WEBVTT 00:00:00.030 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:04.000 Commons license. 00:00:04.000 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high-quality educational resources for free. 00:00:10.690 --> 00:00:13.310 To make a donation or view additional materials 00:00:13.310 --> 00:00:17.025 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.025 --> 00:00:17.650 at ocw.mit.edu. 00:00:26.360 --> 00:00:27.954 LORNA GIBSON: All right. 00:00:27.954 --> 00:00:30.120 And I really wanted to show you my little hook video 00:00:30.120 --> 00:00:32.380 and I downloaded it so I thought we'd start just 00:00:32.380 --> 00:00:37.040 by watching that and then I'll pick up about modeling phones. 00:00:37.040 --> 00:00:39.330 So this takes like nine or 10 minutes, 00:00:39.330 --> 00:00:40.805 but I just thought it was cute. 00:00:40.805 --> 00:00:42.430 And I made it and I want you to see it. 00:00:42.430 --> 00:00:44.594 So let's do that to start. 00:00:44.594 --> 00:00:45.260 [VIDEO PLAYBACK] 00:00:45.260 --> 00:00:47.780 We're here at the Harvard University Botany Library, 00:00:47.780 --> 00:00:51.070 looking at a first edition of Robert Hooke's Micrographia, 00:00:51.070 --> 00:00:51.850 published-- 00:00:51.850 --> 00:00:53.920 How do I get rid of the bar, Greg? 00:00:53.920 --> 00:00:54.710 Oh, there it is. 00:00:54.710 --> 00:00:57.855 show the microscopic structure of materials. 00:00:57.855 --> 00:01:00.930 And it has a number of remarkable drawings in it. 00:01:00.930 --> 00:01:02.870 Here we see drawings of silk. 00:01:02.870 --> 00:01:04.120 These are two different silks. 00:01:04.120 --> 00:01:07.450 On the top here, we have a fine-waled silk. 00:01:07.450 --> 00:01:10.240 And in this more details drawing down here, 00:01:10.240 --> 00:01:13.510 you can see the patterned weaving of the silk. 00:01:13.510 --> 00:01:16.960 The bottom image here is a drawing of watered silk. 00:01:16.960 --> 00:01:20.760 And over here, there's another higher magnification image. 00:01:20.760 --> 00:01:24.480 And you can see the pattern here is more sharply angled. 00:01:24.480 --> 00:01:27.070 And it appears that this sharper angle here 00:01:27.070 --> 00:01:32.570 gives the different texture to the surface finish of the silk. 00:01:32.570 --> 00:01:34.912 So here we see a drawing of charred wood. 00:01:34.912 --> 00:01:37.370 And one of the things I find interesting about this drawing 00:01:37.370 --> 00:01:40.945 is how similar it is to modern electron micrographs which 00:01:40.945 --> 00:01:42.130 we've seen before. 00:01:42.130 --> 00:01:45.190 And in this drawing, we can see two of the main features. 00:01:45.190 --> 00:01:48.340 We see these small cells, which are fibers that provide 00:01:48.340 --> 00:01:49.960 structural support to the tree. 00:01:49.960 --> 00:01:51.620 And we see these larger cells here, 00:01:51.620 --> 00:01:53.330 which are vessels which allow fluids 00:01:53.330 --> 00:01:55.490 to go up and down the tree. 00:01:55.490 --> 00:01:58.730 And here we see a drawing of the surface of a rosemary leaf, 00:01:58.730 --> 00:02:01.085 with the unexpected, tiny, little bars. 00:02:01.085 --> 00:02:02.710 And this is something that you can only 00:02:02.710 --> 00:02:03.830 see with the microscope. 00:02:03.830 --> 00:02:05.960 You wouldn't expect to see those when you just 00:02:05.960 --> 00:02:07.860 feel the surface of the rosemary leaf. 00:02:07.860 --> 00:02:10.068 So it's kind of interesting that with the microscope, 00:02:10.068 --> 00:02:14.250 you can see these features that are invisible to the naked eye. 00:02:14.250 --> 00:02:16.260 One of the main themes of material science 00:02:16.260 --> 00:02:18.030 is that the property of materials 00:02:18.030 --> 00:02:19.776 are related to their structure. 00:02:19.776 --> 00:02:21.540 And so being able to see the structure 00:02:21.540 --> 00:02:24.560 at a microscopic scale is very helpful. 00:02:24.560 --> 00:02:28.221 And today, we can even see the structure at the atomic scale. 00:02:28.221 --> 00:02:30.600 Robert Hooke understood this idea. 00:02:30.600 --> 00:02:32.620 And in the description of the cork, 00:02:32.620 --> 00:02:34.820 Hooke states, "I no sooner discerned 00:02:34.820 --> 00:02:38.100 these-- which were the first microscopical pores I ever 00:02:38.100 --> 00:02:42.420 saw-- but methought that I had with the discovery of them, 00:02:42.420 --> 00:02:46.630 perfectly hinted to me the true and intelligible reason for all 00:02:46.630 --> 00:02:48.620 of the phenomena of cork." 00:02:48.620 --> 00:02:51.200 So what he's saying here is that by looking at the structure 00:02:51.200 --> 00:02:53.850 and looking at the cells here in the drawing, 00:02:53.850 --> 00:02:56.780 he thinks he can understand the properties of cork 00:02:56.780 --> 00:02:59.000 or the phenomena of cork. 00:02:59.000 --> 00:03:00.600 What was it about Robert Hooke that 00:03:00.600 --> 00:03:02.110 allowed him to make this book? 00:03:02.110 --> 00:03:04.110 Why was it him and not somebody else? 00:03:04.110 --> 00:03:06.560 Well, Robert Hooke had kind of an interesting history. 00:03:06.560 --> 00:03:08.160 He grew up on the Isle of Wight. 00:03:08.160 --> 00:03:10.110 And as a boy, he loved making drawings. 00:03:10.110 --> 00:03:12.500 And he got quite skilled at making drawings. 00:03:12.500 --> 00:03:15.420 The other thing was, he loved making models of things. 00:03:15.420 --> 00:03:17.030 He made models of ships. 00:03:17.030 --> 00:03:19.560 He made a wooden clock that was a working 00:03:19.560 --> 00:03:21.370 clock when he was a kid. 00:03:21.370 --> 00:03:23.800 And as a teenager, he moved to London. 00:03:23.800 --> 00:03:26.080 And he became an apprentice to Sir Peter Lilley, 00:03:26.080 --> 00:03:27.680 who was a famous painter of the time. 00:03:27.680 --> 00:03:30.560 So his drawing was good enough that he would be working 00:03:30.560 --> 00:03:33.370 with a very well-known painter. 00:03:33.370 --> 00:03:36.410 After he did that, he went to the Westminister School. 00:03:36.410 --> 00:03:37.810 And he studied classics. 00:03:37.810 --> 00:03:39.020 He studied mathematics. 00:03:39.020 --> 00:03:40.880 But he also learned to use a lathe. 00:03:40.880 --> 00:03:43.920 And this was also very helpful in him making 00:03:43.920 --> 00:03:46.520 various sorts of apparatus. 00:03:46.520 --> 00:03:48.300 And as a student at Oxford, he worked 00:03:48.300 --> 00:03:50.410 in the lab of Robert Boyle. 00:03:50.410 --> 00:03:54.060 And his job in that lab was to develop scientific apparatus. 00:03:54.060 --> 00:03:56.390 And he did things like he built pumps 00:03:56.390 --> 00:03:59.270 that allowed Robert Boyle to do the experiments that 00:03:59.270 --> 00:04:01.800 led to Boyle's Law. 00:04:01.800 --> 00:04:04.150 When he returned to London after Oxford, 00:04:04.150 --> 00:04:05.950 he became the Curator of Experiments 00:04:05.950 --> 00:04:07.430 at the Royal Society. 00:04:07.430 --> 00:04:10.590 And one of the things he did was he got a microscope. 00:04:10.590 --> 00:04:13.904 He improved that microscope, increasing their magnification, 00:04:13.904 --> 00:04:16.279 which was what allowed him to make the beautiful drawings 00:04:16.279 --> 00:04:17.630 that we see today. 00:04:17.630 --> 00:04:19.450 And here in the preface of the book, 00:04:19.450 --> 00:04:21.890 we see that he even made a drawing of his microscope. 00:04:21.890 --> 00:04:26.310 So this thing down here-- this is Robert Hooke's microscope. 00:04:26.310 --> 00:04:27.950 The development of new microscopes 00:04:27.950 --> 00:04:29.750 with higher and higher magnifications 00:04:29.750 --> 00:04:31.540 continues to this day. 00:04:31.540 --> 00:04:35.090 Scanning electron microscopes were invented in the 1960s. 00:04:35.090 --> 00:04:37.600 And today, we have transmission electron microscopes 00:04:37.600 --> 00:04:40.220 and atomic force microscopes with even higher 00:04:40.220 --> 00:04:41.760 magnifications. 00:04:41.760 --> 00:04:43.360 At these higher magnifications, we 00:04:43.360 --> 00:04:45.960 can see details that Hooke was unable to see because 00:04:45.960 --> 00:04:48.110 of the limitations of the microscope 00:04:48.110 --> 00:04:50.290 that he had-- the optical microscope. 00:04:50.290 --> 00:04:51.960 But it's interesting to see today 00:04:51.960 --> 00:04:53.760 the images we see in a scanning electron 00:04:53.760 --> 00:04:56.920 microscope at a similar magnification to those 00:04:56.920 --> 00:04:59.580 that Hooke saw in his optical microscope. 00:04:59.580 --> 00:05:01.860 And it's remarkable to see how many of the features 00:05:01.860 --> 00:05:05.000 that we see in these much more fancy microscopes 00:05:05.000 --> 00:05:07.220 that he was able to capture in his drawings 00:05:07.220 --> 00:05:10.270 with his simple optical microscope. 00:05:10.270 --> 00:05:11.770 So here we have a picture of cork. 00:05:11.770 --> 00:05:15.240 We have Hooke's drawings showing two perpendicular planes. 00:05:15.240 --> 00:05:18.130 We also have this nice, little drawing of a cork branch here. 00:05:18.130 --> 00:05:21.340 Cork is the bark from the cork oak tree. 00:05:21.340 --> 00:05:23.330 And in Hooke's drawings of the microstructure, 00:05:23.330 --> 00:05:26.990 we can see these cells here are roughly box-like. 00:05:26.990 --> 00:05:28.970 They're more or less rectangular. 00:05:28.970 --> 00:05:31.257 And these cells here look more or less circular. 00:05:31.257 --> 00:05:33.090 So there's these two different perpendicular 00:05:33.090 --> 00:05:34.750 planes in the cork. 00:05:34.750 --> 00:05:37.220 And when we look at these scanning electron micrographs, 00:05:37.220 --> 00:05:39.390 we can see very similar structure. 00:05:39.390 --> 00:05:41.750 There are some cells that are roughly boxlike, 00:05:41.750 --> 00:05:46.030 and others that are more or less hexagonal or roughly rounded. 00:05:46.030 --> 00:05:48.850 One feature that Hooke was not able to see, though, 00:05:48.850 --> 00:05:51.440 that you do see on the scanning electron micrographs, 00:05:51.440 --> 00:05:53.580 is the waviness in the cell walls. 00:05:53.580 --> 00:05:55.830 And that was because the resolution of his microscope 00:05:55.830 --> 00:05:59.070 was insufficient to see that level of detail. 00:05:59.070 --> 00:06:01.690 And here in this illustration on the bottom 00:06:01.690 --> 00:06:03.740 here is a drawing of sponge. 00:06:03.740 --> 00:06:06.380 And when we look at the scanning electron micrograph, 00:06:06.380 --> 00:06:09.320 we see that the structure is remarkably similar to what 00:06:09.320 --> 00:06:10.870 Hooke has drawn. 00:06:10.870 --> 00:06:13.270 So here we have Hooke's drawing of feathers. 00:06:13.270 --> 00:06:16.420 And we can see he's made several drawings at different length 00:06:16.420 --> 00:06:17.530 scales. 00:06:17.530 --> 00:06:20.630 And if we look at this one here, we see the barbule. 00:06:20.630 --> 00:06:24.400 And you can see these little hooked regions there. 00:06:24.400 --> 00:06:28.310 And those hooks lock into the little feathers 00:06:28.310 --> 00:06:32.050 over on this side over here of the adjacent barbule. 00:06:32.050 --> 00:06:34.070 And in the higher magnification picture, 00:06:34.070 --> 00:06:38.000 you can see on one barbule, there's hooks on one side 00:06:38.000 --> 00:06:39.690 but not on the other. 00:06:39.690 --> 00:06:43.740 And it's this hooking of the two sections together 00:06:43.740 --> 00:06:46.060 that allows the feathers to maintain 00:06:46.060 --> 00:06:49.967 a smooth surface for the wing when the bird is flying. 00:06:49.967 --> 00:06:51.550 And you can see the same sort of thing 00:06:51.550 --> 00:06:53.258 when you look at the electron micrograph. 00:06:53.258 --> 00:06:57.320 So you can see the little hooks on one side of the barbules. 00:06:57.320 --> 00:06:59.830 And you can see how they interconnect together 00:06:59.830 --> 00:07:01.570 with the next barb. 00:07:01.570 --> 00:07:04.580 One of the most reproduced images from Hooke's book 00:07:04.580 --> 00:07:07.037 is that of the flea-- this image we see here. 00:07:07.037 --> 00:07:07.870 And you can see why. 00:07:07.870 --> 00:07:09.460 It's a gorgeous image. 00:07:09.460 --> 00:07:12.330 And it shows details that people had never seen before. 00:07:12.330 --> 00:07:15.450 People were amazed to see that the little flea that they might 00:07:15.450 --> 00:07:17.540 have found on their dog or something 00:07:17.540 --> 00:07:20.580 was actually made up of this compound body, 00:07:20.580 --> 00:07:23.570 with all these little plates and little hairs here. 00:07:23.570 --> 00:07:26.530 And you can see these little tiny claws on the legs, 00:07:26.530 --> 00:07:28.220 and the legs have all these hairs. 00:07:28.220 --> 00:07:30.850 Nobody had any idea that this is what a flea actually 00:07:30.850 --> 00:07:31.840 looked like. 00:07:31.840 --> 00:07:33.640 And so it was an amazing drawing. 00:07:33.640 --> 00:07:37.090 And it was something that people were just stunned by when 00:07:37.090 --> 00:07:38.620 Hooke's book came out. 00:07:38.620 --> 00:07:41.030 And if we look at a modern electron micrograph, 00:07:41.030 --> 00:07:43.210 we can see it's remarkably similar if we 00:07:43.210 --> 00:07:45.280 look at the same magnification. 00:07:45.280 --> 00:07:48.170 So Hooke showed many of the same details, 00:07:48.170 --> 00:07:50.750 showed some of the same hairs on the legs, 00:07:50.750 --> 00:07:53.760 showed the same sorts of plates, showed the claws 00:07:53.760 --> 00:07:55.450 at the ends of the legs. 00:07:55.450 --> 00:07:58.340 And our modern image is probably from a different species 00:07:58.340 --> 00:07:58.840 of flea. 00:07:58.840 --> 00:08:00.820 We don't know what species of flea 00:08:00.820 --> 00:08:02.160 that Hooke actually looked at. 00:08:02.160 --> 00:08:04.118 But you can see there's a tremendous similarity 00:08:04.118 --> 00:08:05.297 between the two images. 00:08:05.297 --> 00:08:07.130 And it's remarkable how many of the features 00:08:07.130 --> 00:08:09.730 that Hooke was able to capture in his drawing. 00:08:09.730 --> 00:08:12.220 And here we have the compound eye of the fly. 00:08:12.220 --> 00:08:15.450 And this, again, was astonishing to people in Hooke's day. 00:08:15.450 --> 00:08:17.960 And even today, people look at this image, 00:08:17.960 --> 00:08:22.230 and they're pretty amazed at the detail in this drawing. 00:08:22.230 --> 00:08:24.750 And again, we can compare this with a modern electron 00:08:24.750 --> 00:08:25.850 micrograph. 00:08:25.850 --> 00:08:27.880 And again, you can see the similarities 00:08:27.880 --> 00:08:30.090 between what Hooke saw and what we 00:08:30.090 --> 00:08:32.909 see in a modern scanning electron microscope 00:08:32.909 --> 00:08:35.030 at a similar magnification. 00:08:35.030 --> 00:08:37.600 In the 1980s, atomic force microscopes 00:08:37.600 --> 00:08:40.320 were invented, which have a resolution down to tens 00:08:40.320 --> 00:08:41.710 of nanometers. 00:08:41.710 --> 00:08:44.380 And today, there's transmission electron microscopes, 00:08:44.380 --> 00:08:46.820 which allow you to see the atomic structure. 00:08:46.820 --> 00:08:48.750 So for instance in a crystal lattice, 00:08:48.750 --> 00:08:52.090 you can see the individual atoms and the regular crystal 00:08:52.090 --> 00:08:54.040 structure. 00:08:54.040 --> 00:08:56.990 Today, most experimental studies of materials 00:08:56.990 --> 00:08:59.790 include photographs of the microscopic structure 00:08:59.790 --> 00:09:03.420 of the material taken through some sort of microscope. 00:09:03.420 --> 00:09:06.090 And the remarkable thing is that all of these studies 00:09:06.090 --> 00:09:08.540 really trace back to this book here 00:09:08.540 --> 00:09:11.947 that we're looking at today-- to Robert Hooke's Micrographia. 00:09:11.947 --> 00:09:12.530 [END PLAYBACK] 00:09:12.530 --> 00:09:13.350 There you have it. 00:09:13.350 --> 00:09:14.530 So I just thought that was kind of cute. 00:09:14.530 --> 00:09:15.405 You might enjoy that. 00:09:15.405 --> 00:09:18.170 So that was that. 00:09:18.170 --> 00:09:21.820 All right, let's get out of there. 00:09:21.820 --> 00:09:22.320 Stop. 00:09:25.630 --> 00:09:28.000 So let's go back to the foams. 00:09:28.000 --> 00:09:29.690 So I think last time, we got as far 00:09:29.690 --> 00:09:32.400 as talking about the linear elastic behavior of foams 00:09:32.400 --> 00:09:33.785 and modeling that. 00:09:33.785 --> 00:09:36.060 But we didn't quite get to looking 00:09:36.060 --> 00:09:37.949 at the compressive strength of the foam. 00:09:37.949 --> 00:09:39.490 So I think we got as far as comparing 00:09:39.490 --> 00:09:42.020 the models with these equations here, 00:09:42.020 --> 00:09:45.720 and these plots of the data. 00:09:45.720 --> 00:09:47.494 And what I wanted to pick up with today 00:09:47.494 --> 00:09:49.160 was looking at the compressive strength. 00:09:49.160 --> 00:09:52.730 And we'll look at the fracture toughness as well in tension. 00:09:52.730 --> 00:09:54.520 So we're going to start with nonlinear 00:09:54.520 --> 00:09:57.055 elasticity and the elastic collapse stress. 00:10:10.170 --> 00:10:15.750 So if we have an open-cell foam, the derivation 00:10:15.750 --> 00:10:17.910 for the elastic collapse stress is really 00:10:17.910 --> 00:10:19.370 pretty straightforward. 00:10:19.370 --> 00:10:21.840 We say the elastic collapse occurs 00:10:21.840 --> 00:10:23.120 when the cell walls buckles. 00:10:23.120 --> 00:10:25.980 So in this schematic here, you can see the vertical cell 00:10:25.980 --> 00:10:27.120 walls have buckled. 00:10:27.120 --> 00:10:29.490 And so there's going to be some Euler load that's 00:10:29.490 --> 00:10:31.310 related to that buckling. 00:10:31.310 --> 00:10:33.920 And that's just the usual Euler load-- n squared, the n 00:10:33.920 --> 00:10:36.780 constraint factor, pi squared E. This 00:10:36.780 --> 00:10:40.110 is going to be E of the solid, I over l 00:10:40.110 --> 00:10:43.300 squared-- the length of the member. 00:10:43.300 --> 00:10:46.310 And then the stress that corresponds to that 00:10:46.310 --> 00:10:49.810 is just going to be proportional to that buckling load 00:10:49.810 --> 00:10:52.440 over the area of the cell, which is just 00:10:52.440 --> 00:10:56.000 l squared, so just P critical over l squared. 00:10:56.000 --> 00:10:58.820 So that just goes as Es. 00:10:58.820 --> 00:11:00.760 I is going to go as t to the fourth, 00:11:00.760 --> 00:11:04.737 because we have that square sectioned member. 00:11:04.737 --> 00:11:06.320 And now this is going to be l squared. 00:11:06.320 --> 00:11:07.850 And that's an l squared. 00:11:07.850 --> 00:11:09.830 So that's l to the fourth. 00:11:09.830 --> 00:11:12.030 And so if I combine all of that together, 00:11:12.030 --> 00:11:14.800 I can say that the elastic buckling stress is going 00:11:14.800 --> 00:11:17.420 to be some constant-- and I think we're up to C4 00:11:17.420 --> 00:11:21.650 now-- times the Young's modulus of the solid times 00:11:21.650 --> 00:11:26.130 the relative density of the foam squared. 00:11:26.130 --> 00:11:33.270 So that's our equation for the elastic buckling stress. 00:11:33.270 --> 00:11:35.900 And if you compare this with data, 00:11:35.900 --> 00:11:39.050 you can make an estimate of what C4 is. 00:11:39.050 --> 00:11:45.190 And we find that C4 is about 0.05. 00:11:45.190 --> 00:11:47.130 And you can also say that 0.05 really 00:11:47.130 --> 00:11:50.220 corresponds to the strain at which the buckling occurs. 00:11:50.220 --> 00:11:53.470 Because the Young's modulus goes as the constants 1 times 00:11:53.470 --> 00:11:55.310 Es times the relative density squared. 00:11:55.310 --> 00:11:58.030 So the strain's just going to be the stress over the modulus. 00:11:58.030 --> 00:11:59.615 So that does correspond to the strain. 00:12:16.820 --> 00:12:19.650 So that's saying that buckling compressive stress 00:12:19.650 --> 00:12:21.145 occurs at a strain of about 5%. 00:12:25.540 --> 00:12:28.470 So that's open cells. 00:12:28.470 --> 00:12:31.670 And then if we look at closed cells, 00:12:31.670 --> 00:12:33.829 if you recall when we looked at the moduli 00:12:33.829 --> 00:12:35.370 we looked at a couple of extra terms. 00:12:35.370 --> 00:12:38.609 One was associated with face stretching for the modulus. 00:12:38.609 --> 00:12:40.650 And the other was associated with the compression 00:12:40.650 --> 00:12:41.920 of the gas. 00:12:41.920 --> 00:12:43.834 For the buckling, the faces don't really 00:12:43.834 --> 00:12:46.250 contribute that much, because typically the faces are very 00:12:46.250 --> 00:12:48.240 thin relative to the struts. 00:12:48.240 --> 00:12:49.850 And because they're so thin, they 00:12:49.850 --> 00:12:52.730 buckle at a much lower load, and they don't contribute too much. 00:12:52.730 --> 00:12:55.450 So we're not going to worry about that contribution. 00:12:55.450 --> 00:12:58.360 So I'm just going to say that the thickness of the face 00:12:58.360 --> 00:13:01.960 is often small compared to the edges. 00:13:09.200 --> 00:13:18.060 And that really is from the surface tension in processing 00:13:18.060 --> 00:13:20.525 that draws material away from the face and into the edges. 00:13:45.240 --> 00:13:49.550 There can be some contribution from the internal pressure. 00:13:49.550 --> 00:13:52.220 So if the internal pressure is greater 00:13:52.220 --> 00:13:55.350 than atmospheric pressure, then the cell walls 00:13:55.350 --> 00:13:57.870 are pre-tensioned, and you'd have to account for that. 00:14:07.070 --> 00:14:10.310 So the buckling would have to overcome that pressure as well. 00:14:21.490 --> 00:14:24.370 So then you would have the buckling stress 00:14:24.370 --> 00:14:27.920 would just be what we have up there-- C4 times 00:14:27.920 --> 00:14:31.560 Es times the relative density squared. 00:14:31.560 --> 00:14:38.070 And then we just add on that factor P0 minus P atmospheric. 00:14:38.070 --> 00:14:40.336 The thing with the gas which tends to affect more 00:14:40.336 --> 00:14:41.710 than the buckling stress, though, 00:14:41.710 --> 00:14:43.860 is the post-collapse behavior. 00:14:43.860 --> 00:14:46.830 So let me just show you a couple of things here. 00:14:46.830 --> 00:14:50.290 So here's some data for the elastic collapse stress. 00:14:50.290 --> 00:14:52.600 And you can see on the y-axis, we've 00:14:52.600 --> 00:14:56.460 got the stress normalized by the Young's modulus of the solid. 00:14:56.460 --> 00:15:00.140 And on the x-axis, we've got the relative density. 00:15:00.140 --> 00:15:04.660 And that solid line there-- sort of solid, dark line-- 00:15:04.660 --> 00:15:07.610 is that equation there, which is the same as this one up here. 00:15:07.610 --> 00:15:11.237 And you can see the data lie fairly close to that. 00:15:11.237 --> 00:15:13.320 But what's interesting is if you look at the-- why 00:15:13.320 --> 00:15:14.153 is this not working? 00:15:16.832 --> 00:15:19.610 Maybe my batteries finally died. 00:15:19.610 --> 00:15:23.600 If we look at the post-collapse behavior, 00:15:23.600 --> 00:15:27.450 you can see if these are the stress-strain curves, 00:15:27.450 --> 00:15:29.110 they're not flat here. 00:15:29.110 --> 00:15:31.920 They have some rise to them. 00:15:31.920 --> 00:15:33.692 And this is a closed-cell foam. 00:15:33.692 --> 00:15:35.400 And you can imagine as you're compressing 00:15:35.400 --> 00:15:37.910 the closed-cell foam, you're reducing 00:15:37.910 --> 00:15:39.200 the volume of the cell. 00:15:39.200 --> 00:15:41.408 And as you doing that, you're increasing the pressure 00:15:41.408 --> 00:15:43.060 inside the cell from the gas. 00:15:43.060 --> 00:15:44.880 And you can calculate what that is. 00:15:44.880 --> 00:15:46.540 And I'll do that in a second. 00:15:46.540 --> 00:15:49.810 And if you subtract off that gas pressure contribution, 00:15:49.810 --> 00:15:52.090 that works out to this line here. 00:15:52.090 --> 00:15:56.320 Then these lines will be more flat, like this. 00:15:56.320 --> 00:15:58.270 And we already really pretty much worked 00:15:58.270 --> 00:15:59.521 out that gas contribution. 00:16:03.200 --> 00:16:13.360 So I'll just say for the post-collapse behavior, 00:16:13.360 --> 00:16:15.900 the stress rises due to the gas compression. 00:16:27.120 --> 00:16:29.201 And that's as long as the faces don't rupture. 00:16:33.252 --> 00:16:34.710 So if you have an elastomeric foam, 00:16:34.710 --> 00:16:35.918 typically they don't rupture. 00:16:40.940 --> 00:16:43.760 And what we had worked out before 00:16:43.760 --> 00:16:46.185 was that that pressure-- we called 00:16:46.185 --> 00:16:51.360 it P prime-- it was P0 minus P atmospheric-- that 00:16:51.360 --> 00:17:00.340 was equal to P0 times the amount of strain, 00:17:00.340 --> 00:17:06.609 epsilon, times 1 minus 2 times the Poisson's ratio 00:17:06.609 --> 00:17:12.795 divided by 1 minus epsilon times 1 minus 2 nu 00:17:12.795 --> 00:17:13.920 minus the relative density. 00:17:21.140 --> 00:17:24.530 And once you get to the buckling stress, 00:17:24.530 --> 00:17:26.573 then the Poisson's ratio becomes 0. 00:17:36.770 --> 00:17:39.270 So if you take a foam-- so I brought a little foam in so you 00:17:39.270 --> 00:17:41.580 can play around with this one-- so if you take a foam like this 00:17:41.580 --> 00:17:44.090 and you compress it, once you've buckled it like this, 00:17:44.090 --> 00:17:45.632 it's not getting any wider this way. 00:17:45.632 --> 00:17:47.340 And part of the reason for that is you've 00:17:47.340 --> 00:17:48.540 got all these pores in here. 00:17:48.540 --> 00:17:50.660 And the cells just collapse into the pores. 00:17:50.660 --> 00:17:52.755 They don't really need to move out sideways. 00:17:52.755 --> 00:17:54.380 So you can smush that yourself, and try 00:17:54.380 --> 00:17:57.480 to convince yourself that the Poisson's ratio is just 0. 00:17:57.480 --> 00:17:58.770 Yes, Matt. 00:17:58.770 --> 00:18:05.405 AUDIENCE: [INAUDIBLE] I guess I want to measure [INAUDIBLE] 00:18:05.405 --> 00:18:06.787 the gas contribution? 00:18:06.787 --> 00:18:08.620 LORNA GIBSON: Yes, so there is a strain rate 00:18:08.620 --> 00:18:09.940 effect with these things. 00:18:09.940 --> 00:18:11.648 But I wasn't going to get into that here. 00:18:11.648 --> 00:18:13.900 If you look in the book, it's described in the book. 00:18:13.900 --> 00:18:15.320 So I think there's two things. 00:18:15.320 --> 00:18:18.537 One is that the solid itself can have a rate dependency. 00:18:18.537 --> 00:18:20.370 And then there could be something connected. 00:18:20.370 --> 00:18:21.570 AUDIENCE: [INAUDIBLE]. 00:18:21.570 --> 00:18:24.194 LORNA GIBSON: Yeah, I mean, I'm not going to go into that here. 00:18:24.194 --> 00:18:27.720 But one could look at that. 00:18:27.720 --> 00:18:29.790 So let me just write down one more thing here, 00:18:29.790 --> 00:18:32.180 because if we let nu be 0, then this thing here 00:18:32.180 --> 00:18:33.035 becomes simpler. 00:18:50.180 --> 00:18:53.330 So we could say the stress post collapse 00:18:53.330 --> 00:19:03.190 as a function of strain would be our buckling stress and then 00:19:03.190 --> 00:19:04.566 plus this factor here. 00:19:27.970 --> 00:19:32.350 So that curve on the bottom over here-- 00:19:32.350 --> 00:19:34.350 if this is the stress-strain curve-- this little 00:19:34.350 --> 00:19:37.210 dashed line here-- that's the gas contribution. 00:19:37.210 --> 00:19:39.257 And that is this term here. 00:19:39.257 --> 00:19:41.340 So you can kind of see how the shape of the curves 00:19:41.340 --> 00:19:43.035 reflects that gas contribution. 00:19:43.035 --> 00:19:44.410 And when you subtract it out, you 00:19:44.410 --> 00:19:47.245 get pretty much a horizontal plateau over here. 00:19:51.280 --> 00:19:53.230 Are we happy? 00:19:53.230 --> 00:19:54.230 Yeah? 00:19:54.230 --> 00:19:55.870 AUDIENCE: [INAUDIBLE]? 00:19:55.870 --> 00:19:57.620 LORNA GIBSON: This is for the closed cell. 00:19:57.620 --> 00:19:59.120 Because the closed cell are the ones 00:19:59.120 --> 00:20:01.118 that are going to have the gas pressure. 00:20:01.118 --> 00:20:03.576 If it's open cells, the gas can just move out of the cells. 00:20:03.576 --> 00:20:04.550 AUDIENCE: [INAUDIBLE]? 00:20:04.550 --> 00:20:07.059 LORNA GIBSON: Oh, sorry, that was to show you 00:20:07.059 --> 00:20:08.350 that the Poisson's ratio was 0. 00:20:08.350 --> 00:20:09.725 And that's true for both of them. 00:20:33.320 --> 00:20:36.250 So then we can look at the plastic collapse stress. 00:20:36.250 --> 00:20:38.710 Say we had a metal foam. 00:20:38.710 --> 00:20:40.800 And we do a calculation a little bit 00:20:40.800 --> 00:20:43.080 like the one we did for the honeycombs, too. 00:20:43.080 --> 00:20:46.130 So we say the failure occurs when the applied moment equals 00:20:46.130 --> 00:20:47.210 the plastic moment. 00:20:55.630 --> 00:20:57.950 And the applied moment is proportional 00:20:57.950 --> 00:21:00.736 to the applied stress times the length cubed. 00:21:04.640 --> 00:21:06.940 So I'm going to call that applied stress-- our strength 00:21:06.940 --> 00:21:09.920 sigma star plastic times the length cubed. 00:21:09.920 --> 00:21:13.760 So if you think of, say, the little schematic up here, 00:21:13.760 --> 00:21:16.230 the force is going to go with stress times 00:21:16.230 --> 00:21:17.439 the length squared. 00:21:17.439 --> 00:21:19.480 And the moment's going to force times the length. 00:21:19.480 --> 00:21:26.100 So it's the stress times the length cubed. 00:21:26.100 --> 00:21:33.010 And then the plastic moment goes as the yield strength 00:21:33.010 --> 00:21:34.300 times the thickness cubed. 00:21:37.410 --> 00:21:41.950 And then if I just combine those, 00:21:41.950 --> 00:21:45.760 I get that the plastic collapse stress in compression 00:21:45.760 --> 00:21:49.550 is another constant-- I'm going to call it C5-- times the yield 00:21:49.550 --> 00:21:55.921 strength times the relative density to the 3/2 power. 00:22:00.680 --> 00:22:03.610 And if we look at data, we find that the constant 00:22:03.610 --> 00:22:11.080 is about equal to 0.3. 00:22:11.080 --> 00:22:16.290 And if I go to the next slide, here's 00:22:16.290 --> 00:22:22.480 a plot of the yield strength or the plastic collapse strength 00:22:22.480 --> 00:22:24.840 of the foam divided by the yield strength of the solid, 00:22:24.840 --> 00:22:26.900 plotted against the relative density. 00:22:26.900 --> 00:22:30.410 And that dark, bold line is this equation here. 00:22:30.410 --> 00:22:33.187 And you can see the data lie pretty well on that line. 00:22:33.187 --> 00:22:35.520 There's one data set that's a little bit above the line. 00:22:35.520 --> 00:22:38.470 But you can see the slope of that data set is still 00:22:38.470 --> 00:22:39.116 about 3/2. 00:23:21.560 --> 00:23:23.240 OK, and the same as in the honeycombs, 00:23:23.240 --> 00:23:25.640 we could say that we can get elastic collapse 00:23:25.640 --> 00:23:28.070 before the plastic collapse if we were at a low density. 00:23:28.070 --> 00:23:30.100 You can get the same thing in the foams. 00:23:30.100 --> 00:23:33.510 And you calculate out what the critical relative density is 00:23:33.510 --> 00:23:36.280 for that the same kind of way. 00:23:36.280 --> 00:23:42.690 So we can say we can get elastic collapse precedes 00:23:42.690 --> 00:23:51.450 the plastic collapse if the elastic buckling 00:23:51.450 --> 00:23:57.190 stress is less than the plastic collapse stress. 00:23:57.190 --> 00:24:02.772 So all we do is make those two things equal to figure out 00:24:02.772 --> 00:24:04.230 the critical relative density where 00:24:04.230 --> 00:24:06.104 you get the transition from one to the other. 00:24:23.690 --> 00:24:28.000 So the relative density has to be less than 36 times the yield 00:24:28.000 --> 00:24:30.910 strength of the solid over the Young's modulus 00:24:30.910 --> 00:24:38.070 of the solid squared in order to get buckling before yielding. 00:24:38.070 --> 00:24:39.570 And let's see, where can I put that? 00:24:44.410 --> 00:24:47.390 So for rigid polymers, that ratio 00:24:47.390 --> 00:24:51.140 of the strength of the solid over the modulus of the solid 00:24:51.140 --> 00:24:52.528 is about one over 30. 00:24:55.040 --> 00:25:00.420 And so the critical relative density for the transition 00:25:00.420 --> 00:25:03.130 is about 0.04. 00:25:03.130 --> 00:25:05.130 So you'd have to have a pretty low-density foam, 00:25:05.130 --> 00:25:06.850 but it's possible. 00:25:06.850 --> 00:25:18.300 And for metals, that ratio is about 1/1,000. 00:25:18.300 --> 00:25:23.960 And then the critical transition density is less than 10 00:25:23.960 --> 00:25:25.437 to the minus 5. 00:25:25.437 --> 00:25:27.645 So essentially, it never happens for the metal foams. 00:25:47.250 --> 00:25:49.100 And then for the closed-cell foams, 00:25:49.100 --> 00:25:51.860 we could include the terms for face stretching 00:25:51.860 --> 00:25:53.080 and for the gas. 00:25:53.080 --> 00:25:56.690 But in practice, the faces don't really contribute very much. 00:25:56.690 --> 00:25:59.900 And typically for foams like say metal foams 00:25:59.900 --> 00:26:03.616 or a rigid polymer that had a yield point, the faces rupture. 00:26:03.616 --> 00:26:05.240 And then if the faces rupture, then you 00:26:05.240 --> 00:26:07.986 don't get the gas compression term, either. 00:26:07.986 --> 00:26:09.610 So I'll just write the full thing down. 00:26:09.610 --> 00:26:12.475 But typically, you don't need to use it. 00:26:59.600 --> 00:27:02.110 So the first term would be from the edges bending. 00:27:06.439 --> 00:27:08.730 And the second term would be from the faces stretching. 00:27:13.430 --> 00:27:16.160 And this would be from the gas. 00:27:16.160 --> 00:27:18.410 But in practice, the first term is really the only one 00:27:18.410 --> 00:27:20.120 that is significant. 00:27:52.210 --> 00:27:54.699 So for closed-cell foam, this equation 00:27:54.699 --> 00:27:56.240 works pretty well, too-- the same one 00:27:56.240 --> 00:27:57.537 as for the open-cell foams. 00:28:49.850 --> 00:28:53.850 OK, so if we had, say, a ceramic foam that was brittle, 00:28:53.850 --> 00:28:55.790 there'd be a brittle crushing strength. 00:28:55.790 --> 00:28:57.790 And then we get failure when the applied moment 00:28:57.790 --> 00:29:01.630 M is equal to the fracture moment Mf. 00:29:01.630 --> 00:29:05.810 And this works very similar to the plastic yield strength. 00:29:05.810 --> 00:29:08.090 So we find the applied moment goes 00:29:08.090 --> 00:29:13.310 as the global stress times the length cubed. 00:29:13.310 --> 00:29:18.230 And the fracture moment goes into the cell wall strength 00:29:18.230 --> 00:29:20.946 times the cell wall thickness cubed. 00:29:24.030 --> 00:29:26.460 So the brittle crushing strength goes 00:29:26.460 --> 00:29:30.580 as another constant-- let's call it C6-- times the wall 00:29:30.580 --> 00:29:35.580 strength times the relative density to the 3/2 again. 00:29:39.650 --> 00:29:47.330 And C6 is about equal to 0.2. 00:29:47.330 --> 00:29:49.620 And typically, ceramic foams have open cells. 00:29:49.620 --> 00:29:53.920 So I'm just going to leave it at the open-celled formula there. 00:29:53.920 --> 00:29:56.850 So there's one last thing for the compressive behavior, 00:29:56.850 --> 00:29:59.210 and that's the densification strain. 00:29:59.210 --> 00:30:01.080 And we just have an empirical relationship 00:30:01.080 --> 00:30:02.740 for the densification strain. 00:30:18.610 --> 00:30:22.450 So if you compress the foam and you get to very large strains, 00:30:22.450 --> 00:30:24.770 then the cell walls start to touch, 00:30:24.770 --> 00:30:26.825 and the stress starts to rise steeply. 00:30:26.825 --> 00:30:28.700 And there's some strain at which that occurs. 00:30:28.700 --> 00:30:30.970 And we call that the densification strain. 00:30:30.970 --> 00:30:32.970 And in the limit, the modulus at that point 00:30:32.970 --> 00:30:34.830 would go to the modulus of the solid. 00:30:34.830 --> 00:30:37.150 If you could completely squeeze all the pores out, 00:30:37.150 --> 00:30:41.269 the stiffness of that would go to the modulus of the solid. 00:30:41.269 --> 00:30:43.810 And you might expect that that densification strain is just 1 00:30:43.810 --> 00:30:45.930 minus the relative density, but it actually 00:30:45.930 --> 00:30:47.680 occurs at a slightly smaller strain. 00:30:56.200 --> 00:30:59.110 So in a large compressive stress, or strain, 00:30:59.110 --> 00:31:02.205 I guess we could say, cell walls touch, 00:31:02.205 --> 00:31:03.830 and we start to get this densification. 00:31:32.030 --> 00:31:33.500 So the modulus in the limit would 00:31:33.500 --> 00:31:36.900 go to the modulus of the solid. 00:31:36.900 --> 00:31:41.800 And you might expect that the densification 00:31:41.800 --> 00:31:47.570 strain was just equal to 1 minus the relative density. 00:31:47.570 --> 00:31:50.100 But it occurs at a little bit less than that. 00:31:50.100 --> 00:31:55.780 So empirically, we find that it's just 1 minus 1.4 times 00:31:55.780 --> 00:31:58.850 the relative density. 00:31:58.850 --> 00:32:00.640 And then I have this plot here, which 00:32:00.640 --> 00:32:03.565 is really just fitting a line to that data for densification 00:32:03.565 --> 00:32:04.065 strain. 00:32:09.300 --> 00:32:12.060 So those equations describe the compressive stress 00:32:12.060 --> 00:32:13.940 or the compressive behavior of the foam. 00:32:13.940 --> 00:32:17.670 So we've got the moduli, we've got the three compressive 00:32:17.670 --> 00:32:20.210 strengths, and we've got the densification strain. 00:32:22.840 --> 00:32:25.420 So what we're going to do later on in the course is 00:32:25.420 --> 00:32:28.010 we'll use those models to look at how 00:32:28.010 --> 00:32:30.220 we can use foams and things like sandwich panels 00:32:30.220 --> 00:32:32.102 and looking at energy absorption. 00:32:32.102 --> 00:32:34.060 And we'll also look at these equations in terms 00:32:34.060 --> 00:32:37.080 of some biomedical materials-- looking at trabecular bone, 00:32:37.080 --> 00:32:39.979 and looking at tissue engineering scaffolds. 00:32:39.979 --> 00:32:42.020 So there's one last property I wanted to go over, 00:32:42.020 --> 00:32:43.610 and that's the fracture toughness. 00:32:43.610 --> 00:32:45.360 So if we were pulling the foam in tension, 00:32:45.360 --> 00:32:46.894 and we had a crack in the foam, we'd 00:32:46.894 --> 00:32:48.810 want to know what the fracture toughness would 00:32:48.810 --> 00:32:50.170 be for a brittle foam. 00:32:50.170 --> 00:32:52.550 And this follows the same sort of argument 00:32:52.550 --> 00:32:53.752 as we had for the honeycomb. 00:32:53.752 --> 00:32:55.460 So all of these equations really are just 00:32:55.460 --> 00:32:58.411 following the same kinds of arguments. 00:32:58.411 --> 00:33:00.410 But you can kind of see how having the honeycomb 00:33:00.410 --> 00:33:05.810 calculations makes it easier to do the foam ones. 00:33:05.810 --> 00:33:09.340 So we'll do the fracture toughness calculation, 00:33:09.340 --> 00:33:12.970 and then I want to talk a little bit about material selection 00:33:12.970 --> 00:33:15.830 and selection charts for foams. 00:33:15.830 --> 00:33:17.210 So that's less equation-y. 00:33:47.070 --> 00:33:49.700 OK, and we're just going to look at open cells here. 00:33:52.640 --> 00:33:54.660 So imagine we have a crack of length 2a. 00:33:59.860 --> 00:34:02.890 And we have some remote stress applied, 00:34:02.890 --> 00:34:09.510 so remote tensile stress, so I'm going to call that sigma 00:34:09.510 --> 00:34:11.830 infinity-- the far-away stress. 00:34:11.830 --> 00:34:16.090 And then we have a local stress on the cell walls. 00:34:16.090 --> 00:34:19.730 I'm going to call that signal local. 00:34:19.730 --> 00:34:21.552 So I have a little schematic that kind of 00:34:21.552 --> 00:34:22.510 shows what we're doing. 00:34:22.510 --> 00:34:24.260 So we're pulling on it. 00:34:24.260 --> 00:34:25.139 There's some crack. 00:34:25.139 --> 00:34:28.580 The crack length is large compared to the cell size. 00:34:28.580 --> 00:34:32.389 And we want to know what the fracture toughness is. 00:34:32.389 --> 00:34:36.110 So we can say from fracture mechanics the local stress is 00:34:36.110 --> 00:34:38.540 going to be equal to some constant times 00:34:38.540 --> 00:34:43.389 the faraway stress times the square root of pi a 00:34:43.389 --> 00:34:46.380 over the square root of 2 pi r. 00:34:46.380 --> 00:34:49.329 And that's at a distance r from the head of the crack tip. 00:35:03.130 --> 00:35:05.190 And if we look at our little schematic here, 00:35:05.190 --> 00:35:08.580 we could say it's hard to say exactly where the crack tip is, 00:35:08.580 --> 00:35:10.100 but it would be somewhere in here. 00:35:10.100 --> 00:35:12.160 And we'd say this next unbroken cell 00:35:12.160 --> 00:35:14.810 wall is a distance r ahead of the crack tip. 00:35:14.810 --> 00:35:16.580 And that r is going to be related to l. 00:35:16.580 --> 00:35:19.560 It's going to be some function of l. 00:35:19.560 --> 00:35:34.000 So I can say the next unbroken wall ahead of the crack tip 00:35:34.000 --> 00:35:39.280 at some distance r is going to be related to l. 00:35:39.280 --> 00:35:45.950 And that's subject to a force, which 00:35:45.950 --> 00:35:48.125 is going to be the local stress times l squared. 00:35:53.240 --> 00:35:56.500 So that force is going to go as local stress times l squared. 00:35:56.500 --> 00:35:59.140 And the local stress-- I can substitute this thing here 00:35:59.140 --> 00:36:00.910 in-- that's going to be proportional 00:36:00.910 --> 00:36:03.160 to the faraway stress. 00:36:03.160 --> 00:36:05.874 And I'm going to get rid of the pi's. 00:36:05.874 --> 00:36:07.290 And I'm going to substitute for r. 00:36:07.290 --> 00:36:08.690 I'm going to put in l. 00:36:08.690 --> 00:36:10.940 So it's going to be proportional to the faraway stress 00:36:10.940 --> 00:36:15.907 times the root of a over l and times l squared there. 00:36:20.862 --> 00:36:22.320 And then we're going to say, again, 00:36:22.320 --> 00:36:24.430 the edges are going to fail when the applied moment equals 00:36:24.430 --> 00:36:25.300 the fracture moment. 00:36:46.420 --> 00:36:48.590 And the fracture moment is going to go 00:36:48.590 --> 00:36:53.640 as the modulus of rupture of the cell walls times t cubed. 00:36:53.640 --> 00:36:57.480 And the applied moment is going to go as f times l. 00:36:57.480 --> 00:36:59.900 And I've got f from up there, so that 00:36:59.900 --> 00:37:03.910 goes as the faraway stress, sigma infinite, 00:37:03.910 --> 00:37:06.200 times the root of a over l. 00:37:06.200 --> 00:37:08.340 And now I've got l cubed, because there's 00:37:08.340 --> 00:37:10.740 an l squared there and there's an l down here. 00:37:15.210 --> 00:37:17.960 And then if I just equate those, then this 00:37:17.960 --> 00:37:23.110 is going to go as sigma fs times t cubed, like that. 00:37:23.110 --> 00:37:26.780 So then I can say the fracture strength 00:37:26.780 --> 00:37:29.090 is equal to my faraway stress. 00:37:29.090 --> 00:37:32.560 That's going to go as my modulus of rupture times 00:37:32.560 --> 00:37:40.396 the root of l for a times t over l cubed. 00:37:40.396 --> 00:37:41.770 And then my fracture toughness is 00:37:41.770 --> 00:37:45.150 going to be this tensile stress times the root of pi a. 00:37:55.360 --> 00:37:57.610 So there's going to be some other constant here, which 00:37:57.610 --> 00:37:59.405 I'm going to call that C8. 00:37:59.405 --> 00:38:03.070 We've got the modulus of rupture of the solid. 00:38:03.070 --> 00:38:05.030 I've got the square root of l, and I'm 00:38:05.030 --> 00:38:07.317 going to multiply it by pi so it's like other fraction 00:38:07.317 --> 00:38:08.525 mechanics kinds of equations. 00:38:13.054 --> 00:38:15.220 And then we multiply that times the relative density 00:38:15.220 --> 00:38:16.484 to the 3/2 power. 00:38:19.210 --> 00:38:23.320 And here, if we look at data, we find that that constant 00:38:23.320 --> 00:38:28.840 is about equal to 0.65. 00:38:28.840 --> 00:38:30.760 And here's another one of these plots. 00:38:30.760 --> 00:38:34.240 So here I've normalized the fracture toughness of the foams 00:38:34.240 --> 00:38:35.920 by the modulus of rupture of the cell 00:38:35.920 --> 00:38:37.770 walls times the root of pi l. 00:38:37.770 --> 00:38:40.334 So I've taken the cell size into account here, 00:38:40.334 --> 00:38:42.250 and I've plotted against the relative density. 00:38:42.250 --> 00:38:43.720 And that equation there is the same 00:38:43.720 --> 00:38:45.900 as this equation I've got down on the board. 00:38:48.610 --> 00:38:50.360 And this is the only one of the properties 00:38:50.360 --> 00:38:53.570 that we've looked at that depends on the cell size. 00:38:53.570 --> 00:38:55.350 There's a cell size dependence here. 00:39:07.020 --> 00:39:11.755 All right, so I think that's all the modeling of the foams. 00:39:17.570 --> 00:39:19.910 Are we good? 00:39:19.910 --> 00:39:21.240 I gave you a lot of equations. 00:39:21.240 --> 00:39:21.850 We're good? 00:39:21.850 --> 00:39:22.350 All right. 00:40:13.950 --> 00:40:16.410 So I want to talk about how we might design foams 00:40:16.410 --> 00:40:17.625 to improve their properties. 00:40:17.625 --> 00:40:19.000 And then I want to talk about how 00:40:19.000 --> 00:40:21.890 we might select foams for certain applications 00:40:21.890 --> 00:40:24.100 and look at selection charts. 00:40:24.100 --> 00:40:26.350 So when we've been talking about the foams, especially 00:40:26.350 --> 00:40:28.080 the open-cell foams, we've been saying 00:40:28.080 --> 00:40:32.990 their deformation is largely by bending of the cell edges. 00:40:32.990 --> 00:40:34.720 And if we could do something to increase 00:40:34.720 --> 00:40:37.710 the stiffness of the edges or the strength of the edges, 00:40:37.710 --> 00:40:41.180 then that would increase the overall properties of the foam. 00:40:41.180 --> 00:40:44.150 And there's a couple of ways to think about doing that. 00:40:44.150 --> 00:40:46.600 So the foam properties-- if the foam 00:40:46.600 --> 00:40:48.270 is controlled by bending of the edges, 00:40:48.270 --> 00:40:50.190 and the edges have some flexural rigidity, 00:40:50.190 --> 00:40:53.790 EI, if we could increase that EI of the edges, 00:40:53.790 --> 00:40:56.730 we would increase the properties of the foam. 00:40:56.730 --> 00:40:59.980 And one way to do that is by making the edges hollow. 00:40:59.980 --> 00:41:03.540 So if we had hollow edges, and you had a tube, 00:41:03.540 --> 00:41:05.270 then that would increase the EI. 00:41:05.270 --> 00:41:08.870 And we can work out how much it's going to increase them. 00:41:08.870 --> 00:41:16.520 And I have a little example here of-- a natural example 00:41:16.520 --> 00:41:20.150 of hollow foam struts. 00:41:23.010 --> 00:41:24.677 So this is a grass. 00:41:24.677 --> 00:41:26.260 I don't know what kind of grass it is. 00:41:26.260 --> 00:41:27.379 I just saw this grass. 00:41:27.379 --> 00:41:28.920 And we picked some different grasses, 00:41:28.920 --> 00:41:31.440 and we took some SEM pictures. 00:41:31.440 --> 00:41:35.270 And it has a really kind of common structure for grasses. 00:41:35.270 --> 00:41:37.730 It's very common for grass stems to have 00:41:37.730 --> 00:41:42.130 sort of a solid outer part and then a foam-like inner part. 00:41:42.130 --> 00:41:44.310 It's so common that botanists have a name for it. 00:41:44.310 --> 00:41:46.060 They call it the core-rind structure. 00:41:46.060 --> 00:41:48.440 And if you take one of these grass 00:41:48.440 --> 00:41:51.780 stems, and you look at the sort of foamy bit in the middle, 00:41:51.780 --> 00:41:54.426 and you do a SEM picture of that, 00:41:54.426 --> 00:41:56.550 you can see that the little cell walls are actually 00:41:56.550 --> 00:41:57.740 little hollow tubes. 00:41:57.740 --> 00:42:00.930 So one of these things-- it's a little hollow tube. 00:42:00.930 --> 00:42:02.610 So what I wanted to do is work out 00:42:02.610 --> 00:42:05.374 how much the modulus of the foam would 00:42:05.374 --> 00:42:07.040 increase if you could make all the edges 00:42:07.040 --> 00:42:09.406 into little hollow tubes. 00:42:09.406 --> 00:42:10.780 So we're going to start by saying 00:42:10.780 --> 00:42:13.825 the foam behavior is dominated by cell bending, 00:42:13.825 --> 00:42:14.860 so edge bending. 00:42:32.170 --> 00:42:36.120 And the foam properties can be increased by increasing 00:42:36.120 --> 00:42:37.620 the EI of the cell wall. 00:42:52.256 --> 00:42:54.130 So there's a couple of ways we could do that. 00:42:54.130 --> 00:42:57.242 So the first one is looking at hollow walls. 00:43:03.940 --> 00:43:06.280 So imagine I have a thin-walled tube-- just 00:43:06.280 --> 00:43:07.867 a circular, thin-walled tube. 00:43:15.120 --> 00:43:17.130 There's my little wall there. 00:43:17.130 --> 00:43:23.460 It has some radius little r, and a wall thickness t. 00:43:23.460 --> 00:43:25.510 And then imagine I have the same amount of mass, 00:43:25.510 --> 00:43:29.340 but now I have a solid circular section. 00:43:29.340 --> 00:43:35.520 And I'm going to say the radius of that is big R. 00:43:35.520 --> 00:43:42.590 So for our thin-walled tube, the moment of inertia 00:43:42.590 --> 00:43:48.540 is pi r cubed times the thickness, t, if it's thin. 00:43:48.540 --> 00:43:57.554 And for our solid circular section, 00:43:57.554 --> 00:44:04.180 I is going to be pi big R to the 4th over 4. 00:44:04.180 --> 00:44:06.600 And if I say I want to set this up so that the masses are 00:44:06.600 --> 00:44:09.740 equal, then the areas of the cross-sections 00:44:09.740 --> 00:44:12.541 have to be equal-- say it's from the same material. 00:44:15.212 --> 00:44:16.670 So the masses are going to be equal 00:44:16.670 --> 00:44:24.280 if pi R squared is equal to 2 pi r t. 00:44:24.280 --> 00:44:26.610 So I'm going to solve here for R. 00:44:26.610 --> 00:44:28.780 So the pi's are going to cancel out. 00:44:28.780 --> 00:44:33.210 So the masses are equal if R is equal to the square root of 2 00:44:33.210 --> 00:44:34.637 times r times t. 00:44:38.284 --> 00:44:39.700 And then what we're going to do is 00:44:39.700 --> 00:44:42.080 see how the big is the moment of inertia 00:44:42.080 --> 00:44:45.910 of the tube relative to the solid. 00:44:45.910 --> 00:44:50.040 And the tube is pi little r cubed t. 00:44:50.040 --> 00:44:56.355 And the solid was pi R to the 4th, divided by 4. 00:44:56.355 --> 00:44:57.730 And I'm going to get rid of the R 00:44:57.730 --> 00:44:59.811 here, and get rid of the pi's there. 00:45:03.870 --> 00:45:10.270 So R to the 4th is going to be 4r squared t squared. 00:45:10.270 --> 00:45:11.860 So the 4s are going to go. 00:45:11.860 --> 00:45:16.850 And this boils down to r over t. 00:45:16.850 --> 00:45:21.340 So if I had a thin-walled tube, the moment of inertia 00:45:21.340 --> 00:45:25.830 is going to be r over t bigger than if I had the same mass 00:45:25.830 --> 00:45:27.472 in a solid circular section. 00:45:27.472 --> 00:45:28.930 So you can see for the little plant 00:45:28.930 --> 00:45:30.460 here, by making a thin-walled tube, 00:45:30.460 --> 00:45:32.310 you're increasing the stiffness of the foam 00:45:32.310 --> 00:45:34.060 with the same amount of material. 00:45:34.060 --> 00:45:37.120 That's the idea. 00:45:37.120 --> 00:45:39.887 And you can do a similar kind of analysis for other properties. 00:45:56.360 --> 00:45:59.440 So that's if we have hollow tubes. 00:45:59.440 --> 00:46:01.840 So another option is we could have cell walls that 00:46:01.840 --> 00:46:03.390 are sandwich structures. 00:46:03.390 --> 00:46:05.440 So imagine if the cell walls themselves 00:46:05.440 --> 00:46:07.120 were little, tiny sandwich structures. 00:46:43.440 --> 00:46:48.400 So when you have a sandwich beam, what 00:46:48.400 --> 00:46:50.960 you have is too stiff, strong faces 00:46:50.960 --> 00:46:53.180 that are separated by some sort of porous core, 00:46:53.180 --> 00:46:56.330 like a honeycomb or a foam or balsa wood. 00:46:56.330 --> 00:46:59.220 And the idea with the sandwich structure-- 00:46:59.220 --> 00:47:06.080 if I draw a little sketch of the sandwich, here's my faces. 00:47:06.080 --> 00:47:07.730 So imagine those are solid. 00:47:07.730 --> 00:47:09.800 So they might be aluminum sheets, 00:47:09.800 --> 00:47:12.567 or they might be fiber reinforced composites. 00:47:12.567 --> 00:47:14.400 And then we have some sort of cellular thing 00:47:14.400 --> 00:47:19.080 here as the core. 00:47:19.080 --> 00:47:23.750 And the idea is, that's analogous to an I-beam. 00:47:23.750 --> 00:47:28.690 So in the sandwich beam, we have two, 00:47:28.690 --> 00:47:37.600 stiff, strong faces separated by a lightweight core. 00:47:43.120 --> 00:47:48.770 So the core is typically a honeycomb, or a foam, 00:47:48.770 --> 00:47:49.980 or balsa wood. 00:47:54.400 --> 00:47:56.320 And the idea is, you increase the moment 00:47:56.320 --> 00:47:59.337 of inertia of the cross-section with little increase in weight. 00:48:13.550 --> 00:48:16.550 And if you think of an I-beam, an I-beam 00:48:16.550 --> 00:48:19.060 has a large moment of inertia, because you're separating 00:48:19.060 --> 00:48:20.840 the flanges by the web. 00:48:20.840 --> 00:48:22.900 And the sandwich beam works in the same way. 00:48:22.900 --> 00:48:25.359 You're separating the faces by the core. 00:48:25.359 --> 00:48:26.900 But the core doesn't weigh very much, 00:48:26.900 --> 00:48:29.410 because it's a cellular thing. 00:48:29.410 --> 00:48:33.780 So the faces of the sandwich are like the flanges in the I-beam. 00:48:40.170 --> 00:48:41.810 And then the core is like the web. 00:48:48.440 --> 00:48:51.465 So the idea is to make something called a micro-sandwich foam. 00:48:57.530 --> 00:49:00.660 So what you want to do is make the cell walls into sandwiches. 00:49:00.660 --> 00:49:03.760 And one way to do that is to disperse a large volume 00:49:03.760 --> 00:49:06.680 fraction of thin-walled spheres into the foam. 00:49:23.210 --> 00:49:27.130 And you have to get the geometry right to make it work. 00:49:27.130 --> 00:49:32.070 So let me draw a little kind of sketch here of how it works. 00:49:32.070 --> 00:49:33.818 So here's our thin-walled spheres. 00:49:41.000 --> 00:49:44.880 And then you're going to distribute those in a foam. 00:49:44.880 --> 00:49:47.135 Here's another sphere over here. 00:49:47.135 --> 00:49:48.445 The spheres are not perfect. 00:49:51.570 --> 00:49:53.195 Let's say there's another one in here. 00:49:58.170 --> 00:50:00.934 And then the idea is this stuff in here would be the foam. 00:50:05.680 --> 00:50:07.206 So these guys are hollow spheres. 00:50:13.250 --> 00:50:17.100 And say the spheres have a diameter D. 00:50:17.100 --> 00:50:21.570 And say they have a wall thickness here of t. 00:50:21.570 --> 00:50:24.930 And say that the separation of the spheres I'm 00:50:24.930 --> 00:50:27.290 going to call c. 00:50:27.290 --> 00:50:28.870 You can see that there. 00:50:28.870 --> 00:50:32.710 And then the cell size of the foam I'm going to call e. 00:50:37.450 --> 00:50:41.060 So there's a bunch of parameters you have to kind of play with 00:50:41.060 --> 00:50:42.044 to get this to work. 00:50:46.100 --> 00:50:50.080 So you have to have thin-walled spheres so the faces are thin. 00:50:50.080 --> 00:50:52.640 The sandwich panels work best when the faces are thin. 00:50:52.640 --> 00:50:56.570 So you need the thickness of the sphere to be much less than D. 00:50:56.570 --> 00:50:59.770 You need the faces to be stiff relative to the foam. 00:50:59.770 --> 00:51:02.800 So you need the modulus of the sphere material 00:51:02.800 --> 00:51:05.110 to be greater than the modulus of the foam. 00:51:07.810 --> 00:51:10.910 And you need the volume fraction of the spheres 00:51:10.910 --> 00:51:13.550 to be relatively high to get the spheres close enough 00:51:13.550 --> 00:51:15.777 together for this to work. 00:51:15.777 --> 00:51:17.360 So you want that volume fraction to be 00:51:17.360 --> 00:51:20.690 something like 50% to 60%. 00:51:20.690 --> 00:51:23.850 And for the foam, you need to have the foam cell 00:51:23.850 --> 00:51:27.075 size less than the separation between the spheres. 00:51:27.075 --> 00:51:28.950 You need to have a number of-- you can't just 00:51:28.950 --> 00:51:30.090 have one pore in here. 00:51:30.090 --> 00:51:31.660 That's not really like a foam. 00:51:31.660 --> 00:51:34.180 It won't behave like a foam as a continuum. 00:51:34.180 --> 00:51:36.700 So you need to have a number of different cell sizes 00:51:36.700 --> 00:51:40.010 in between each sphere. 00:51:40.010 --> 00:51:41.830 And so you need the cell size of the foam 00:51:41.830 --> 00:51:44.670 to be a lot less than the separation of the spheres 00:51:44.670 --> 00:51:45.810 there, c. 00:51:45.810 --> 00:51:47.950 But if you can control this geometry, 00:51:47.950 --> 00:51:49.750 you can get the sandwich effect. 00:51:49.750 --> 00:51:51.992 And you can get improved properties by doing that. 00:51:51.992 --> 00:51:53.450 So there's ways you can play around 00:51:53.450 --> 00:51:58.297 with the structure of the foams to improve their properties. 00:51:58.297 --> 00:51:59.880 So that was one thing I wanted to say. 00:52:32.250 --> 00:52:34.730 Another way to improve the properties 00:52:34.730 --> 00:52:37.570 of a foam-like material is to use 00:52:37.570 --> 00:52:39.420 one of those lattice materials. 00:52:39.420 --> 00:52:41.570 So we've been talking about ways to improve 00:52:41.570 --> 00:52:42.580 the bending stiffness. 00:52:42.580 --> 00:52:43.650 But if you could get rid of the bending 00:52:43.650 --> 00:52:46.410 altogether and have axial deformation in the cell walls, 00:52:46.410 --> 00:52:47.950 that would be much stiffer. 00:52:47.950 --> 00:52:49.330 And you can get axial deformation 00:52:49.330 --> 00:52:52.520 by having those 3D truss kind of materials. 00:52:52.520 --> 00:52:54.880 So I have a picture of this. 00:52:54.880 --> 00:52:57.430 There we go, so there's one of those 3D truss materials. 00:52:57.430 --> 00:52:59.890 So another alternative is to sort of get rid of the bending 00:52:59.890 --> 00:53:02.642 altogether, and to try to make a truss-type material. 00:53:41.297 --> 00:53:42.880 So there's various ways to make these. 00:53:42.880 --> 00:53:45.510 I think that we talked about a few of them earlier on. 00:53:45.510 --> 00:53:49.640 And you can analyze them as truss-type structures. 00:53:49.640 --> 00:53:51.380 And I can just run through a sort 00:53:51.380 --> 00:53:54.300 of little dimensional argument to get the modulus. 00:53:54.300 --> 00:53:59.410 So the modulus is going to go as the stress over the strain. 00:53:59.410 --> 00:54:03.630 The stress is going to go as a force over a length squared. 00:54:03.630 --> 00:54:06.350 The strain's going to go as a deformation over l. 00:54:06.350 --> 00:54:09.370 So this is just like what we had before for the foams. 00:54:09.370 --> 00:54:11.640 But in this case, the deformation 00:54:11.640 --> 00:54:13.400 is going to go with the force times 00:54:13.400 --> 00:54:18.210 the length over the area of the cross-section divided 00:54:18.210 --> 00:54:22.840 by Es, because we're pulling it or pushing it axially. 00:54:22.840 --> 00:54:30.640 So that goes as Fl over t squared Es. 00:54:30.640 --> 00:54:32.740 And if I just put that back in the equation 00:54:32.740 --> 00:54:37.180 here for the modulus, I get that we've got F over l. 00:54:37.180 --> 00:54:43.910 And I've got delta here, so that's F l t squared Es. 00:54:43.910 --> 00:54:45.930 And you just get the modulus goes 00:54:45.930 --> 00:54:51.940 as the modulus of the solid times t over l squared. 00:54:51.940 --> 00:54:57.020 And that goes as the modulus of the solid times 00:54:57.020 --> 00:54:58.770 the relative density. 00:54:58.770 --> 00:55:00.920 So for the open-celled foams, the modulus 00:55:00.920 --> 00:55:02.690 went as the relative density squared. 00:55:02.690 --> 00:55:06.330 So if it was 10% solid, the modulus would be 0.01. 00:55:06.330 --> 00:55:09.140 And this is saying if it's 10% solid, the modulus is 0.1. 00:55:09.140 --> 00:55:11.480 So it's much bigger. 00:55:11.480 --> 00:55:13.970 So this is all sort of well and good. 00:55:13.970 --> 00:55:18.740 The only difficulty is that when you look at the modulus, 00:55:18.740 --> 00:55:20.020 you can do reasonably well. 00:55:20.020 --> 00:55:21.820 But when you look at the strength, some of the members 00:55:21.820 --> 00:55:23.570 are going to be inevitably in compression. 00:55:23.570 --> 00:55:25.224 When you have these truss materials, 00:55:25.224 --> 00:55:26.890 some members are going to be in tension. 00:55:26.890 --> 00:55:28.723 Some members are going to be in compression. 00:55:28.723 --> 00:55:30.790 And the compression members tend to buckle. 00:55:30.790 --> 00:55:32.530 And once the compression members buckle, 00:55:32.530 --> 00:55:35.260 then you're back to the same kind of strength relationship 00:55:35.260 --> 00:55:36.690 that you have for the foam. 00:55:36.690 --> 00:55:39.280 So that's one of the difficulties of this. 00:55:41.800 --> 00:55:48.450 So let me say that the strength-- so if the strength 00:55:48.450 --> 00:55:50.770 was controlled by uni-axial yield, 00:55:50.770 --> 00:55:52.934 it would go linearly with relative density. 00:55:52.934 --> 00:55:55.100 But if it goes with buckling, it goes as the square. 00:56:17.830 --> 00:56:20.678 So I'll just say the compression members can buckle. 00:56:35.568 --> 00:56:40.680 And say you had a metal lattice. 00:56:40.680 --> 00:56:43.890 Then there's some interaction between the plastic behavior 00:56:43.890 --> 00:56:44.670 and the buckling. 00:56:44.670 --> 00:56:47.620 And you use what's called the tangent modulus instead 00:56:47.620 --> 00:56:49.680 of just the Young's modulus. 00:56:49.680 --> 00:56:52.040 And the tangent modulus is lower. 00:56:52.040 --> 00:56:54.400 And there's also what's called knock-down factors that 00:56:54.400 --> 00:56:55.433 can be large, too. 00:56:58.810 --> 00:57:01.183 So the knock-down factor can be like 50%. 00:57:07.170 --> 00:57:09.790 So the measured strength can be half of what 00:57:09.790 --> 00:57:12.820 you thought it was going to be. 00:57:12.820 --> 00:57:14.861 This should be a squared over here. 00:57:14.861 --> 00:57:15.360 Sorry. 00:57:18.450 --> 00:57:21.600 So even though the stiffness of these 3D trusses 00:57:21.600 --> 00:57:25.150 can be quite good, the strength often isn't quite as good 00:57:25.150 --> 00:57:26.270 as one might hope. 00:57:26.270 --> 00:57:30.825 So that's one of the issues with them. 00:57:30.825 --> 00:57:31.325 All right. 00:57:43.474 --> 00:57:45.140 So do you see the idea, though, with all 00:57:45.140 --> 00:57:48.040 these different micro structures, 00:57:48.040 --> 00:57:49.970 is that you can control the structure in a way 00:57:49.970 --> 00:57:51.770 to try to increase the bending stiffness 00:57:51.770 --> 00:57:53.228 or get rid of the bending stiffness 00:57:53.228 --> 00:57:54.620 and increase the axial stiffness? 00:57:54.620 --> 00:57:56.870 So there's things you can do to play around with that. 00:58:20.110 --> 00:58:22.860 And I wanted to talk a bit today about material selection 00:58:22.860 --> 00:58:23.880 charts for foams. 00:58:23.880 --> 00:58:26.460 So when we talked about woods, we started talking about this. 00:58:26.460 --> 00:58:29.320 Remember, I derived a little performance index. 00:58:29.320 --> 00:58:31.370 We said if we had a material and we 00:58:31.370 --> 00:58:32.880 wanted to have a given stiffness, 00:58:32.880 --> 00:58:34.338 and we wanted to minimize the mass, 00:58:34.338 --> 00:58:37.110 we had that performance index that was E to the 1/2 over rho. 00:58:37.110 --> 00:58:39.910 And we had a chart of modulus versus density. 00:58:39.910 --> 00:58:41.530 And we saw that wood was really good. 00:58:41.530 --> 00:58:44.310 You can do that for other sorts of properties, not just 00:58:44.310 --> 00:58:44.810 modulus. 00:58:44.810 --> 00:58:49.140 So you can make-- depending on what the mechanical requirement 00:58:49.140 --> 00:58:52.050 is, you can work out different performance indices. 00:58:52.050 --> 00:58:54.500 So I want to go into that in a little bit more detail. 00:58:54.500 --> 00:58:56.880 So the question is, how do we select 00:58:56.880 --> 00:59:01.032 the best material for some mechanical requirement? 00:59:33.522 --> 00:59:35.730 So in the wood section, we looked at the minimum mass 00:59:35.730 --> 00:59:38.216 of a beam of a given stiffness. 00:59:42.240 --> 00:59:45.300 And we saw that the performance index was E to the 1/2 00:59:45.300 --> 00:59:46.094 over rho. 00:59:48.767 --> 00:59:50.850 So let me do another one of these little examples, 00:59:50.850 --> 00:59:52.558 and then I'll show you some more of them. 00:59:52.558 --> 00:59:55.620 So another example would be what material-- minimize 00:59:55.620 --> 00:59:58.130 the mass of a beam of a given strength or a given failure 00:59:58.130 --> 00:59:58.630 load. 01:00:22.020 --> 01:00:26.560 So we'll call the failure load Pf. 01:00:26.560 --> 01:00:29.450 And we can see the maximum stress in the beam 01:00:29.450 --> 01:00:32.370 is going to be the moment in the beam times the distance 01:00:32.370 --> 01:00:34.600 from the neutral axis y, and divided 01:00:34.600 --> 01:00:36.170 by the moment of inertia. 01:00:36.170 --> 01:00:40.274 So here, M is the maximum moment in the beam. 01:00:45.800 --> 01:00:48.535 And y is the maximum distance from the neutral axis. 01:00:57.680 --> 01:01:00.560 And I is the moment of inertia. 01:01:00.560 --> 01:01:05.800 And I'm going to say i goes as t to the 4. 01:01:05.800 --> 01:01:08.830 And I'm going to define a failure stress of the material 01:01:08.830 --> 01:01:09.610 sigma f. 01:01:19.440 --> 01:01:24.250 So sigma max is going to go as my failure 01:01:24.250 --> 01:01:25.490 load times the length. 01:01:25.490 --> 01:01:27.080 That would be the moment. 01:01:27.080 --> 01:01:30.750 The distance from the neutral axis is going to go as t. 01:01:30.750 --> 01:01:33.576 And the moment of inertia is going to go as t to the 4th. 01:01:36.470 --> 01:01:39.740 And that's going to be the failure strength there. 01:01:39.740 --> 01:01:41.626 So I can solve this for t. 01:01:41.626 --> 01:01:43.750 And then I'm going to write the mass in terms of t, 01:01:43.750 --> 01:01:45.640 and put that in there. 01:01:45.640 --> 01:01:55.410 So here t goes as Pf l divided by sigma f. 01:01:55.410 --> 01:01:57.530 And that's going to be to the 1/3 power. 01:02:03.720 --> 01:02:05.107 I guess I can scoot over here. 01:02:27.080 --> 01:02:32.470 Then we can say that the mass M goes as the density of times 01:02:32.470 --> 01:02:35.520 t squared times l. 01:02:35.520 --> 01:02:48.950 So the mass M is going to go as rho times l times t squared. 01:02:48.950 --> 01:02:53.240 So that whole thing goes to the 2/3 power. 01:02:53.240 --> 01:02:55.450 So if we look at the material properties, 01:02:55.450 --> 01:03:01.010 the mass goes as the density times the failure stress 01:03:01.010 --> 01:03:02.642 raised to the 2/3 power. 01:03:02.642 --> 01:03:04.100 So if we want to minimize the mass, 01:03:04.100 --> 01:03:07.450 we want to minimize rho over sigma f to the 2/3, 01:03:07.450 --> 01:03:12.020 or we want to maximize sigma F to the 2/3 over rho. 01:03:19.150 --> 01:03:21.456 So that's the performance index for that case. 01:03:25.460 --> 01:03:27.950 So we can obtain these performance indices 01:03:27.950 --> 01:03:29.960 for different loading configurations 01:03:29.960 --> 01:03:32.330 and different mechanical requirements. 01:03:32.330 --> 01:03:34.530 And I don't want to go through a whole lot of them, 01:03:34.530 --> 01:03:37.840 but I'm going to put this up with the notes. 01:03:37.840 --> 01:03:40.135 So this is from Mike Ashby's book on Material Selection 01:03:40.135 --> 01:03:41.720 in Mechanical Design. 01:03:41.720 --> 01:03:44.680 And this is a whole series of these performance indices 01:03:44.680 --> 01:03:47.560 for different situations, for things 01:03:47.560 --> 01:03:49.760 loaded in torsion, for columns and buckling, 01:03:49.760 --> 01:03:51.640 for panels and bending. 01:03:51.640 --> 01:03:53.210 So these ones are all for stiffness. 01:03:53.210 --> 01:03:56.220 And they all involve a modulus raised to some power divided 01:03:56.220 --> 01:03:57.530 by a density. 01:03:57.530 --> 01:04:01.800 So a tie in tension, c over rho, the beam in bending 01:04:01.800 --> 01:04:03.580 is E to the 1/2 over rho. 01:04:03.580 --> 01:04:06.370 A plate in bending is E to 1/3 over rho. 01:04:06.370 --> 01:04:08.420 So you don't need to memorize those. 01:04:08.420 --> 01:04:11.850 But you can see you can derive these for different situations. 01:04:11.850 --> 01:04:14.710 And here's another one for strength-limited design. 01:04:14.710 --> 01:04:19.450 So the shaft is, depending on what the specifications are, 01:04:19.450 --> 01:04:22.780 it's the strength raised to the 2/3 power over rho. 01:04:22.780 --> 01:04:25.740 The beam loaded in bending-- the top one there-- sigma f 01:04:25.740 --> 01:04:26.790 to the 2/3 over rho. 01:04:26.790 --> 01:04:27.940 That's what we just did. 01:04:27.940 --> 01:04:30.600 So there's all these different kind of performance indices. 01:04:30.600 --> 01:04:32.670 So depending on what your situation is, 01:04:32.670 --> 01:04:34.770 you would pick one of these indices. 01:04:34.770 --> 01:04:37.440 And then what you can do is use these material selection 01:04:37.440 --> 01:04:40.580 charts, which plot one property against another 01:04:40.580 --> 01:04:42.247 on log-log scales. 01:04:42.247 --> 01:04:44.080 And because all of these performance indices 01:04:44.080 --> 01:04:46.160 involve a power, they always end up 01:04:46.160 --> 01:04:48.940 being a straight line on your log-log plot. 01:04:48.940 --> 01:04:50.630 And here this one, I think, is the same 01:04:50.630 --> 01:04:52.850 as what I showed you for the wood. 01:04:52.850 --> 01:04:57.080 This one's the modulus here plotted against density. 01:04:57.080 --> 01:04:58.260 So foams are down here. 01:04:58.260 --> 01:05:00.487 And other engineering materials are over here. 01:05:03.229 --> 01:05:05.520 And these guidelines here are the different performance 01:05:05.520 --> 01:05:05.810 indices. 01:05:05.810 --> 01:05:06.955 So this one's E over rho. 01:05:06.955 --> 01:05:08.670 This one's E to the 1/2 over rho. 01:05:08.670 --> 01:05:10.530 This one's E to the 1/3 over rho. 01:05:10.530 --> 01:05:12.850 And for this case here, as you move the lines up 01:05:12.850 --> 01:05:15.930 to the top left-hand corner, E is getting bigger. 01:05:15.930 --> 01:05:17.090 Rho is getting smaller. 01:05:17.090 --> 01:05:19.100 And so the actual value of the performance index 01:05:19.100 --> 01:05:20.200 is getting bigger. 01:05:20.200 --> 01:05:22.630 So you can use this to select a material. 01:05:22.630 --> 01:05:24.950 So we've made these charts for foams as well. 01:05:27.530 --> 01:05:30.017 So here's a couple of charts for foams. 01:05:30.017 --> 01:05:32.350 And I think what I'm going to do is just go through them 01:05:32.350 --> 01:05:32.854 quickly. 01:05:32.854 --> 01:05:34.520 And there aren't really that many notes, 01:05:34.520 --> 01:05:36.120 so I'll just put the notes on the website. 01:05:36.120 --> 01:05:38.170 And you can come and write all the notes down. 01:05:38.170 --> 01:05:40.330 And then we can finish this today. 01:05:40.330 --> 01:05:42.980 So this one here is the Young's modulus versus density. 01:05:42.980 --> 01:05:45.380 And these are all sorts of different foams. 01:05:45.380 --> 01:05:47.970 So the low modulus ones tend to be flexible. 01:05:47.970 --> 01:05:50.950 The higher modulus ones tend to be more rigid. 01:05:50.950 --> 01:05:55.420 And you could use this to select foams, if you wanted. 01:05:55.420 --> 01:05:57.730 You can also see what the range of values is. 01:05:57.730 --> 01:05:59.780 So the values of the modulus here 01:05:59.780 --> 01:06:03.930 goes from a little less than a 100-- 01:06:03.930 --> 01:06:06.030 because this is two orders of magnitude here, 01:06:06.030 --> 01:06:09.266 I think, each one of these-- down to about 10 to the minus 4 01:06:09.266 --> 01:06:10.580 or a little less than that. 01:06:10.580 --> 01:06:11.580 So there's a huge range. 01:06:11.580 --> 01:06:13.538 There's almost a range of a factor of a million 01:06:13.538 --> 01:06:15.339 in those moduli. 01:06:15.339 --> 01:06:16.880 And the same with the strengths here. 01:06:16.880 --> 01:06:19.980 The strengths go from 10 to the minus 3 01:06:19.980 --> 01:06:22.880 mega-pascals up to about maybe 30 mega-pascals, 01:06:22.880 --> 01:06:24.570 something like that. 01:06:24.570 --> 01:06:27.420 And you can see for the modulus and the strength, 01:06:27.420 --> 01:06:29.050 things like the metal foams are good. 01:06:29.050 --> 01:06:29.987 The balsa's good. 01:06:29.987 --> 01:06:31.910 Here's the balsa up here. 01:06:31.910 --> 01:06:33.370 Metal foam's up there. 01:06:33.370 --> 01:06:38.940 So you can kind of see the range of properties 01:06:38.940 --> 01:06:41.040 that you could get. 01:06:41.040 --> 01:06:45.740 And then you could also-- need a drink, hang on. 01:06:50.860 --> 01:06:53.840 You can also plot the specific property. 01:06:53.840 --> 01:06:55.460 So here's the compressive strength 01:06:55.460 --> 01:06:58.580 divided by the density plotted against the Young's modulus 01:06:58.580 --> 01:07:00.006 divided by the density. 01:07:00.006 --> 01:07:01.630 And here you want to be up at this end. 01:07:01.630 --> 01:07:04.260 So you would have a high strength and a high stiffness. 01:07:04.260 --> 01:07:07.790 So the balsa and the metal foams are good up here. 01:07:07.790 --> 01:07:10.695 This next plot-- this is the compressive stress 01:07:10.695 --> 01:07:13.100 at 25% strain. 01:07:13.100 --> 01:07:15.530 And this is the densification strain. 01:07:15.530 --> 01:07:21.020 And if you think of having your stress-strain curve 01:07:21.020 --> 01:07:24.560 looks like this, something like that, 01:07:24.560 --> 01:07:27.780 so you could say that's a strain of 0.25 01:07:27.780 --> 01:07:30.920 and that's the stress that corresponds to that. 01:07:30.920 --> 01:07:34.520 So that stress times the densification strain, 01:07:34.520 --> 01:07:37.730 which is out here someplace, is an estimate 01:07:37.730 --> 01:07:41.230 of the energy underneath the stress-strain curve. 01:07:41.230 --> 01:07:45.240 So you can think of this right-hand plot here-- 01:07:45.240 --> 01:07:48.920 those dashed lines-- these lines like this and this and this-- 01:07:48.920 --> 01:07:51.630 each one of those corresponds to how much energy you 01:07:51.630 --> 01:07:54.100 would absorb under the stress-strain curve. 01:07:54.100 --> 01:07:57.840 So points that lie on here would have 01:07:57.840 --> 01:08:01.500 an energy of 0.001 megajoules per cubic meter. 01:08:01.500 --> 01:08:04.130 And over here, we're at 10 joules per cubic meter. 01:08:04.130 --> 01:08:07.600 So again, the balsa and the metal foams are good over here. 01:08:07.600 --> 01:08:10.660 So you can use these plots to try to identify foams 01:08:10.660 --> 01:08:12.700 for particular applications. 01:08:12.700 --> 01:08:14.890 And I think there's a couple more. 01:08:14.890 --> 01:08:16.740 It doesn't have to be mechanical properties. 01:08:16.740 --> 01:08:19.699 Here is thermal conductivity versus compressive strength. 01:08:19.699 --> 01:08:21.740 So you can imagine if you wanted some insulation, 01:08:21.740 --> 01:08:24.130 you wanted to have a certain thermal conductivity value, 01:08:24.130 --> 01:08:26.550 you probably also need at least some minimal compressive 01:08:26.550 --> 01:08:27.729 strength. 01:08:27.729 --> 01:08:30.130 You could also have something like a maximum service 01:08:30.130 --> 01:08:32.074 temperature, that maybe the foam is going 01:08:32.074 --> 01:08:33.240 to melt at some temperature. 01:08:33.240 --> 01:08:35.670 You can't go beyond that. 01:08:35.670 --> 01:08:37.850 So there's some property there. 01:08:37.850 --> 01:08:40.420 And I think there's one more here. 01:08:40.420 --> 01:08:42.090 You can look at things like the density 01:08:42.090 --> 01:08:43.729 in terms of the buoyancy of a foam, 01:08:43.729 --> 01:08:45.840 if you have some buoyancy application. 01:08:45.840 --> 01:08:48.560 And you can look at cell size on this one here. 01:08:48.560 --> 01:08:50.510 And cell size can be important for things 01:08:50.510 --> 01:08:52.250 like filtration and catalysis. 01:08:52.250 --> 01:08:55.510 So the amount of surface area goes as 1 over the cell size-- 01:08:55.510 --> 01:08:57.680 the surface area per unit volume. 01:08:57.680 --> 01:09:00.010 And so the cell size can be important for those sorts 01:09:00.010 --> 01:09:01.132 of applications. 01:09:01.132 --> 01:09:03.340 So the idea is, you can make these material selection 01:09:03.340 --> 01:09:04.130 charts for foam. 01:09:04.130 --> 01:09:06.287 And you can put data on there. 01:09:06.287 --> 01:09:07.370 And you can compare foams. 01:09:07.370 --> 01:09:09.745 And you can use these performance indices. 01:09:09.745 --> 01:09:11.120 So I'm going to leave it at that. 01:09:11.120 --> 01:09:12.760 There is a little bit more notes. 01:09:12.760 --> 01:09:14.460 But I'll just put them on the website, 01:09:14.460 --> 01:09:17.050 and you can get them from there. 01:09:17.050 --> 01:09:19.409 So I think we're good for today.