 2Dx1x2 plus 2Ex1x3 plus 2Fx2x3. 106 00:07:29,490 --> 00:07:31,171 And that would all equal 1. 107 00:07:31,171 --> 00:07:33,212 And you can write those coefficients as a matrix. 108 00:07:55,570 --> 00:07:58,000 And if you do that, the first three, A, B, C, 109 00:07:58,000 --> 00:08:03,860 are the diagonals, and D, E, F are the off diagonals. 110 00:08:10,230 --> 00:08:13,090 And you can also represent this is a tensor. 111 00:08:13,090 --> 00:08:15,045 And if you do, it's called the fabric tensor. 112 00:08:24,410 --> 00:08:28,000 So the fabric-- I think they use that term for other types 113 00:08:28,000 --> 00:08:30,250 of materials-- it says something about the orientation 114 00:08:30,250 --> 00:08:32,270 of the material. 115 00:08:32,270 --> 00:08:34,830 And if you have this matrix here, 116 00:08:34,830 --> 00:08:39,330 if D, E, and F are all of 0, then A, B, and C 117 00:08:39,330 --> 00:08:41,516 are the principal dimensions of the cell. 118 00:08:45,410 --> 00:08:55,620 So if all non-diagonal elements are 0, 119 00:08:55,620 --> 00:09:00,980 then A, B, and C are the principal dimensions. 120 00:09:14,600 --> 00:09:15,960 Are we good with how this works? 121 00:09:15,960 --> 00:09:19,954 So it's kind of a crank and churn kind of thing. 122 00:09:19,954 --> 00:09:21,620 But it gives you a way of characterizing 123 00:09:21,620 --> 00:09:23,440 the cell shape and the orientation 124 00:09:23,440 --> 00:09:24,925 of the cells in the foams. 125 00:09:50,334 --> 00:09:52,000 So the next thing I wanted to talk about 126 00:09:52,000 --> 00:09:54,810 was the connectivity of the cells. 127 00:09:54,810 --> 00:09:57,190 So imagine we have an array of cells. 128 00:09:57,190 --> 00:09:59,760 And if we have an array we have vertices 129 00:09:59,760 --> 00:10:01,400 that are connected by edges. 130 00:10:01,400 --> 00:10:03,220 And edges surround faces. 131 00:10:03,220 --> 00:10:05,730 And the faces enclose the cells. 132 00:10:05,730 --> 00:10:08,450 And there's something called the edge connectivity, which 133 00:10:08,450 --> 00:10:10,840 is usually given the symbol Ze. 134 00:10:10,840 --> 00:10:13,051 And that's the number of edges that meet at a vertex. 135 00:10:13,051 --> 00:10:15,300 And there's a face connectivity, and that's the number 136 00:10:15,300 --> 00:10:17,290 of faces that meet at an edge. 137 00:10:17,290 --> 00:10:20,610 And it's very common for honeycombs 138 00:10:20,610 --> 00:10:21,710 to be three connected. 139 00:10:21,710 --> 00:10:22,850 So Ze is 3. 140 00:10:22,850 --> 00:10:25,870 And it's common for foams to be four connected. 141 00:10:25,870 --> 00:10:27,410 And there's some topological laws 142 00:10:27,410 --> 00:10:28,618 that are kind of interesting. 143 00:10:28,618 --> 00:10:32,540 And so we'll get into those after this bit on connectivity. 144 00:10:32,540 --> 00:10:37,800 So for the connectivity we have vertices, 145 00:10:37,800 --> 00:10:49,750 which are connected by edges, which surround 146 00:10:49,750 --> 00:11:00,780 faces, which enclose cells. 147 00:11:00,780 --> 00:11:05,400 So we're going to talk about the vertices, the edges, the faces, 148 00:11:05,400 --> 00:11:06,130 and the cells. 149 00:11:14,850 --> 00:11:19,470 So the edge connectivity, Ze, is the number 150 00:11:19,470 --> 00:11:21,030 of edges that meet at a vertex. 151 00:11:36,070 --> 00:11:48,060 And for a honeycomb, say a hexagonal honeycomb, Ze is 3. 152 00:11:48,060 --> 00:11:49,689 So I have a little honeycomb here. 153 00:11:49,689 --> 00:11:51,980 If we look at the number of edges, there's three edges, 154 00:11:51,980 --> 00:11:53,060 connect at a vertex. 155 00:11:53,060 --> 00:11:54,790 So Ze is 3. 156 00:11:54,790 --> 00:12:00,920 And for a foam, Ze is typically four. 157 00:12:00,920 --> 00:12:03,230 So I'll just say typically here. 158 00:12:03,230 --> 00:12:13,750 And if I go back, if you look at the sketches here 159 00:12:13,750 --> 00:12:17,770 of the rhombic dodecahedra and the tetrakaidecadedra, 160 00:12:17,770 --> 00:12:19,970 if you look at the tetrakaidecadedra, 161 00:12:19,970 --> 00:12:21,470 you can see the number of edges. 162 00:12:21,470 --> 00:12:22,590 So here's one edge. 163 00:12:22,590 --> 00:12:24,160 Here's another one here. 164 00:12:24,160 --> 00:12:25,202 There's another one here. 165 00:12:25,202 --> 00:12:26,409 And there's another one here. 166 00:12:26,409 --> 00:12:28,860 So that's the four edges that are meeting at that vertex 167 00:12:28,860 --> 00:12:29,420 there. 168 00:12:29,420 --> 00:12:30,940 So if you look at arrays of cells, 169 00:12:30,940 --> 00:12:35,670 you can convince yourself that Ze is typically 4 for a foam. 170 00:12:35,670 --> 00:12:44,864 And then we also have the face connectivity, Zf. 171 00:12:44,864 --> 00:12:47,030 And that's the number of faces that meet at an edge. 172 00:13:01,330 --> 00:13:02,925 And that's typically three for foams. 173 00:13:14,530 --> 00:13:16,280 And so again if you look at these pictures 174 00:13:16,280 --> 00:13:23,110 you can sort of see how the face connectivity is three. 175 00:13:23,110 --> 00:13:27,350 So if you look at this bottom one here, there's faces here. 176 00:13:27,350 --> 00:13:31,725 And then there's another face-- well, let's see if we can-- 177 00:13:31,725 --> 00:13:35,210 it's kind of hard to show. 178 00:13:35,210 --> 00:13:37,210 If you look at this one here, there's this cell. 179 00:13:37,210 --> 00:13:37,950 There's that face there. 180 00:13:37,950 --> 00:13:40,180 And then there's another one coming out of the page, 181 00:13:40,180 --> 00:13:41,450 I think. 182 00:13:41,450 --> 00:13:42,040 OK. 183 00:13:42,040 --> 00:13:42,998 So that's connectivity. 184 00:14:08,430 --> 00:14:11,195 And the next thing are some of these topological laws. 185 00:14:15,254 --> 00:14:16,920 So the first one I'm going to talk about 186 00:14:16,920 --> 00:14:18,410 is called Euler's law. 187 00:14:18,410 --> 00:14:21,850 If you remember Euler from buckling, same guy. 188 00:14:21,850 --> 00:14:25,980 So Euler's law relates the number of vertices and faces 189 00:14:25,980 --> 00:14:31,410 and cells and edges for a large array of cells. 190 00:14:31,410 --> 00:14:41,790 So we can relate the total number of edges, 191 00:14:41,790 --> 00:14:52,940 so I'm going to call that E, vertices V, faces F, 192 00:14:52,940 --> 00:15:00,710 and cells C. That total number is 193 00:15:00,710 --> 00:15:05,290 related by Euler's law for a large aggregate of cells. 194 00:15:21,020 --> 00:15:24,380 So in 2D Euler's law is that the total number 195 00:15:24,380 --> 00:15:28,020 of faces minus the number of edges 196 00:15:28,020 --> 00:15:30,943 plus the number of vertices is equal to 1. 197 00:15:30,943 --> 00:15:38,160 And in 3D, you just add a minus the number of cells in front. 198 00:15:38,160 --> 00:15:40,040 So minus the number of cells, plus the number 199 00:15:40,040 --> 00:15:43,130 of faces, minus the number of edges, 200 00:15:43,130 --> 00:15:45,710 plus the number of vertices is equal to 1. 201 00:15:56,764 --> 00:15:58,180 So there's some interesting things 202 00:15:58,180 --> 00:16:04,080 you can figure out using Euler's law. 203 00:16:04,080 --> 00:16:06,000 And one of the things I wanted to look at 204 00:16:06,000 --> 00:16:08,980 was if we had an irregular honeycomb that 205 00:16:08,980 --> 00:16:10,220 was three connected. 206 00:16:10,220 --> 00:16:13,160 So say we have a honeycomb that's not just 207 00:16:13,160 --> 00:16:16,430 a regular hexagonal honeycomb, or not even one that's 208 00:16:16,430 --> 00:16:18,460 got repeating hexagonal cells. 209 00:16:18,460 --> 00:16:20,990 But say we had a honeycomb where some of the cells 210 00:16:20,990 --> 00:16:26,500 had 5 sides to them, some of them had 7, some of them had 6. 211 00:16:26,500 --> 00:16:29,750 You can ask, what's the average number of sides per face? 212 00:16:29,750 --> 00:16:32,340 And you can use Euler's law to figure that out. 213 00:16:32,340 --> 00:16:34,900 So we're going to look at an irregular three connected 214 00:16:34,900 --> 00:16:37,180 honeycomb. 215 00:16:37,180 --> 00:16:39,840 Let's see, I can start that up here. 216 00:16:46,890 --> 00:16:48,440 So when I say irregular, I mean we've 217 00:16:48,440 --> 00:16:50,710 got cells with different numbers of sides. 218 00:17:04,500 --> 00:17:05,969 So we have an irregular-- oops. 219 00:17:05,969 --> 00:17:07,094 You OK, Greg? 220 00:17:21,920 --> 00:17:23,700 --that is three connected. 221 00:17:23,700 --> 00:17:26,426 And what is the average number of sides per face? 222 00:17:38,390 --> 00:17:40,290 And I'm going to call that n bar. 223 00:17:45,450 --> 00:17:48,300 So we're told that it's three connected. 224 00:17:48,300 --> 00:17:50,760 So that's saying that the edge connectivity is 3. 225 00:17:50,760 --> 00:17:53,720 So there's three edges coming to each vertex. 226 00:17:53,720 --> 00:17:58,560 So imagine that you had just a regular hexagonal honeycomb, 227 00:17:58,560 --> 00:18:00,870 like this. 228 00:18:00,870 --> 00:18:03,050 Here's our vertex here. 229 00:18:03,050 --> 00:18:06,030 And there's our kind of three edges that come into it. 230 00:18:06,030 --> 00:18:09,530 And each vertex is going to have half of each edge, right? 231 00:18:09,530 --> 00:18:11,410 Because the next vertex-- each edge 232 00:18:11,410 --> 00:18:15,130 is shared between two vertices. 233 00:18:15,130 --> 00:18:20,770 So if Ze is equal to 3, then the number of edges per vertex 234 00:18:20,770 --> 00:18:22,199 is 3/2. 235 00:18:22,199 --> 00:18:24,240 Because each edge is shared between two vertices. 236 00:18:39,720 --> 00:18:45,310 And I'm also going to define something I'm going to call Fn. 237 00:18:45,310 --> 00:18:48,692 And Fn is going to be the number of faces with n sides. 238 00:18:58,160 --> 00:19:01,000 So I could have some number of the cells have six sides, 239 00:19:01,000 --> 00:19:03,650 some number have five sides, some number have seven sides. 240 00:19:03,650 --> 00:19:07,320 So these are-- Fn is the number of faces 241 00:19:07,320 --> 00:19:11,190 with some particular number of sides, n. 242 00:19:11,190 --> 00:19:14,610 So if we have Fn is the number of faces with n sides, 243 00:19:14,610 --> 00:19:19,690 then if I sum up n times Fn and divide it by 2, 244 00:19:19,690 --> 00:19:21,400 that's the number of edges. 245 00:19:21,400 --> 00:19:23,100 So imagine I have all these faces. 246 00:19:23,100 --> 00:19:25,980 I take n times Fn. 247 00:19:25,980 --> 00:19:29,100 So say we had four five-sided faces. 248 00:19:29,100 --> 00:19:32,340 That would be four of those. 249 00:19:32,340 --> 00:19:35,820 There's be 20 edges associated with that, 20 sides. 250 00:19:35,820 --> 00:19:37,556 Then I have some number of six-sided, 251 00:19:37,556 --> 00:19:38,680 some number of seven-sided. 252 00:19:38,680 --> 00:19:39,730 I add them all up. 253 00:19:39,730 --> 00:19:42,350 And I have to divide by 2 to get the number of edges, 254 00:19:42,350 --> 00:19:45,367 because each edge separates two faces. 255 00:19:58,040 --> 00:19:59,490 OK. 256 00:19:59,490 --> 00:20:00,960 And then I can use Euler's law. 257 00:20:07,300 --> 00:20:13,980 So I can say F minus E plus V is going to equal 1. 258 00:20:13,980 --> 00:20:19,720 And here I've got an F minus E plus-- I can put V here. 259 00:20:19,720 --> 00:20:26,470 V's going to be 2/3 of E. So I've got that. 260 00:20:26,470 --> 00:20:35,050 So that's the same as F minus 1/3 of E is equal to 1, 261 00:20:35,050 --> 00:20:36,080 like that. 262 00:20:36,080 --> 00:20:38,960 And then for E I can substitute this thing here in. 263 00:20:38,960 --> 00:20:48,870 So that gives me F minus 1/3 of the sum of n times Fn over 2 264 00:20:48,870 --> 00:20:51,290 is equal to 1. 265 00:20:51,290 --> 00:20:53,680 And then what I'm going to do is multiply everything by 6 266 00:20:53,680 --> 00:20:56,010 so I can get rid of my denominator here. 267 00:20:56,010 --> 00:21:02,150 So 6F minus sum of nFn is going to be 6. 268 00:21:02,150 --> 00:21:05,230 And then I'm going to divide that through by F. 269 00:21:05,230 --> 00:21:14,710 So that's 6 minus sum of nFn over F is equal to 6/F. 270 00:21:14,710 --> 00:21:17,530 And then if I let F go to a large number-- so say 271 00:21:17,530 --> 00:21:19,270 I've got a large aggregate of cell. 272 00:21:19,270 --> 00:21:20,780 I've got lots of faces. 273 00:21:20,780 --> 00:21:23,060 I'm going to let F become large. 274 00:21:29,400 --> 00:21:33,650 So if F becomes large, then 6/F is going to tend to 0. 275 00:21:36,530 --> 00:21:37,890 Let me get the other rubber. 276 00:21:40,668 --> 00:21:41,600 AUDIENCE: Professor? 277 00:21:41,600 --> 00:21:42,944 LORNA GIBSON: Mhm? 278 00:21:42,944 --> 00:21:46,501 AUDIENCE: That's-- F is like the total number of faces, 279 00:21:46,501 --> 00:21:48,610 not the total number of faces per cell. 280 00:21:48,610 --> 00:21:50,490 LORNA GIBSON: F is the total number of faces. 281 00:21:50,490 --> 00:21:53,360 And Fn is the number of faces with n sides. 282 00:21:53,360 --> 00:21:55,949 We've got some number with 5 sides, some number with 6, 283 00:21:55,949 --> 00:21:56,740 some number with 7. 284 00:22:03,810 --> 00:22:06,940 So we're almost at the end here. 285 00:22:06,940 --> 00:22:08,860 So 6/F goes to 0. 286 00:22:08,860 --> 00:22:15,280 And so that says that the sum of n times Fn over F 287 00:22:15,280 --> 00:22:16,970 is equal to 6. 288 00:22:16,970 --> 00:22:18,640 And that just is n bar. 289 00:22:18,640 --> 00:22:21,970 That is the average number of sides per face. 290 00:22:21,970 --> 00:22:24,250 This is your-- kind of your total number 291 00:22:24,250 --> 00:22:26,190 of sides in the whole thing. 292 00:22:26,190 --> 00:22:28,500 And you're dividing by the total number of faces. 293 00:22:28,500 --> 00:22:31,550 So that is the average number of sides per face. 294 00:22:41,290 --> 00:22:43,460 So what this is saying is if you have 295 00:22:43,460 --> 00:22:46,280 a three connected honeycomb, the average number 296 00:22:46,280 --> 00:22:48,770 of sides per face is always 6. 297 00:22:48,770 --> 00:22:51,300 So that if you introduce a cell with 5 sides, 298 00:22:51,300 --> 00:22:53,480 somewhere you have to introduce a cell with 7 sides, 299 00:22:53,480 --> 00:22:55,605 so that they compensate and you come back out to 6. 300 00:23:18,150 --> 00:23:20,440 And I have a little soap bubble picture here 301 00:23:20,440 --> 00:23:23,460 that kind of illustrates that. 302 00:23:23,460 --> 00:23:25,142 So here we have a soap honeycomb. 303 00:23:25,142 --> 00:23:27,100 So you can make these just by putting two glass 304 00:23:27,100 --> 00:23:31,460 sheets close together with a soap bubble froth 305 00:23:31,460 --> 00:23:32,590 in between them. 306 00:23:32,590 --> 00:23:34,810 And you can kind of see in this picture here, 307 00:23:34,810 --> 00:23:38,570 they've numbered how many sides each cell or each face has. 308 00:23:38,570 --> 00:23:41,150 So here's one with 5, here's one with 7. 309 00:23:41,150 --> 00:23:43,531 And I'm not going to add these up all together. 310 00:23:43,531 --> 00:23:45,530 But you can kind of see that the 5's and the 7's 311 00:23:45,530 --> 00:23:46,946 kind of compensate for each other, 312 00:23:46,946 --> 00:23:49,650 and that the average works out to about 6. 313 00:23:49,650 --> 00:23:52,292 And this is true for a large aggregate of cells. 314 00:23:52,292 --> 00:23:53,750 So if you only have a few, it's not 315 00:23:53,750 --> 00:23:54,640 going to work out perfectly. 316 00:23:54,640 --> 00:23:56,764 You need to have a large aggregate to have it work. 317 00:23:56,764 --> 00:23:57,820 Yeah? 318 00:23:57,820 --> 00:23:59,892 AUDIENCE: And that 5, 7 balance doesn't 319 00:23:59,892 --> 00:24:02,290 have to be touching, right? 320 00:24:02,290 --> 00:24:04,430 LORNA GIBSON: No, no, no, overall. 321 00:24:04,430 --> 00:24:06,000 Yeah, overall. 322 00:24:06,000 --> 00:24:08,910 And, you know, you could have one with 4 sides, 323 00:24:08,910 --> 00:24:12,190 and you'd need two 7's or one 8 or something. 324 00:24:12,190 --> 00:24:14,470 But you'd need to have that balance out and match up. 325 00:24:17,750 --> 00:24:19,170 OK. 326 00:24:19,170 --> 00:24:23,380 So that's the Euler law. 327 00:24:23,380 --> 00:24:26,400 Now there's a couple more of these kinds of things. 328 00:24:26,400 --> 00:24:28,910 There's something called the Aboav-Weaire law. 329 00:24:38,330 --> 00:24:41,736 And so the Aboav-Weaire is sort of related to the Euler 330 00:24:41,736 --> 00:24:43,110 because it's looking at this idea 331 00:24:43,110 --> 00:24:45,690 that if you have a three connected honeycomb, if you 332 00:24:45,690 --> 00:24:48,590 introduce a 5-sided cell, you have to introduce 333 00:24:48,590 --> 00:24:50,980 a 7-sided cell to compensate. 334 00:24:50,980 --> 00:24:54,000 And what Aboav noticed was that generally 335 00:24:54,000 --> 00:24:56,530 cells with more sides than average 336 00:24:56,530 --> 00:24:58,535 have neighbors with fewer sides than average. 337 00:25:01,560 --> 00:25:02,456 Let's see. 338 00:25:09,270 --> 00:25:27,380 So we'll say the introduction of a 5-sided cell 339 00:25:27,380 --> 00:25:31,290 requires the introduction of a 7-sided cell. 340 00:25:31,290 --> 00:25:34,280 And I'll just put-- this is all for a 3 connected net, a 3 341 00:25:34,280 --> 00:25:35,175 connected honeycomb. 342 00:25:48,940 --> 00:25:53,310 So that generally, cells with more sides than average 343 00:25:53,310 --> 00:25:55,310 have neighbors with fewer sides than average. 344 00:26:28,550 --> 00:26:31,140 And in 3D, you could say the same thing about the faces. 345 00:26:31,140 --> 00:26:33,630 The cells that have more faces than average have neighbors 346 00:26:33,630 --> 00:26:35,690 with fewer faces than average. 347 00:26:35,690 --> 00:26:38,280 So Aboav made observations of this. 348 00:26:38,280 --> 00:26:40,940 And it was Dennis Weaire who made a proof of it. 349 00:27:06,840 --> 00:27:09,270 And the equation that they came up with 350 00:27:09,270 --> 00:27:14,400 relates the average number of sides in the cell surrounding 351 00:27:14,400 --> 00:27:15,980 a candidate cell. 352 00:27:15,980 --> 00:27:18,230 Let's see-- are we going to be able to put it in here? 353 00:27:18,230 --> 00:27:20,150 Maybe. 354 00:27:20,150 --> 00:27:29,670 So say you have a candidate cell and say it has n sides. 355 00:27:29,670 --> 00:27:32,740 So you look at one particular cell and you say, count up, 356 00:27:32,740 --> 00:27:34,540 it's got n sides around it. 357 00:27:34,540 --> 00:27:37,870 And then you count up the number of sides of all the cells 358 00:27:37,870 --> 00:27:38,820 surrounding it. 359 00:27:38,820 --> 00:27:41,490 It has n neighbors because it has n sides, 360 00:27:41,490 --> 00:27:46,500 so then the average number of sides of the n neighbors 361 00:27:46,500 --> 00:27:48,795 is called m bar. 362 00:27:48,795 --> 00:27:50,392 It has n sides. 363 00:27:54,890 --> 00:28:13,520 So then the average number of sides of its n neighbors 364 00:28:13,520 --> 00:28:16,490 is m bar. 365 00:28:16,490 --> 00:28:25,940 And m bar is equal to 5 plus 6 over n for a 2D honeycomb 366 00:28:25,940 --> 00:28:29,950 kind of cell structure. 367 00:28:29,950 --> 00:28:32,110 OK, so that's the Aboav-Weaire law. 368 00:28:36,146 --> 00:28:37,895 And then there's one more of these things. 369 00:29:10,920 --> 00:29:12,915 The last one is called Lewis' rule. 370 00:29:17,850 --> 00:29:22,000 And Lewis looked at biological cells and 2D cell patterns, 371 00:29:22,000 --> 00:29:24,110 and he found that the area of the cell 372 00:29:24,110 --> 00:29:26,730 varied linearly with the number of the sides. 373 00:30:12,970 --> 00:30:16,580 And he found just an empirical relationship. 374 00:30:16,580 --> 00:30:18,341 It looks like this. 375 00:30:18,341 --> 00:30:20,090 We can say what everything is in a minute. 376 00:30:24,260 --> 00:30:28,730 So he found that the area of a cell with n sides, that's A n, 377 00:30:28,730 --> 00:30:31,040 was linearly related to the number of sides, 378 00:30:31,040 --> 00:30:34,280 so this n over here, and naught's just a constant 379 00:30:34,280 --> 00:30:39,620 and A n bar is the area of the cell with the average number 380 00:30:39,620 --> 00:30:41,220 of sides. 381 00:30:41,220 --> 00:30:56,253 So here, A n is the area of cells with n sides and A n bar. 382 00:31:08,810 --> 00:31:10,470 And then n naught is just a constant. 383 00:31:13,400 --> 00:31:18,450 And he found that for 2D cells n naught was equal to 2. 384 00:31:22,805 --> 00:31:24,430 And if you look at voronoi honeycombs-- 385 00:31:24,430 --> 00:31:26,430 remember last time we talked about those voronoi 386 00:31:26,430 --> 00:31:29,370 honeycombs-- you can show that this 387 00:31:29,370 --> 00:31:30,860 holds for voronoi honeycombs. 388 00:31:57,362 --> 00:32:02,210 And then you could write a 3D version of this as well. 389 00:32:02,210 --> 00:32:05,860 And in 3D, it's the volume and faces instead of the areas 390 00:32:05,860 --> 00:32:07,330 and the number of sides. 391 00:32:07,330 --> 00:32:11,210 So you can say that the volume of the cells with f 392 00:32:11,210 --> 00:32:14,010 faces relative to the volumes of the cells 393 00:32:14,010 --> 00:32:16,070 with the average number of faces, 394 00:32:16,070 --> 00:32:20,000 they vary linearly with the number of faces. 395 00:32:20,000 --> 00:32:22,677 So there's sort of an exactly analogous expression here. 396 00:32:44,810 --> 00:32:47,790 And here, this f naught is another constant. 397 00:32:47,790 --> 00:32:51,320 And in 3D it's about equal to 3. 398 00:32:51,320 --> 00:32:55,190 So these are all just kind of interesting topological rules 399 00:32:55,190 --> 00:32:57,070 that are nice to know about. 400 00:33:00,306 --> 00:33:00,805 OK. 401 00:33:15,090 --> 00:33:16,696 Yeah? 402 00:33:16,696 --> 00:33:17,770 AUDIENCE: Basic question. 403 00:33:17,770 --> 00:33:20,752 In two dimensions, what is a face? 404 00:33:20,752 --> 00:33:21,990 Like what does that refer to? 405 00:33:21,990 --> 00:33:24,770 LORNA GIBSON: So-- and let me put my little slides again. 406 00:33:30,940 --> 00:33:35,570 So say we have some honeycombs like this, then 407 00:33:35,570 --> 00:33:36,570 say we look at this guy. 408 00:33:36,570 --> 00:33:39,764 So that's a vertex there, that's an edge, 409 00:33:39,764 --> 00:33:41,680 and this thing in the middle here is the face. 410 00:33:41,680 --> 00:33:45,690 So in 2D, the face and the cell is kind of the same thing. 411 00:33:45,690 --> 00:33:46,310 All right? 412 00:33:46,310 --> 00:33:46,810 And then-- 413 00:33:46,810 --> 00:33:49,602 AUDIENCE: [INAUDIBLE] sides, what's different with the edge? 414 00:33:49,602 --> 00:33:50,810 LORNA GIBSON: It's not quite. 415 00:33:50,810 --> 00:33:53,060 Like if I count up the number of edges, 416 00:33:53,060 --> 00:33:56,230 I say one, two, three, four, five, 417 00:33:56,230 --> 00:33:58,310 and I count those up like that. 418 00:33:58,310 --> 00:34:01,540 But if I say sides of the face-- see that's a face 419 00:34:01,540 --> 00:34:04,060 or it's a cell-- I would say that had six sides. 420 00:34:04,060 --> 00:34:05,320 Right? 421 00:34:05,320 --> 00:34:08,690 So typically when people say it has-- a cell 422 00:34:08,690 --> 00:34:11,840 or a face has so many sides, they count up how many 423 00:34:11,840 --> 00:34:13,130 all around it. 424 00:34:13,130 --> 00:34:15,406 But because each side is-- or each edge 425 00:34:15,406 --> 00:34:17,489 is-- shared between two faces, the number of edges 426 00:34:17,489 --> 00:34:19,440 is actually half that. 427 00:34:19,440 --> 00:34:20,040 Right? 428 00:34:20,040 --> 00:34:25,275 Because there's one edge, but if I count up the number of sides 429 00:34:25,275 --> 00:34:27,900 for this face and I count up the number of sides for that face, 430 00:34:27,900 --> 00:34:30,760 I'm counting that twice. 431 00:34:30,760 --> 00:34:33,250 So it's a little bit-- so I try to see edges 432 00:34:33,250 --> 00:34:35,070 when I'm talking about adding them all up 433 00:34:35,070 --> 00:34:38,070 and I try to say sides when I'm talking about, 434 00:34:38,070 --> 00:34:40,610 here's a face, how many sides does it have. 435 00:34:40,610 --> 00:34:41,110 OK? 436 00:34:41,110 --> 00:34:42,276 It's a little bit confusing. 437 00:34:44,419 --> 00:34:44,960 Anybody else? 438 00:34:44,960 --> 00:34:51,361 AUDIENCE: [INAUDIBLE] what is n bar naught? 439 00:34:51,361 --> 00:34:53,610 LORNA GIBSON: n bar naught, did I put an n bar naught? 440 00:34:53,610 --> 00:34:56,709 Oh, that's where I didn't erase this enough. 441 00:34:56,709 --> 00:34:57,250 There you go. 442 00:34:57,250 --> 00:34:58,210 It's just n naught. 443 00:35:01,926 --> 00:35:02,425 OK. 444 00:35:22,660 --> 00:35:24,470 So we've been talking about the structure 445 00:35:24,470 --> 00:35:26,530 of the honeycombs in the foams. 446 00:35:26,530 --> 00:35:27,990 And what we ultimately want to do 447 00:35:27,990 --> 00:35:29,984 is be able to model the mechanical behavior 448 00:35:29,984 --> 00:35:31,900 or the thermal behavior, some sort of behavior 449 00:35:31,900 --> 00:35:33,660 of the cellular material. 450 00:35:33,660 --> 00:35:35,420 And for looking at mechanical behavior, 451 00:35:35,420 --> 00:35:38,150 there's three main approaches that people take. 452 00:35:38,150 --> 00:35:40,700 So you have to model the structure somehow. 453 00:35:40,700 --> 00:35:42,076 So there's three main approaches. 454 00:35:48,920 --> 00:35:51,503 And the first one is to use the unit cell. 455 00:35:54,040 --> 00:35:59,300 So say, for example, for the honeycombs, 456 00:35:59,300 --> 00:36:01,380 if you have a hexagonal honeycomb you'd just 457 00:36:01,380 --> 00:36:03,260 use that unit hexagonal cell. 458 00:36:06,390 --> 00:36:08,990 And that's what we're going to do probably starting later 459 00:36:08,990 --> 00:36:10,722 on today. 460 00:36:10,722 --> 00:36:12,930 So for the hexagonal honeycomb, it's kind of obvious. 461 00:36:12,930 --> 00:36:16,070 You would use a unit cell, the thing's periodic, 462 00:36:16,070 --> 00:36:20,290 you figure out how that unit cell behaves, you're all set. 463 00:36:20,290 --> 00:36:24,490 For a foam, it was not so obvious a unit cell to use. 464 00:36:24,490 --> 00:36:26,560 But the thing-- people use different things, 465 00:36:26,560 --> 00:36:29,060 but one of the common ones they use is a tetrakaidecahedron. 466 00:36:35,020 --> 00:36:37,000 So the nice thing about the tetrakaidecahedron 467 00:36:37,000 --> 00:36:38,980 is that it packs to fill space. 468 00:36:38,980 --> 00:36:40,690 So it's a repeating single cell. 469 00:36:40,690 --> 00:36:42,320 Packs to fill space. 470 00:36:42,320 --> 00:36:45,190 But, in fact, real foams aren't tetrakaidecahedrons, 471 00:36:45,190 --> 00:36:46,880 so this is a bit of an idealization. 472 00:36:46,880 --> 00:36:52,580 So we'll just say foam cells are not tetrakaidecahedrons. 473 00:37:05,730 --> 00:37:09,070 OK, so that's one approach. 474 00:37:09,070 --> 00:37:11,270 And then a second approach is to use something 475 00:37:11,270 --> 00:37:12,530 called dimensional analysis. 476 00:37:19,470 --> 00:37:22,050 So in dimensional analysis, what you 477 00:37:22,050 --> 00:37:23,730 want to do here, with this technique, 478 00:37:23,730 --> 00:37:26,830 is model the mechanisms of deformation and failure 479 00:37:26,830 --> 00:37:28,420 in the structure. 480 00:37:28,420 --> 00:37:31,220 But you don't necessarily represent the cell geometry 481 00:37:31,220 --> 00:37:32,290 exactly. 482 00:37:32,290 --> 00:37:34,630 And so what you do is, you say one thing 483 00:37:34,630 --> 00:37:37,060 is proportional to another and there's 484 00:37:37,060 --> 00:37:38,870 some constant of proportionality, 485 00:37:38,870 --> 00:37:41,820 and you just wrap all of those constants of proportionality up 486 00:37:41,820 --> 00:37:43,020 at the end. 487 00:37:43,020 --> 00:37:46,900 So for instance, when people look at foams, 488 00:37:46,900 --> 00:37:49,470 the geometry of the foams is kind of complicated. 489 00:37:49,470 --> 00:37:52,090 There's cells with different number of faces, 490 00:37:52,090 --> 00:37:54,880 there's different sizes of cells, it's kind of a mess. 491 00:37:54,880 --> 00:37:57,540 So we could just say the geometry is complex 492 00:37:57,540 --> 00:37:58,800 and it's difficult to model. 493 00:38:09,470 --> 00:38:13,130 And with dimensional analysis, instead what we do 494 00:38:13,130 --> 00:38:16,840 is we model the deformation mechanisms and failure 495 00:38:16,840 --> 00:38:19,190 mechanisms. 496 00:38:19,190 --> 00:38:22,700 And you can get quite a bit out of just modeling those. 497 00:38:46,780 --> 00:38:49,594 So when we look at modeling foams, we're going to do this, 498 00:38:49,594 --> 00:38:50,760 and you'll see how it works. 499 00:39:15,610 --> 00:39:18,350 And then the third method is to use finite element analysis. 500 00:39:24,970 --> 00:39:28,800 So this is a numerical technique and it's a very 501 00:39:28,800 --> 00:39:31,570 standard numerical technique. 502 00:39:31,570 --> 00:39:33,270 And one of the nice things about this 503 00:39:33,270 --> 00:39:35,064 is you can apply it to random structures. 504 00:39:35,064 --> 00:39:36,730 So for example, those voronoi structures 505 00:39:36,730 --> 00:39:39,400 we saw, if you want to try to see-- 506 00:39:39,400 --> 00:39:42,190 say you had a certain amount of material 507 00:39:42,190 --> 00:39:45,140 and you had a honeycomb that was a regular hexagonal honeycomb. 508 00:39:45,140 --> 00:39:46,100 You had the same amount of material 509 00:39:46,100 --> 00:39:47,120 and you put it in a voronoi honeycomb, 510 00:39:47,120 --> 00:39:50,140 and you want to know how does having a random material affect 511 00:39:50,140 --> 00:39:52,800 the properties relative to the uniform material? 512 00:39:52,800 --> 00:39:55,350 You could figure that out using finite element analysis. 513 00:39:55,350 --> 00:39:57,058 So you can apply it to random structures. 514 00:40:13,060 --> 00:40:15,820 And another thing you can do with finite element analysis-- 515 00:40:15,820 --> 00:40:18,560 and people in the orthopedics end of the world 516 00:40:18,560 --> 00:40:21,610 often do this-- if you're interested in trabecular bone, 517 00:40:21,610 --> 00:40:24,720 that porous type of bone I showed you the first day, 518 00:40:24,720 --> 00:40:28,400 people can take micro-computed tomography images of trabecular 519 00:40:28,400 --> 00:40:32,520 bone and they get a file which basically says, every voxel, 520 00:40:32,520 --> 00:40:34,560 says if it's solid or it's void. 521 00:40:34,560 --> 00:40:36,870 And you can use those files as input 522 00:40:36,870 --> 00:40:38,910 to a finite element analysis. 523 00:40:38,910 --> 00:40:41,910 And so you can analyze exactly how a piece of a trabecular 524 00:40:41,910 --> 00:40:43,520 bone would deform. 525 00:40:43,520 --> 00:40:44,580 So it's nice for that. 526 00:41:12,164 --> 00:41:14,330 So one of the things we're going to talk about later 527 00:41:14,330 --> 00:41:17,610 is we'll look at the structure of trabecular bone 528 00:41:17,610 --> 00:41:19,630 and the sort of mechanics of trabecular bone, 529 00:41:19,630 --> 00:41:22,510 and I'll show you some results that a former student of mine 530 00:41:22,510 --> 00:41:24,690 did, where we look at-- say you have 531 00:41:24,690 --> 00:41:27,730 a sort of intact structure of a certain density 532 00:41:27,730 --> 00:41:30,460 and then you reduce the density by fitting the cell walls 533 00:41:30,460 --> 00:41:32,890 or you reduce the density by removing cell walls-- because 534 00:41:32,890 --> 00:41:36,140 in osteoporosis sometimes the walls resorb all together-- 535 00:41:36,140 --> 00:41:37,890 and you can see what residual strength you 536 00:41:37,890 --> 00:41:40,630 would have for some given amount of density loss. 537 00:41:40,630 --> 00:41:43,690 So it's good for looking at that kind of a thing. 538 00:41:43,690 --> 00:41:45,735 You can also use it for looking at local effects. 539 00:41:50,790 --> 00:41:53,707 So you can look at defects, for instance. 540 00:41:53,707 --> 00:41:55,290 So if you think of the trabecular bone 541 00:41:55,290 --> 00:41:57,802 as having missing trabeculae, that 542 00:41:57,802 --> 00:41:59,260 would be a defect in the structure. 543 00:41:59,260 --> 00:42:00,710 You can look at that. 544 00:42:00,710 --> 00:42:02,230 You can look at size effects. 545 00:42:06,905 --> 00:42:09,740 So when we were studying some of those metal foams, some of them 546 00:42:09,740 --> 00:42:11,910 have very large cells, and if you 547 00:42:11,910 --> 00:42:14,390 have a small sample relative to the cell size, 548 00:42:14,390 --> 00:42:17,870 you may only have four or five cells across the dimension. 549 00:42:17,870 --> 00:42:19,360 Where you've cut the cells, you've 550 00:42:19,360 --> 00:42:21,000 got edges that are less constrained 551 00:42:21,000 --> 00:42:23,704 than in the bulk of the material because at the outer edge 552 00:42:23,704 --> 00:42:24,370 you've cut them. 553 00:42:24,370 --> 00:42:26,030 They're not connected to anything else. 554 00:42:26,030 --> 00:42:29,300 And so you can look at edge effects that relate to the size 555 00:42:29,300 --> 00:42:32,712 effects in foams as well. 556 00:42:53,200 --> 00:42:55,030 So these are basically the three approaches 557 00:42:55,030 --> 00:42:58,462 that people use for modeling cellular materials. 558 00:42:58,462 --> 00:42:59,920 And what we're going to do is we're 559 00:42:59,920 --> 00:43:02,700 going to start with looking at honeycombs 560 00:43:02,700 --> 00:43:05,660 and we're going to look at these hexagonal kind of honeycombs 561 00:43:05,660 --> 00:43:06,870 like this. 562 00:43:06,870 --> 00:43:09,210 And one reason to start with them 563 00:43:09,210 --> 00:43:11,400 is that they have this unit cell structure. 564 00:43:11,400 --> 00:43:14,440 And if you can analyze how that unit cell deforms and fails, 565 00:43:14,440 --> 00:43:16,770 you can say something about the whole structure. 566 00:43:16,770 --> 00:43:18,830 And it turns out that the honeycombs, 567 00:43:18,830 --> 00:43:21,960 they deform and fail by the same mechanisms as the foams. 568 00:43:21,960 --> 00:43:24,680 So if you can understand through this simple structure, 569 00:43:24,680 --> 00:43:27,370 it gives you a lot of insight into how the foams behave. 570 00:43:27,370 --> 00:43:30,644 So if I deform this a little bit, the cell walls bend. 571 00:43:30,644 --> 00:43:33,060 And you can show that if you deform this guy a little bit, 572 00:43:33,060 --> 00:43:34,430 the cell walls bend. 573 00:43:34,430 --> 00:43:37,000 And so you can learn a lot by looking at the honeycombs 574 00:43:37,000 --> 00:43:39,435 and then sort of applying that to the foams. 575 00:43:39,435 --> 00:43:41,060 So we're going to start off-- so that's 576 00:43:41,060 --> 00:43:43,740 the end of the section on sort of the structure 577 00:43:43,740 --> 00:43:45,920 of the cellular materials-- and now we're 578 00:43:45,920 --> 00:43:48,617 going to look at modeling the mechanical properties 579 00:43:48,617 --> 00:43:50,450 and we're going to start with the honeycombs 580 00:43:50,450 --> 00:43:52,210 and then we're going to do foams. 581 00:43:52,210 --> 00:43:56,450 So when we talk about the honeycombs, 582 00:43:56,450 --> 00:43:59,080 we've got this hexagonal structure here 583 00:43:59,080 --> 00:44:00,700 and we're going to call properties 584 00:44:00,700 --> 00:44:03,090 in this plane the in-plane property. 585 00:44:03,090 --> 00:44:05,130 So if I load it this way on or that way on, 586 00:44:05,130 --> 00:44:07,940 those are in-plane, and if I load it that way on, 587 00:44:07,940 --> 00:44:09,050 those are out of plane. 588 00:44:09,050 --> 00:44:11,060 So think of the cells are in the plane 589 00:44:11,060 --> 00:44:13,520 and the prismatic direction is the out of plane. 590 00:44:13,520 --> 00:44:15,290 And clearly, the honeycombs are going 591 00:44:15,290 --> 00:44:18,150 to have similar properties this way and that way, 592 00:44:18,150 --> 00:44:20,900 but they are going to have very different properties that way. 593 00:44:20,900 --> 00:44:23,024 So we're going to start with the in-plane behavior. 594 00:44:47,770 --> 00:44:54,380 So the honeycombs have these prismatic cells 595 00:44:54,380 --> 00:45:04,590 and they're widely available in different materials, polymers, 596 00:45:04,590 --> 00:45:05,860 metals, ceramics. 597 00:45:15,226 --> 00:45:17,280 And they're used in a variety of applications. 598 00:45:20,020 --> 00:45:22,540 So one of the most common is to use them in sandwich panels. 599 00:45:25,340 --> 00:45:28,410 So I brought a couple of sandwich panels with me today. 600 00:45:28,410 --> 00:45:30,340 So here's a couple of sandwich panels 601 00:45:30,340 --> 00:45:31,560 that have honeycomb cores. 602 00:45:31,560 --> 00:45:33,550 This one's an aircraft flooring panel. 603 00:45:33,550 --> 00:45:38,870 It's got carbon fiber faces and a Nomex core, honeycomb core. 604 00:45:38,870 --> 00:45:41,000 And this is an aluminum honeycomb. 605 00:45:41,000 --> 00:45:43,860 It's got aluminum honeycomb core and then aluminum faces, 606 00:45:43,860 --> 00:45:47,020 and it's kind of amazing how stiff that little panel is. 607 00:45:47,020 --> 00:45:49,840 And each of the pieces is really not that stiff at all, 608 00:45:49,840 --> 00:45:51,670 but the thing put together is quite stiff, 609 00:45:51,670 --> 00:45:55,630 and we'll talk more about that when we get to sandwich panels. 610 00:45:55,630 --> 00:45:57,047 So it's used in sandwich panels. 611 00:45:57,047 --> 00:45:58,505 They're used for energy absorption. 612 00:46:03,400 --> 00:46:08,170 So sometimes you'll see there's some natural disaster area 613 00:46:08,170 --> 00:46:09,930 and they fly in helicopters and they 614 00:46:09,930 --> 00:46:13,977 drop big crates of supplies, and they 615 00:46:13,977 --> 00:46:16,060 will have it like a pallet with a big crate thing. 616 00:46:16,060 --> 00:46:19,450 Often they have a honeycomb, like a metal honeycomb, that 617 00:46:19,450 --> 00:46:20,930 is kind of oriented this way. 618 00:46:20,930 --> 00:46:24,174 So the pallet would be like this and the honeycomb's like that. 619 00:46:24,174 --> 00:46:25,840 And the idea is that when they drop it-- 620 00:46:25,840 --> 00:46:29,150 they sort of bring it down as close as they can-- 621 00:46:29,150 --> 00:46:31,340 and then the honeycomb absorbs some 622 00:46:31,340 --> 00:46:33,300 of the energy from the impact. 623 00:46:33,300 --> 00:46:36,977 And they're also used, I think, sometimes in car bumpers. 624 00:46:36,977 --> 00:46:38,560 So they're used for energy absorption. 625 00:46:38,560 --> 00:46:40,374 And they're used as carriers for catalysts. 626 00:46:46,020 --> 00:46:49,400 So the catalytic converter in your car 627 00:46:49,400 --> 00:46:51,840 looks like this, this is kind of the material that's 628 00:46:51,840 --> 00:46:54,400 used in the catalytic converter in your car. 629 00:46:54,400 --> 00:46:58,610 And the way that works is, the cell walls here are actually 630 00:46:58,610 --> 00:47:01,680 porous and they're coated in the platinum, which 631 00:47:01,680 --> 00:47:05,980 is the catalyst, and half of the cells 632 00:47:05,980 --> 00:47:07,250 are blocked off on this end. 633 00:47:07,250 --> 00:47:09,440 So every other cell on this end is blocked off 634 00:47:09,440 --> 00:47:12,280 and then every other cell over here, the opposite ones, 635 00:47:12,280 --> 00:47:13,300 are blocked off. 636 00:47:13,300 --> 00:47:16,320 And so the gas is forced down a channel 637 00:47:16,320 --> 00:47:18,830 but then through the wall and then out the next channel. 638 00:47:18,830 --> 00:47:21,450 And that's where the reaction actually occurs, 639 00:47:21,450 --> 00:47:22,970 is in the cell wall. 640 00:47:22,970 --> 00:47:25,440 So they're used as carriers there. 641 00:47:25,440 --> 00:47:33,160 And some natural materials also have a cellular structure, 642 00:47:33,160 --> 00:47:34,285 have a honeycomb structure. 643 00:47:44,500 --> 00:47:49,380 So for example, things like woods and cork 644 00:47:49,380 --> 00:47:51,232 have a honeycomb structure, too. 645 00:47:54,210 --> 00:48:00,760 So I said the mechanisms of deformation and failure 646 00:48:00,760 --> 00:48:04,060 in the hexagonal honeycombs parallel those in foams. 647 00:48:29,140 --> 00:48:30,980 So we can learn a lot about foams 648 00:48:30,980 --> 00:48:32,790 by understanding the honeycombs. 649 00:48:32,790 --> 00:48:34,460 And then, similarly, the mechanisms 650 00:48:34,460 --> 00:48:43,540 of deformation and failure in triangulated honeycombs 651 00:48:43,540 --> 00:48:46,268 parallel those in the lattice materials. 652 00:48:51,050 --> 00:48:53,200 Remember I brought those lattice materials in? 653 00:48:53,200 --> 00:48:55,550 The sort of trust type materials. 654 00:48:55,550 --> 00:48:57,450 The triangular honeycombs have a behavior 655 00:48:57,450 --> 00:49:01,030 similar to those lattice materials. 656 00:49:01,030 --> 00:49:01,700 OK. 657 00:49:01,700 --> 00:49:03,470 So let me scoot over here. 658 00:49:30,371 --> 00:49:30,870 OK. 659 00:49:49,500 --> 00:49:50,910 So let me scoot out of this one. 660 00:49:57,420 --> 00:50:00,030 OK. 661 00:50:00,030 --> 00:50:03,130 So here is our kind of hexagonal geometry. 662 00:50:03,130 --> 00:50:06,000 This is kind of an idealized geometry here. 663 00:50:06,000 --> 00:50:09,120 And, as we talked about in this section on the structure, 664 00:50:09,120 --> 00:50:10,900 we're going to call these vertical walls, 665 00:50:10,900 --> 00:50:12,840 we're going to say they have a length h, 666 00:50:12,840 --> 00:50:17,094 the inclined walls have a length l, the wall thickness is t, 667 00:50:17,094 --> 00:50:19,760 and there's going to be an angle between the horizontal and that 668 00:50:19,760 --> 00:50:22,110 inclined wall of theta. 669 00:50:22,110 --> 00:50:24,370 And I'm going to define three axes here, 670 00:50:24,370 --> 00:50:28,130 an x1, an x2, and an x3 axis. 671 00:50:28,130 --> 00:50:32,900 So the x1, x2 plane is the in-plane and x3 is out 672 00:50:32,900 --> 00:50:33,630 of plane. 673 00:50:33,630 --> 00:50:34,130 OK? 674 00:50:36,590 --> 00:50:39,750 So if we load our honeycomb up, we 675 00:50:39,750 --> 00:50:42,250 get stress strain curves that look like this. 676 00:50:42,250 --> 00:50:44,690 So the ones on the left here, over here these three 677 00:50:44,690 --> 00:50:47,250 are all in compression, and the ones on the right, 678 00:50:47,250 --> 00:50:48,810 these three are all in tension. 679 00:50:48,810 --> 00:50:52,490 OK, so let's talk about the compression ones first. 680 00:50:52,490 --> 00:50:54,630 And let's start at the top. 681 00:50:54,630 --> 00:50:56,870 This is in elastomeric material, so 682 00:50:56,870 --> 00:50:58,720 like one of these rubber honeycombs. 683 00:50:58,720 --> 00:51:00,870 This would be a material that yields plastically, 684 00:51:00,870 --> 00:51:02,850 so like an aluminum honeycomb. 685 00:51:02,850 --> 00:51:06,240 And this would be a honeycomb that fails in a brittle manner, 686 00:51:06,240 --> 00:51:07,860 like one of those ceramic honeycombs. 687 00:51:07,860 --> 00:51:09,510 OK? 688 00:51:09,510 --> 00:51:11,290 So if we go up to the elastomeric one, 689 00:51:11,290 --> 00:51:15,770 up here, if I compress it I get a linear elastic part first, 690 00:51:15,770 --> 00:51:18,630 and then at some point, I get a stress plateau 691 00:51:18,630 --> 00:51:21,430 where the stress is almost constant for strains 692 00:51:21,430 --> 00:51:22,940 that are quite large. 693 00:51:22,940 --> 00:51:25,160 And then finally, the stress starts 694 00:51:25,160 --> 00:51:27,800 to rise quite sharply at the end here. 695 00:51:27,800 --> 00:51:30,610 So the strain here goes from 0 to 1. 696 00:51:30,610 --> 00:51:32,150 So that's a strain of 100%. 697 00:51:32,150 --> 00:51:34,930 You've completely flattened the thing there. 698 00:51:34,930 --> 00:51:36,400 So that's a large strain. 699 00:51:36,400 --> 00:51:40,150 So initially, when we're loading it up to smaller strains 700 00:51:40,150 --> 00:51:44,210 like this, we've got linear elastic behavior and these cell 701 00:51:44,210 --> 00:51:45,510 walls bend. 702 00:51:45,510 --> 00:51:47,617 And you can relate the modulus here-- 703 00:51:47,617 --> 00:51:49,200 let's see where'd my little arrow go-- 704 00:51:49,200 --> 00:51:52,200 you can relate this Young's modulus here to the bending 705 00:51:52,200 --> 00:51:53,420 of those cell walls. 706 00:51:53,420 --> 00:51:55,350 And we're going to-- I don't know 707 00:51:55,350 --> 00:51:58,180 if we'll finished that today-- but we'll start that today. 708 00:51:58,180 --> 00:52:00,986 Then-- oops, lost my arrow, where'd my arrow 709 00:52:00,986 --> 00:52:04,060 go-- then at the stress plateau here, 710 00:52:04,060 --> 00:52:07,470 that plateau is related to collapse of the cells. 711 00:52:07,470 --> 00:52:09,610 So if I have my little rubber honeycomb 712 00:52:09,610 --> 00:52:12,160 and I load it, at some point, the cell walls buckle. 713 00:52:12,160 --> 00:52:13,810 So you see how they've buckled there? 714 00:52:13,810 --> 00:52:17,100 And that stress plateau, I can smush that 715 00:52:17,100 --> 00:52:20,170 to quite large strains that are roughly constant stress. 716 00:52:20,170 --> 00:52:20,920 OK? 717 00:52:20,920 --> 00:52:26,449 So what's happening here is that we're buckling the cells. 718 00:52:26,449 --> 00:52:28,490 And as we go along here, the buckling deformation 719 00:52:28,490 --> 00:52:29,910 gets bigger and bigger. 720 00:52:29,910 --> 00:52:34,270 And then if I smush it and I've got it kind of like that, 721 00:52:34,270 --> 00:52:36,470 at some point it becomes much harder to press it 722 00:52:36,470 --> 00:52:38,740 together again because the cell walls are now touching 723 00:52:38,740 --> 00:52:40,980 each other and they're pressing against themselves, 724 00:52:40,980 --> 00:52:42,563 and to get a certain amount of strain, 725 00:52:42,563 --> 00:52:44,720 it gets much more difficult to do that. 726 00:52:44,720 --> 00:52:47,340 And the stress required to do that gets much bigger. 727 00:52:47,340 --> 00:52:50,350 And that's what leads to this last piece of the stress strain 728 00:52:50,350 --> 00:52:53,650 curve here, which is called densification because you've 729 00:52:53,650 --> 00:52:55,060 almost eliminated the pores. 730 00:52:55,060 --> 00:52:56,780 It's hard to eliminate them entirely 731 00:52:56,780 --> 00:52:58,780 but the porosity has gone way down 732 00:52:58,780 --> 00:53:00,480 by the time you're up there. 733 00:53:00,480 --> 00:53:02,560 So this type of stress strain curve 734 00:53:02,560 --> 00:53:04,570 where you've got linear elasticity and then 735 00:53:04,570 --> 00:53:07,880 a stress plateau and then the densification 736 00:53:07,880 --> 00:53:12,000 is classic for compressive behavior of cellular materials. 737 00:53:12,000 --> 00:53:14,740 So elastomeric ones will have a plateau that's 738 00:53:14,740 --> 00:53:16,220 related to elastic buckling. 739 00:53:16,220 --> 00:53:17,630 I lost my arrow again. 740 00:53:17,630 --> 00:53:18,960 There we go. 741 00:53:18,960 --> 00:53:21,000 If we had a metal honeycomb, we'd 742 00:53:21,000 --> 00:53:24,560 again have linear elasticity related to cell wall bending. 743 00:53:24,560 --> 00:53:26,280 This plateau here would be related 744 00:53:26,280 --> 00:53:27,850 to yielding of the cell walls. 745 00:53:27,850 --> 00:53:29,320 So say we had aluminum honeycomb, 746 00:53:29,320 --> 00:53:31,810 the aluminum could yield, and that would again 747 00:53:31,810 --> 00:53:33,330 cause this stress plateau. 748 00:53:33,330 --> 00:53:35,230 And then we get densification. 749 00:53:35,230 --> 00:53:38,030 If I had a brittle honeycomb like the ceramic one, 750 00:53:38,030 --> 00:53:40,076 we'd have initial linear elasticity. 751 00:53:40,076 --> 00:53:41,700 Then you've got a stress plateau that's 752 00:53:41,700 --> 00:53:43,690 kind of a lot of up and down here. 753 00:53:43,690 --> 00:53:47,650 And the serrated nature is related to the fracture 754 00:53:47,650 --> 00:53:49,154 of individual cell walls. 755 00:53:49,154 --> 00:53:51,570 So it goes up and down because when you break a cell wall, 756 00:53:51,570 --> 00:53:54,900 the stress drops off, and then the other cell walls will try 757 00:53:54,900 --> 00:53:56,520 to pick the stress up again. 758 00:53:56,520 --> 00:53:58,550 And so each one of those little up and downs 759 00:53:58,550 --> 00:54:00,849 corresponds to breaking a cell wall. 760 00:54:00,849 --> 00:54:02,640 But if you kind of took an average of that, 761 00:54:02,640 --> 00:54:04,220 you can see there's a stress plateau 762 00:54:04,220 --> 00:54:06,300 and then there's the densification region. 763 00:54:06,300 --> 00:54:10,740 So in compression, the shape of the curves is very similar 764 00:54:10,740 --> 00:54:14,660 and the mechanism of the plateau varies a little bit. 765 00:54:14,660 --> 00:54:17,322 So let's see if I can write some of that down. 766 00:54:22,650 --> 00:54:25,785 So in compression, we can say we have three regimes of behavior. 767 00:54:32,050 --> 00:54:37,785 So we've got the linear elastic regime initially 768 00:54:37,785 --> 00:54:40,243 and we're going to see that's related to cell wall bending. 769 00:54:43,380 --> 00:54:45,030 Oh, thanks. 770 00:54:45,030 --> 00:54:46,661 And then we've got a stress plateau. 771 00:54:50,880 --> 00:54:53,840 And for elastomeric materials, that's 772 00:54:53,840 --> 00:54:56,500 caused by buckling of the walls. 773 00:54:56,500 --> 00:55:01,170 For metals, it would be caused by yielding. 774 00:55:01,170 --> 00:55:04,435 And for ceramics, it would be caused by a brittle crushing. 775 00:55:10,286 --> 00:55:11,661 And then we've got densification. 776 00:55:21,844 --> 00:55:23,760 And that's related to the cell walls touching. 777 00:55:37,200 --> 00:55:39,960 And if we increase the ratio of the thickness of the cell walls 778 00:55:39,960 --> 00:55:41,950 relative to their length, we're going 779 00:55:41,950 --> 00:55:46,680 to increase the stiffness, so the Young's 780 00:55:46,680 --> 00:55:48,570 modulus of the honeycomb. 781 00:55:48,570 --> 00:55:52,540 The stress plateau I'm going to call sigma star. 782 00:55:52,540 --> 00:55:55,410 And we decrease the strain at which 783 00:55:55,410 --> 00:55:59,290 that densification occurs, which I'm going to call epsilon D. 784 00:55:59,290 --> 00:56:02,590 So you see over on the right hand side of these plots 785 00:56:02,590 --> 00:56:05,515 here, that strain there is the densification strain, epsilon 786 00:56:05,515 --> 00:56:28,110 D. So that's in compression. 787 00:56:28,110 --> 00:56:32,400 And these materials are very often loaded in compression. 788 00:56:32,400 --> 00:56:38,190 In tension, we still get linear elasticity initially. 789 00:56:38,190 --> 00:56:39,940 And that's going to, again, be related 790 00:56:39,940 --> 00:56:41,350 to the bending of the cell walls. 791 00:56:44,430 --> 00:56:50,530 But if we look at the stress plateau, 792 00:56:50,530 --> 00:56:53,770 if you look at these three curves here on the right, 793 00:56:53,770 --> 00:56:56,920 the stress plateau only exists if the material has a yield 794 00:56:56,920 --> 00:56:59,400 point and you get some plastic yielding there. 795 00:56:59,400 --> 00:57:01,710 If you have an elastomer, if I pull on this-- 796 00:57:01,710 --> 00:57:03,840 you don't get buckling in tension, 797 00:57:03,840 --> 00:57:06,500 so you're not going to get a stress plateau in tension. 798 00:57:06,500 --> 00:57:08,170 And for a brittle material like one 799 00:57:08,170 --> 00:57:10,910 of those brittle honeycombs, if you pull that in tension, you 800 00:57:10,910 --> 00:57:13,076 would just get a crack propagate and the thing would 801 00:57:13,076 --> 00:57:15,770 break into two pieces and so you don't get a stress plateau 802 00:57:15,770 --> 00:57:17,020 there. 803 00:57:17,020 --> 00:57:19,460 So the stress plateau in tension only 804 00:57:19,460 --> 00:57:22,640 exists if the material yields. 805 00:57:39,090 --> 00:57:41,230 So you don't get any buckling in tension. 806 00:57:41,230 --> 00:57:44,600 And for a brittle honeycomb, you would just get fracture. 807 00:57:55,625 --> 00:57:56,125 OK? 808 00:58:01,640 --> 00:58:04,150 These inclined walls are going to bend. 809 00:58:04,150 --> 00:58:07,510 We're going to see that in more gory detail in one minute. 810 00:58:07,510 --> 00:58:09,290 If you can wait one minute. 811 00:58:09,290 --> 00:58:10,780 OK. 812 00:58:10,780 --> 00:58:12,930 So are we good with a stress strain curves yet? 813 00:58:12,930 --> 00:58:14,842 AUDIENCE: More of an abstract question, 814 00:58:14,842 --> 00:58:23,490 but is it possible [INAUDIBLE] to get collapse this way 815 00:58:23,490 --> 00:58:25,334 before the plateau? 816 00:58:29,030 --> 00:58:30,960 LORNA GIBSON: I haven't seen that. 817 00:58:30,960 --> 00:58:33,810 But maybe it it's possible, I don't know. 818 00:58:33,810 --> 00:58:35,390 I don't think so, but maybe. 819 00:58:35,390 --> 00:58:37,741 Anybody else? 820 00:58:37,741 --> 00:58:38,240 OK. 821 00:58:40,780 --> 00:58:45,180 OK, so here's some photographs of honeycombs. 822 00:58:45,180 --> 00:58:48,330 So these ones are white but these ones are just the same. 823 00:58:48,330 --> 00:58:51,190 So most of them are just a regular hexagonal honeycomb 824 00:58:51,190 --> 00:58:54,680 and one of them is this funny shaped honeycomb here. 825 00:58:54,680 --> 00:58:58,640 These two guys at the top, these two here, are unloaded. 826 00:58:58,640 --> 00:59:00,720 And then the rest of them have some loading. 827 00:59:00,720 --> 00:59:02,677 So if you look at this one down here-- so here 828 00:59:02,677 --> 00:59:04,510 I'm taking the honeycomb and I'm loading it. 829 00:59:04,510 --> 00:59:07,240 This is the x1 direction, that way on. 830 00:59:07,240 --> 00:59:10,520 And if you look very carefully, you 831 00:59:10,520 --> 00:59:12,720 can see what happens is these vertical walls just 832 00:59:12,720 --> 00:59:15,720 move sideways and that is going to zoot, 833 00:59:15,720 --> 00:59:16,920 with the sound effects. 834 00:59:16,920 --> 00:59:21,510 And then the-- I can't help making sound effects-- 835 00:59:21,510 --> 00:59:23,310 and then these guys here bend. 836 00:59:23,310 --> 00:59:25,750 OK, so this guy here-- it's maybe a little hard 837 00:59:25,750 --> 00:59:27,460 to see it on the image but it's actually 838 00:59:27,460 --> 00:59:29,320 a little bit bent there. 839 00:59:29,320 --> 00:59:31,840 So that's loading it this way on. 840 00:59:31,840 --> 00:59:35,580 And then similarly, if I take it and I load it that way on, 841 00:59:35,580 --> 00:59:37,710 that's the x2 direction. 842 00:59:37,710 --> 00:59:39,770 And now these guys here, the vertical guys, 843 00:59:39,770 --> 00:59:41,430 are just going to compress a little. 844 00:59:41,430 --> 00:59:43,150 But these guys here, the incline guys, 845 00:59:43,150 --> 00:59:44,730 are still going to bend a little bit. 846 00:59:44,730 --> 00:59:46,300 And I've got some schematics that's 847 00:59:46,300 --> 00:59:48,161 going to show that a little bit better. 848 00:59:48,161 --> 00:59:50,410 And then if I shear it, if I took it like this and I-- 849 00:59:50,410 --> 00:59:52,868 I can't do it because my hands aren't glued to the rubber-- 850 00:59:52,868 --> 00:59:56,050 but if I could sort of shear it this way on, 851 00:59:56,050 --> 00:59:59,230 then you would get this kind of deformation here 852 00:59:59,230 --> 01:00:00,590 and that also involves bending. 853 01:00:00,590 --> 01:00:02,131 You can imagine these guys here would 854 01:00:02,131 --> 01:00:04,810 bend if I do that to them. 855 01:00:04,810 --> 01:00:07,560 And then the buckling deformation looks like this. 856 01:00:07,560 --> 01:00:12,140 If I scooch that like that, it should look kind 857 01:00:12,140 --> 01:00:14,030 of like that picture there. 858 01:00:14,030 --> 01:00:14,697 OK? 859 01:00:14,697 --> 01:00:16,530 So that's kind of what the deformation looks 860 01:00:16,530 --> 01:00:19,200 like for these sorts of honeycombs. 861 01:00:19,200 --> 01:00:21,510 So these were elastomer rubbers. 862 01:00:21,510 --> 01:00:24,130 These are just some images from an aluminum honeycomb. 863 01:00:24,130 --> 01:00:26,620 So here's, the top one's the undeformed honeycomb, 864 01:00:26,620 --> 01:00:28,710 the middle one's loading it in the one direction 865 01:00:28,710 --> 01:00:30,600 from left to right, and the bottom one's 866 01:00:30,600 --> 01:00:34,100 loading it in the two direction from top and bottom like that. 867 01:00:34,100 --> 01:00:34,740 OK? 868 01:00:34,740 --> 01:00:39,470 So you can imagine that you've got little cell walls. 869 01:00:39,470 --> 01:00:42,810 If you load it up high enough, those walls are going to yield 870 01:00:42,810 --> 01:00:45,840 and we're going to see that the yielding in the walls 871 01:00:45,840 --> 01:00:47,400 causes the formation of something 872 01:00:47,400 --> 01:00:49,900 called plastic hinges and the walls can rotate 873 01:00:49,900 --> 01:00:52,670 and they can produce these kind of shapes. 874 01:00:52,670 --> 01:00:53,530 OK. 875 01:00:53,530 --> 01:00:57,090 So then this is sort of a schematic stress strain curve 876 01:00:57,090 --> 01:00:57,870 here. 877 01:00:57,870 --> 01:01:00,120 And this is showing what happens if you increase 878 01:01:00,120 --> 01:01:02,959 the thickness of the walls relative to the length, 879 01:01:02,959 --> 01:01:04,500 or you increase the relative density, 880 01:01:04,500 --> 01:01:06,600 you increase the volume per action of solids. 881 01:01:06,600 --> 01:01:09,040 And this kind of shows the different regimes. 882 01:01:09,040 --> 01:01:10,730 So over here-- well let's, first of all, 883 01:01:10,730 --> 01:01:12,480 start with the different relative density. 884 01:01:12,480 --> 01:01:14,840 So this one here is the lowest relative density, 885 01:01:14,840 --> 01:01:17,240 then this is higher, and higher, and higher. 886 01:01:17,240 --> 01:01:18,950 And not too surprisingly, the more you 887 01:01:18,950 --> 01:01:20,860 increase the density, the more material 888 01:01:20,860 --> 01:01:23,580 you've got, the stiffer it is, the stronger it's going to be. 889 01:01:23,580 --> 01:01:25,180 And the more material you've got, 890 01:01:25,180 --> 01:01:26,720 the sooner it's going to densify. 891 01:01:26,720 --> 01:01:28,724 If you've got more solid in here, 892 01:01:28,724 --> 01:01:30,390 you're going to reach that densification 893 01:01:30,390 --> 01:01:32,510 strain at a smaller number. 894 01:01:32,510 --> 01:01:36,550 OK, so the shape of the curves looks like this. 895 01:01:36,550 --> 01:01:38,830 And you can define these three kind of regimes. 896 01:01:38,830 --> 01:01:41,010 So everything in here is linear elastic, 897 01:01:41,010 --> 01:01:42,880 everything in this big sort of envelope 898 01:01:42,880 --> 01:01:46,470 here is the plateau region, and everything up here 899 01:01:46,470 --> 01:01:47,910 is this densification region. 900 01:01:47,910 --> 01:01:51,230 So this is just a bigger picture kind of plot. 901 01:01:51,230 --> 01:01:55,560 OK, so let me skip through all of that. 902 01:01:55,560 --> 01:01:57,830 All right. 903 01:01:57,830 --> 01:02:00,110 OK, where are we? 904 01:02:00,110 --> 01:02:01,200 Chalk? 905 01:02:01,200 --> 01:02:01,980 Here we go. 906 01:02:53,710 --> 01:02:56,210 OK, so I think I mentioned this before, but let 907 01:02:56,210 --> 01:02:58,080 me just go over it again. 908 01:02:58,080 --> 01:03:00,340 So there's three types of things that 909 01:03:00,340 --> 01:03:03,110 affect the properties, the mechanical properties, 910 01:03:03,110 --> 01:03:04,269 of the honeycombs. 911 01:03:04,269 --> 01:03:05,810 And probably the most important thing 912 01:03:05,810 --> 01:03:07,518 is the relative density of the honeycomb. 913 01:03:09,804 --> 01:03:11,970 So remember, we said this was the same as the volume 914 01:03:11,970 --> 01:03:16,560 fraction of solids, and for a hexagonal honeycomb, 915 01:03:16,560 --> 01:03:21,210 you can show that's equal to the thickness to length ratio 916 01:03:21,210 --> 01:03:27,970 times h over l plus 2 divided by 2 cos theta times h over l 917 01:03:27,970 --> 01:03:31,130 plus sin theta. 918 01:03:31,130 --> 01:03:39,150 And if you have regular hexagons, so h over l 919 01:03:39,150 --> 01:03:43,650 is equal to 1, all the sides are of equal length, 920 01:03:43,650 --> 01:03:48,740 theta is equal to 30 degrees, the relevant density 921 01:03:48,740 --> 01:03:55,640 is 2 over 3 times t over l. 922 01:03:55,640 --> 01:03:57,390 So it just goes linearly with t over l. 923 01:03:57,390 --> 01:04:01,720 The thickness to length ratio of the walls. 924 01:04:01,720 --> 01:04:03,570 So it depends on how much solid you've got. 925 01:04:03,570 --> 01:04:05,130 Depends on the solid properties. 926 01:04:17,780 --> 01:04:20,240 So the Young's modulus of the solid, yield strength 927 01:04:20,240 --> 01:04:22,680 if it's a metal, some sort of fracture strength 928 01:04:22,680 --> 01:04:23,360 if it's brittle. 929 01:04:25,970 --> 01:04:32,280 It also depends on the cell geometry, 930 01:04:32,280 --> 01:04:38,760 which we can describe with h over l and theta. 931 01:04:38,760 --> 01:04:53,800 So if we think of a cell here-- that's our edge length h, 932 01:04:53,800 --> 01:05:00,160 that's our edge length l, that's our angle theta, 933 01:05:00,160 --> 01:05:02,570 here's the cell wall thickness t, 934 01:05:02,570 --> 01:05:06,360 and then we've got some set of solid properties here. 935 01:05:10,930 --> 01:05:11,700 OK? 936 01:05:11,700 --> 01:05:13,940 So that's kind of the set up. 937 01:05:13,940 --> 01:05:18,140 And we're going to define x1, x2 axes like that. 938 01:05:52,554 --> 01:05:54,470 And we're going to make a few assumptions just 939 01:05:54,470 --> 01:05:56,320 to make life a little bit simpler. 940 01:05:56,320 --> 01:06:01,230 So we're going to assume that t over l is small, 941 01:06:01,230 --> 01:06:03,685 so that also means the relative density is small. 942 01:06:03,685 --> 01:06:05,060 And what that means is that we're 943 01:06:05,060 --> 01:06:08,100 going to be able to neglect axial and shear deformations. 944 01:06:16,610 --> 01:06:18,550 So you can imagine, if I have a thin wall 945 01:06:18,550 --> 01:06:21,160 and I'm applying loads that produce moments and produce 946 01:06:21,160 --> 01:06:25,010 bending, if the wall is very thin then 947 01:06:25,010 --> 01:06:26,910 the axial and the shear deformations 948 01:06:26,910 --> 01:06:28,720 are going to be small. 949 01:06:28,720 --> 01:06:32,670 I'm also going to assume the deformations are small. 950 01:06:32,670 --> 01:06:37,200 And what that means is that I'm going 951 01:06:37,200 --> 01:06:40,217 to neglect any changes in the geometry of the cell 952 01:06:40,217 --> 01:06:41,175 during the deformation. 953 01:06:52,140 --> 01:06:55,197 And I'm going to assume that the cell wall is linear, elastic, 954 01:06:55,197 --> 01:06:55,780 and isotropic. 955 01:07:20,302 --> 01:07:22,010 And we're going to start off with looking 956 01:07:22,010 --> 01:07:24,370 at in-plane behavior, and we're going 957 01:07:24,370 --> 01:07:27,670 to start with the elastic moduli. 958 01:07:27,670 --> 01:07:30,930 And if we look at the elastic moduli, 959 01:07:30,930 --> 01:07:33,160 we're going to be talking about Hooke's law. 960 01:07:33,160 --> 01:07:36,470 And Hooke's law and the elastic behavior of the material 961 01:07:36,470 --> 01:07:39,330 can be described by a set of elastic constants. 962 01:07:39,330 --> 01:07:43,280 And if you recall, the number of independent elastic constants, 963 01:07:43,280 --> 01:07:45,760 how many constants you need to describe the material, 964 01:07:45,760 --> 01:07:47,530 depends on its symmetry. 965 01:07:47,530 --> 01:07:49,740 And these materials are orthotropic. 966 01:07:49,740 --> 01:07:52,550 So the regular hexagonal honeycomb 967 01:07:52,550 --> 01:07:54,480 is actually transversely isotropic, 968 01:07:54,480 --> 01:07:56,686 but imagine that h was not equal to l, 969 01:07:56,686 --> 01:07:58,870 then it would be orthotropic. 970 01:07:58,870 --> 01:08:01,100 So remember, orthotropic means that you 971 01:08:01,100 --> 01:08:05,830 can rotate the structure 180 degrees about three mutually 972 01:08:05,830 --> 01:08:08,170 perpendicular axes and the structure looks the same. 973 01:08:08,170 --> 01:08:10,396 So if I take this and I do that, it looks the same, 974 01:08:10,396 --> 01:08:12,770 and if I do that, it looks the same, if I-- no matter how 975 01:08:12,770 --> 01:08:16,359 I rotate this, about three mutually perpendicular axes, 976 01:08:16,359 --> 01:08:18,399 the structure remains unchanged. 977 01:08:18,399 --> 01:08:19,446 So it's orthotropic. 978 01:09:38,847 --> 01:09:40,430 So I'm going to write down Hooke's law 979 01:09:40,430 --> 01:09:44,180 for our orthotropic material, and then we'll 980 01:09:44,180 --> 01:09:47,740 talk about the constants that we're going to work out. 981 01:11:16,020 --> 01:11:18,677 OK, so this is Hooke's law for our orthotropic material. 982 01:11:22,150 --> 01:11:26,190 And let me just remind you what our notation is here. 983 01:11:34,920 --> 01:11:38,768 So epsilon 1 is epsilon 1 1. 984 01:11:38,768 --> 01:11:42,576 Epsilon 2 is epsilon 2 2. 985 01:11:42,576 --> 01:11:46,020 Epsilon 3 is epsilon 3 3. 986 01:11:46,020 --> 01:11:50,245 Epsilon 4 is gamma 2 3. 987 01:11:50,245 --> 01:11:53,670 Epsilon 5 is gamma 1 3. 988 01:11:53,670 --> 01:11:56,300 And epsilon 6 is gamma 1 2. 989 01:11:56,300 --> 01:11:59,710 So these are the normal strains here, epsilon 1, 2, and 3, 990 01:11:59,710 --> 01:12:03,010 and these are the shear strains here, epsilon 4, 5, and 6. 991 01:12:03,010 --> 01:12:05,690 And you remember this convention where the subscripts add up 992 01:12:05,690 --> 01:12:06,210 to 9. 993 01:12:06,210 --> 01:12:09,880 So 4 plus 2 plus 3 is 9, 5 plus 1 plus 3 is 994 01:12:09,880 --> 01:12:12,800 9, 6 plus 1 plus 2 is 9. 995 01:12:12,800 --> 01:12:14,654 And then the stresses are a similar thing. 996 01:12:22,616 --> 01:12:26,560 So sigma 1, sigma 2, and sigma 3 are the normal stresses. 997 01:12:26,560 --> 01:12:32,720 And then sigma 4, sigma 5, and sigma 6 are the shear stresses. 998 01:12:41,570 --> 01:12:47,045 And for the in-plane moduli, so we're 999 01:12:47,045 --> 01:12:49,300 dealing with the x1, x2 plane, there's 1000 01:12:49,300 --> 01:12:51,320 four independent elastic constants. 1001 01:13:04,420 --> 01:13:12,420 So we could think of it as E1, E2, a Poisson's ratio 1 2 1002 01:13:12,420 --> 01:13:16,390 and a shear modulus in the 1 2 plane. 1003 01:13:16,390 --> 01:13:17,440 OK? 1004 01:13:17,440 --> 01:13:19,560 And the compliance matrix is symmetric, 1005 01:13:19,560 --> 01:13:22,494 so there's the reciprocal relationship between the moduli 1006 01:13:22,494 --> 01:13:23,535 and the Poisson's ratios. 1007 01:13:56,550 --> 01:14:07,770 And then the notation I'm going to use for Poisson's ratio, 1008 01:14:07,770 --> 01:14:10,310 I'm going to say that mu i j is minus 1009 01:14:10,310 --> 01:14:13,810 the ratio of the strain in the j direction divided 1010 01:14:13,810 --> 01:14:17,201 by the strain in the i direction. 1011 01:14:17,201 --> 01:14:17,700 OK. 1012 01:14:20,750 --> 01:14:22,800 So what we're going to do next is 1013 01:14:22,800 --> 01:14:26,180 we're going to calculate some of the elastic moduli. 1014 01:14:26,180 --> 01:14:29,560 I'm going to show you the derivation for E1 star 1015 01:14:29,560 --> 01:14:33,514 and mu 1 2 star, and you can get the other two in a similar way. 1016 01:14:33,514 --> 01:14:34,930 So I'm not going to do all of them 1017 01:14:34,930 --> 01:14:37,780 but next time we'll do the derivations for the Young's 1018 01:14:37,780 --> 01:14:39,700 modulus in the Poisson's ratio. 1019 01:14:39,700 --> 01:14:40,400 OK? 1020 01:14:40,400 --> 01:14:42,840 And then we're going to talk about the out of plane 1021 01:14:42,840 --> 01:14:46,620 direction later and we'll get the moduli for the out of plane 1022 01:14:46,620 --> 01:14:47,790 direction as well. 1023 01:14:47,790 --> 01:14:48,770 OK? 1024 01:14:48,770 --> 01:14:51,460 So I think I'm going to stop there for today. 1025 01:14:51,460 --> 01:14:55,070 And then we'll start doing the derivations next time.