1 00:00:00,000 --> 00:00:03,710 The following content is provided by MIT OpenCourseWare 2 00:00:03,710 --> 00:00:06,130 under a Creative Commons license. 3 00:00:06,130 --> 00:00:08,740 Additional information about our license and MIT 4 00:00:08,740 --> 00:00:12,090 OpenCourseWare in general is available at ocw.MIT.edu. 5 00:00:15,820 --> 00:00:21,970 PROFESSOR: This, as you know, is the last lecture of thermal 6 00:00:21,970 --> 00:00:27,820 tensor properties of crystals for the semester. 7 00:00:27,820 --> 00:00:30,140 Come on, can't somebody go, aww, just to 8 00:00:30,140 --> 00:00:31,120 make me feel good? 9 00:00:31,120 --> 00:00:35,060 At least nobody said yay, so that makes me feel good, too. 10 00:00:35,060 --> 00:00:38,780 So I guess on balance I come out feeling pretty happy. 11 00:00:38,780 --> 00:00:44,450 What I'm going to do today is to talk a little bit more 12 00:00:44,450 --> 00:00:47,060 about forthright tensor properties, which we've not 13 00:00:47,060 --> 00:00:49,520 said much about. 14 00:00:49,520 --> 00:00:54,450 And I'm going to look at some scalar moduli and look at them 15 00:00:54,450 --> 00:00:59,590 for a couple of different symmetries and restrictions on 16 00:00:59,590 --> 00:01:02,410 the compliances. 17 00:01:02,410 --> 00:01:05,720 The surfaces that you might expect turn out to be really, 18 00:01:05,720 --> 00:01:06,630 really weird. 19 00:01:06,630 --> 00:01:10,800 Because these will be fourth order variation with the 20 00:01:10,800 --> 00:01:12,470 direction cosines. 21 00:01:12,470 --> 00:01:15,310 So when you take trigonometric functions and raise them to 22 00:01:15,310 --> 00:01:18,710 the fourth power, you get severe anisotropies for a 23 00:01:18,710 --> 00:01:22,910 great many of the scalar moduli. 24 00:01:22,910 --> 00:01:28,050 I'm sorry to say, years ago when I made the acquaintance 25 00:01:28,050 --> 00:01:32,220 of a computer programmer in course six who was looking for 26 00:01:32,220 --> 00:01:36,230 something challenging to do in connection with material 27 00:01:36,230 --> 00:01:38,040 science and engineering, I said, wow, have I got 28 00:01:38,040 --> 00:01:39,150 something for you. 29 00:01:39,150 --> 00:01:41,410 How about doing some computer graphics? 30 00:01:41,410 --> 00:01:46,890 And this was a few years ago when such 31 00:01:46,890 --> 00:01:48,870 products were fairly rare. 32 00:01:48,870 --> 00:01:53,010 And how about making a program that would let us see visually 33 00:01:53,010 --> 00:01:56,580 how some of these scalar moduli will vary with 34 00:01:56,580 --> 00:01:59,770 direction and provide the provision so you can turn them 35 00:01:59,770 --> 00:02:03,670 over in space, just as you would pick up a model in your 36 00:02:03,670 --> 00:02:06,490 hands and rotate it around till you finally came to 37 00:02:06,490 --> 00:02:07,680 appreciate it? 38 00:02:07,680 --> 00:02:11,750 And then for something like a triclinic crystal, where there 39 00:02:11,750 --> 00:02:15,870 are almost a couple of dozen different moduli, how about 40 00:02:15,870 --> 00:02:20,290 allowing people to scale up or scale down the value of one of 41 00:02:20,290 --> 00:02:25,340 the elastic constants and let them see 42 00:02:25,340 --> 00:02:26,890 how the shape changes? 43 00:02:26,890 --> 00:02:29,720 And oh, that was-- he went to work on it for a 44 00:02:29,720 --> 00:02:31,400 semester and came back. 45 00:02:31,400 --> 00:02:33,240 And it was a gorgeous thing. 46 00:02:33,240 --> 00:02:38,850 You could take a monoclinic representation of three 47 00:02:38,850 --> 00:02:40,830 dimensions of Young's modulus. 48 00:02:40,830 --> 00:02:46,330 Say you want to change S31. 49 00:02:46,330 --> 00:02:48,010 And it would change by a factor of 10. 50 00:02:48,010 --> 00:02:50,630 You'd see a big nose blow out on the surface and then 51 00:02:50,630 --> 00:02:51,780 contract down again. 52 00:02:51,780 --> 00:02:52,990 And you could turn around. 53 00:02:52,990 --> 00:02:56,180 Anyway, the program having told you how marvelous it is, 54 00:02:56,180 --> 00:02:59,010 it's no longer supported because of the operating 55 00:02:59,010 --> 00:03:01,850 system that it operated under having done defunct. 56 00:03:01,850 --> 00:03:04,140 But anyway, that was a fun thing to play with. 57 00:03:04,140 --> 00:03:06,220 AUDIENCE: What operating system? 58 00:03:06,220 --> 00:03:09,570 PROFESSOR: I don't even remember anymore. 59 00:03:09,570 --> 00:03:12,570 It was a fairly obscure one. 60 00:03:12,570 --> 00:03:15,440 Was not only the system, but it worked through a software 61 00:03:15,440 --> 00:03:17,366 package which operated on that system. 62 00:03:20,870 --> 00:03:22,790 All right. 63 00:03:22,790 --> 00:03:26,020 So we're going to talk today about fourth rank tensor 64 00:03:26,020 --> 00:03:27,270 properties. 65 00:03:29,040 --> 00:03:30,710 The most important-- 66 00:03:30,710 --> 00:03:34,640 in fact, probably the only ones you've ever heard of, are 67 00:03:34,640 --> 00:03:39,740 the elastic, stiffnesses, and compliances, which are ways of 68 00:03:39,740 --> 00:03:42,230 representing strain in terms of stress or 69 00:03:42,230 --> 00:03:44,600 stress in terms of strain. 70 00:03:44,600 --> 00:03:50,300 I think I'll sort of see how the time plays out. 71 00:03:50,300 --> 00:03:54,310 If I finish just about everything that I'm hoping to 72 00:03:54,310 --> 00:03:59,040 say by on the hour, I think I'll just go an extra five or 73 00:03:59,040 --> 00:04:00,140 10 minutes. 74 00:04:00,140 --> 00:04:04,510 And then comes a great, grand, and traditional event at MIT. 75 00:04:04,510 --> 00:04:10,210 All semester long I've been grilling you with problem sets 76 00:04:10,210 --> 00:04:14,610 and quizzes, and you've had to dance through your steps just 77 00:04:14,610 --> 00:04:16,480 because I told you to. 78 00:04:16,480 --> 00:04:20,410 Now how's this for reciprocity-- 79 00:04:20,410 --> 00:04:22,800 you get to evaluate me. 80 00:04:22,800 --> 00:04:24,470 You get to grade me. 81 00:04:24,470 --> 00:04:27,460 MIT, at the end of every semester, has a course 82 00:04:27,460 --> 00:04:28,780 evaluation. 83 00:04:28,780 --> 00:04:32,585 It's submitted anonymously by you, so you could really vent 84 00:04:32,585 --> 00:04:36,250 your spleen without ever having called to task for it. 85 00:04:36,250 --> 00:04:43,090 So when we finish, either for our break or at the end of a 86 00:04:43,090 --> 00:04:48,800 slightly extended first session, I will bow, to 87 00:04:48,800 --> 00:04:51,350 thunderous applause, I hope, and exit. 88 00:04:51,350 --> 00:04:55,790 And then you will be left alone to fill out the course 89 00:04:55,790 --> 00:04:56,680 evaluation. 90 00:04:56,680 --> 00:05:00,250 Corinne will collect them from you, and she will deliver them 91 00:05:00,250 --> 00:05:02,660 over to department headquarters. 92 00:05:02,660 --> 00:05:06,760 The results will be tabulated, and so will the comments. 93 00:05:06,760 --> 00:05:11,120 But they will not be in your hand, all of which I 94 00:05:11,120 --> 00:05:13,930 recognize, having spent hour after hour 95 00:05:13,930 --> 00:05:16,200 grading your quizzes. 96 00:05:16,200 --> 00:05:18,780 So they will be submitted to me in a 97 00:05:18,780 --> 00:05:21,290 completely sanitized fashion. 98 00:05:21,290 --> 00:05:23,866 I won't know who said what. 99 00:05:23,866 --> 00:05:27,390 A final question that I will answer before you ask it, what 100 00:05:27,390 --> 00:05:28,590 about the quizzes? 101 00:05:28,590 --> 00:05:31,310 How are you doing on them? 102 00:05:31,310 --> 00:05:33,390 Well, I have come to hate them. 103 00:05:33,390 --> 00:05:35,730 I think I will give one-question quizzes from now 104 00:05:35,730 --> 00:05:37,160 on the future. 105 00:05:37,160 --> 00:05:39,960 It takes me about five to seven minutes 106 00:05:39,960 --> 00:05:41,860 to grade each question. 107 00:05:41,860 --> 00:05:44,430 And there are about 25 of you. 108 00:05:44,430 --> 00:05:47,710 So it takes me, to grade one question, somewhere between 109 00:05:47,710 --> 00:05:49,810 two and 1/2 and three hours. 110 00:05:49,810 --> 00:05:51,910 And there are 10 questions on the two quizzes 111 00:05:51,910 --> 00:05:53,240 remaining to be graded. 112 00:05:53,240 --> 00:05:56,060 And so I let you do the arithmetic, and you can see 113 00:05:56,060 --> 00:06:00,860 what sort of torment I've been living in the last few days in 114 00:06:00,860 --> 00:06:03,380 return for the torment I put you through 115 00:06:03,380 --> 00:06:04,600 for two brief hours. 116 00:06:04,600 --> 00:06:06,590 So wow. 117 00:06:06,590 --> 00:06:09,470 I will leave it to you to decide who's getting the worst 118 00:06:09,470 --> 00:06:10,830 of this deal. 119 00:06:10,830 --> 00:06:14,170 OK, fourth rank tensors. 120 00:06:14,170 --> 00:06:17,540 Elasticity's probably the only one you can think of. 121 00:06:17,540 --> 00:06:21,130 Let me give you an example in this handout, an example of 122 00:06:21,130 --> 00:06:23,610 another forthright tensor property. 123 00:06:23,610 --> 00:06:28,410 And this is the piezoresistive effect, the way electrical 124 00:06:28,410 --> 00:06:31,760 resistance of the material changes in response to an 125 00:06:31,760 --> 00:06:32,810 applied stress. 126 00:06:32,810 --> 00:06:34,560 It's a pretty exotic property. 127 00:06:34,560 --> 00:06:38,950 I had never heard of it until I was at a meeting of the 128 00:06:38,950 --> 00:06:42,010 American Crystallographic Association and somebody, 129 00:06:42,010 --> 00:06:44,200 believe it or not, from Texas Instruments-- 130 00:06:44,200 --> 00:06:48,080 so you better bet that this property is useful in 131 00:06:48,080 --> 00:06:49,260 technology-- 132 00:06:49,260 --> 00:06:53,500 fellow named Ahmed Amin got up and gave a talk on the 133 00:06:53,500 --> 00:06:55,330 piezoresistive effect. 134 00:06:55,330 --> 00:06:58,840 And he had marvelous slides at the outset that defined the 135 00:06:58,840 --> 00:06:59,370 effect 136 00:06:59,370 --> 00:07:06,430 So what you have here on these few sheets is a few pages that 137 00:07:06,430 --> 00:07:09,460 define the piezoresistance effect. 138 00:07:09,460 --> 00:07:10,990 We're not going to do anything with it. 139 00:07:10,990 --> 00:07:15,890 It's a fourth rank tensor, so it will proceed to transform 140 00:07:15,890 --> 00:07:19,810 like other forthright tensors, such as elastic stiffness and 141 00:07:19,810 --> 00:07:21,060 compliance. 142 00:07:23,050 --> 00:07:28,800 I should warn you about these following pages, which were 143 00:07:28,800 --> 00:07:31,560 slides that he used at his talk. 144 00:07:31,560 --> 00:07:35,680 He seems to have done the lettering with a magic marker. 145 00:07:35,680 --> 00:07:40,840 Because all of the I's and J's are indistinguishable, which 146 00:07:40,840 --> 00:07:44,720 makes interpretation of what he's saying here a little bit 147 00:07:44,720 --> 00:07:45,780 challenging. 148 00:07:45,780 --> 00:07:50,160 And then the other thing that throws one off is that he uses 149 00:07:50,160 --> 00:07:53,220 x to represent stress. 150 00:07:53,220 --> 00:07:56,310 I don't know what field or in what discipline the elements 151 00:07:56,310 --> 00:08:00,280 of stress are called xij, but this is what he does. 152 00:08:00,280 --> 00:08:02,880 And once you figure that out, the interpretation of what he 153 00:08:02,880 --> 00:08:08,900 has said here is fairly apparent if 154 00:08:08,900 --> 00:08:10,120 you think about it. 155 00:08:10,120 --> 00:08:14,970 Anyway, he takes a state of zero stress and strain and 156 00:08:14,970 --> 00:08:17,420 expands it as a series, and then picks off different 157 00:08:17,420 --> 00:08:18,490 coefficients. 158 00:08:18,490 --> 00:08:25,520 And one of them is a set of moduli which he calls pi, pi 159 00:08:25,520 --> 00:08:37,150 subscript ijkl, which relate stress, xkl in his notation, 160 00:08:37,150 --> 00:08:44,910 to a change in the density, delta row ij. 161 00:08:44,910 --> 00:08:50,760 So here's a fourth rank tensor property that will indeed have 162 00:08:50,760 --> 00:08:53,960 to conform to all the restrictions for the moduli 163 00:08:53,960 --> 00:08:57,560 that we have defined for the elastic 164 00:08:57,560 --> 00:08:58,810 stiffnesses and moduli. 165 00:09:01,710 --> 00:09:02,220 All right. 166 00:09:02,220 --> 00:09:07,250 Let me then turn to stress and strain. 167 00:09:07,250 --> 00:09:13,130 We introduced the relation between stress and strain, but 168 00:09:13,130 --> 00:09:20,120 didn't really go into detail on the bizarre absorptions of 169 00:09:20,120 --> 00:09:24,900 factors of two or four that have to be done in order to 170 00:09:24,900 --> 00:09:28,420 make this come out in a nice, clean matrix form. 171 00:09:28,420 --> 00:09:38,460 So let me remind you of where these silly factors came in. 172 00:09:38,460 --> 00:09:43,462 We took our stress tensor, sigma 1 1, sigma 1 2, sigma 1 173 00:09:43,462 --> 00:09:52,190 3, 2 1, 2 2, and 2 3, sigma 3 1, sigma 3 2, and sigma 3 3. 174 00:09:55,982 --> 00:09:59,370 The tensor had to be symmetric. 175 00:09:59,370 --> 00:10:02,720 The off-diagonal term sigma ij had to be equal to 176 00:10:02,720 --> 00:10:04,590 the term sigma ij. 177 00:10:04,590 --> 00:10:09,070 And this was for the reason of mechanical equilibrium. 178 00:10:12,690 --> 00:10:16,260 These off-diagonal terms, sigma ij and sigma ij, we saw, 179 00:10:16,260 --> 00:10:18,160 exerted a torque on a body. 180 00:10:18,160 --> 00:10:20,900 And unless they were equal, the body would undergo an 181 00:10:20,900 --> 00:10:22,830 angular acceleration. 182 00:10:22,830 --> 00:10:27,040 And then we renumbered these according to the convention 183 00:10:27,040 --> 00:10:33,670 that as we would march down the diagonal of the tensor and 184 00:10:33,670 --> 00:10:39,170 take pairs of subscripts, 1 1, 2 2, 3 3, represent them by a 185 00:10:39,170 --> 00:10:45,095 single 1, 2, and 3, then march up the side to define a sigma 186 00:10:45,095 --> 00:10:47,960 4, sigma 5, and a sigma 6. 187 00:10:47,960 --> 00:10:50,410 If you remember that little algorithm you can always keep 188 00:10:50,410 --> 00:10:55,180 straight what these reduced subscripts represent. 189 00:10:55,180 --> 00:10:58,080 So with that definition, then, the stress tensor reduced to 190 00:10:58,080 --> 00:11:09,490 sigma 1, sigma 6, sigma 5, sigma 6, sigma 2, sigma 4, 191 00:11:09,490 --> 00:11:17,755 sigma 5, sigma 4, sigma 3. 192 00:11:20,620 --> 00:11:26,090 And that takes a tensor and degrades it to a matrix. 193 00:11:26,090 --> 00:11:30,520 Because there is no law of transformation for this 194 00:11:30,520 --> 00:11:34,780 representation of the elements of stress. 195 00:11:34,780 --> 00:11:40,860 OK, then we do something very similar with the strain 196 00:11:40,860 --> 00:11:48,990 tensor, epsilon 1 1, 1 2, 1 3, epsilon 2 1, epsilon 2 2, 197 00:11:48,990 --> 00:11:53,970 epsilon 2 3, epsilon 3 1, epsilon 3 2, epsilon 3 3. 198 00:11:57,340 --> 00:12:00,750 And we saw that for physical reasons this tensor also had 199 00:12:00,750 --> 00:12:01,860 to be symmetric. 200 00:12:01,860 --> 00:12:12,430 We had to have epsilon ij identical to epsilon ji, the 201 00:12:12,430 --> 00:12:15,750 reason being that if these off-diagonal shear strains 202 00:12:15,750 --> 00:12:20,840 were not equal, the state that we would be defining was one 203 00:12:20,840 --> 00:12:27,280 of actual deformation combined with rigid body rotation. 204 00:12:27,280 --> 00:12:33,840 So unless this was the case, the definition performed by 205 00:12:33,840 --> 00:12:36,630 the tensor epsilon iij would define rigid 206 00:12:36,630 --> 00:12:40,350 body notation as well. 207 00:12:40,350 --> 00:12:40,830 OK. 208 00:12:40,830 --> 00:12:47,180 We use the same process of renumbering to convert this 209 00:12:47,180 --> 00:12:49,580 from a tensor to a matrix. 210 00:12:49,580 --> 00:12:54,450 And we got this into a form, epsilon 1, epsilon 6, epsilon 211 00:12:54,450 --> 00:13:02,792 5, epsilon 6, epsilon 2, epsilon 4, epsilon 5, epsilon 212 00:13:02,792 --> 00:13:04,510 4, epsilon 3. 213 00:13:08,480 --> 00:13:11,800 But in order to do that, we saw we had to 214 00:13:11,800 --> 00:13:15,140 consume a factor of two. 215 00:13:15,140 --> 00:13:19,690 So there is a factor of two built into the 216 00:13:19,690 --> 00:13:21,410 off-diagonal terms. 217 00:13:21,410 --> 00:13:28,210 And what we have put in here is we converted this to 218 00:13:28,210 --> 00:13:35,010 actually epsilon 1, 1/2, epsilon 6, 1/2, epsilon 5. 219 00:13:35,010 --> 00:13:39,190 And I should have written that this way to begin with. 220 00:13:39,190 --> 00:13:45,480 So all these factors of two appear so that when we combine 221 00:13:45,480 --> 00:13:52,550 these epsilons we get a nice, clean matrix relation between 222 00:13:52,550 --> 00:13:56,430 stress and strain that doesn't involve factors of two. 223 00:13:56,430 --> 00:13:59,190 And that's a great convenience if we're going to be doing all 224 00:13:59,190 --> 00:14:02,510 of our deformation within the framework of 225 00:14:02,510 --> 00:14:03,760 one coordinate system. 226 00:14:10,310 --> 00:14:10,800 OK. 227 00:14:10,800 --> 00:14:17,550 So if we now use these definitions to write the 228 00:14:17,550 --> 00:14:22,830 stress in terms of strain, we will set up our fourth rank 229 00:14:22,830 --> 00:14:24,580 tensor property. 230 00:14:24,580 --> 00:14:30,510 If we do this first for the stress in terms of strain, you 231 00:14:30,510 --> 00:14:35,250 would have a tensor element of stress sigma 1 1, which would 232 00:14:35,250 --> 00:14:47,270 be equal to C1 1 1 1 times epsilon 1 1 plus C1 1 2 2, 233 00:14:47,270 --> 00:14:55,950 epsilon 2 2 plus C1 1 3 3, epsilon 3 3. 234 00:14:55,950 --> 00:15:05,440 And then we would have, in addition, off-diagonal 235 00:15:05,440 --> 00:15:06,560 elements of strain. 236 00:15:06,560 --> 00:15:21,990 We'd have C1 1 2 3 times epsilon 2 3 plus C1 1 3 2 237 00:15:21,990 --> 00:15:34,590 times epsilon 3 2 plus C1 1 2 1, epsilon 2 1 plus C1 1 1 2, 238 00:15:34,590 --> 00:15:36,790 epsilon 1 2 plus C1 1-- 239 00:15:40,230 --> 00:15:44,720 and I should have made this 1 3 so that the other integers 240 00:15:44,720 --> 00:15:45,970 come out right. 241 00:15:51,230 --> 00:16:02,815 And C1 1 1 2 times an epsilon 1 1 1 2 plus a C 1 1 2 1 times 242 00:16:02,815 --> 00:16:05,040 an epsilon 2 1. 243 00:16:05,040 --> 00:16:07,000 So that is one line of the relation 244 00:16:07,000 --> 00:16:10,230 between stress and strain. 245 00:16:10,230 --> 00:16:16,570 And when we condense this to a matrix form, it is surprising, 246 00:16:16,570 --> 00:16:19,530 once we expand it once more, at how cumbersome this 247 00:16:19,530 --> 00:16:21,250 expression is. 248 00:16:21,250 --> 00:16:23,450 Now, why do I remind you of all this? 249 00:16:23,450 --> 00:16:26,710 Isn't it nice to work in the form of a fixed coordinate 250 00:16:26,710 --> 00:16:32,335 system where we can use a matrix for the Cijkl's? 251 00:16:32,335 --> 00:16:37,500 The answer is, fine, unless you want to change the 252 00:16:37,500 --> 00:16:39,050 coordinate system. 253 00:16:39,050 --> 00:16:42,780 And you might want to do that for practical reasons such as 254 00:16:42,780 --> 00:16:49,270 cutting out a specimen which makes it convenient to define 255 00:16:49,270 --> 00:16:51,820 the stiffnesses and compliances relative to a 256 00:16:51,820 --> 00:16:57,770 coordinate system taken along the logical 257 00:16:57,770 --> 00:16:59,640 directions in the specimen. 258 00:16:59,640 --> 00:17:02,140 And the other reason for doing it, and that is what I want to 259 00:17:02,140 --> 00:17:06,089 do a little bit of this afternoon, is to derive the 260 00:17:06,089 --> 00:17:09,710 symmetry restrictions on the stiffnesses and compliances. 261 00:17:09,710 --> 00:17:14,890 You cannot transform a matrix representation of the 262 00:17:14,890 --> 00:17:16,319 stiffness or the compliance. 263 00:17:16,319 --> 00:17:20,160 You can only do this for the full-blown tensor arrangement. 264 00:17:20,160 --> 00:17:22,680 So having collapsed down, which we'll do momentarily 265 00:17:22,680 --> 00:17:26,470 just to remind you of how it goes, we will have to expand 266 00:17:26,470 --> 00:17:30,120 again to derive symmetry restrictions. 267 00:17:30,120 --> 00:17:31,580 Do not exit at this point. 268 00:17:31,580 --> 00:17:35,180 We're not going to do any of these calculations in their 269 00:17:35,180 --> 00:17:37,520 full glory detail. 270 00:17:37,520 --> 00:17:41,020 We'll just set up the problem and then jump immediately to 271 00:17:41,020 --> 00:17:42,660 the outset. 272 00:17:42,660 --> 00:17:47,570 So how does this pay out if we try to make the condensation? 273 00:17:47,570 --> 00:18:00,670 I remind you again that the symbol C stands for stiffness, 274 00:18:00,670 --> 00:18:10,010 and the symbol S, which we'll see later, the Sijkl's are 275 00:18:10,010 --> 00:18:11,260 call compliances. 276 00:18:14,940 --> 00:18:20,630 And the perversity of that semantic description of these 277 00:18:20,630 --> 00:18:25,380 tensor elements is perverse for reasons that I have never 278 00:18:25,380 --> 00:18:27,500 really been able to understand. 279 00:18:27,500 --> 00:18:30,930 So if we go down to matrix form, we'll write instead of 280 00:18:30,930 --> 00:18:33,230 sigma 1 1, simply sigma 1 1. 281 00:18:33,230 --> 00:18:41,880 We'll call this C1 1 times epsilon 1 plus C1 2 282 00:18:41,880 --> 00:18:43,620 times epsilon 2. 283 00:18:43,620 --> 00:18:45,180 So far so good. 284 00:18:45,180 --> 00:18:49,560 C1 3 times epsilon 3. 285 00:18:49,560 --> 00:18:55,580 And now we have problems, because epsilon 2 3 was 286 00:18:55,580 --> 00:19:00,360 defined as 1/2 of epsilon 5. 287 00:19:00,360 --> 00:19:05,990 Now we'll have a C1 5 here, and then a 288 00:19:05,990 --> 00:19:11,930 C1 3 2 is C1 5 again. 289 00:19:11,930 --> 00:19:14,580 And this also is 1/2 of epsilon 5. 290 00:19:17,200 --> 00:19:21,900 And then similarly, this is C1-- 291 00:19:21,900 --> 00:19:22,350 I'm sorry. 292 00:19:22,350 --> 00:19:23,600 This is C1 4. 293 00:19:28,930 --> 00:19:35,540 This is C1 5 times 1/2 of epsilon 5 plus C1 5 times 1/2 294 00:19:35,540 --> 00:19:45,760 of epsilon 5 plus C1 6 times 1/2 of epsilon 6 plus C1 6 295 00:19:45,760 --> 00:19:47,010 times 1/2 of epsilon 6. 296 00:19:50,210 --> 00:19:56,090 So this, given the way in which we had defined matrix 297 00:19:56,090 --> 00:19:58,660 strain, initially in connection with 298 00:19:58,660 --> 00:20:01,680 piezoelectricity when we entered the realm of third 299 00:20:01,680 --> 00:20:04,520 rank tensors, this will play out OK. 300 00:20:04,520 --> 00:20:12,780 This says that sigma 1 is simply C1 epsilon 1 plus C1 2 301 00:20:12,780 --> 00:20:17,280 times epsilon 2 plus C1 3 times epsilon 3. 302 00:20:17,280 --> 00:20:22,310 And now the 1/2 gets absorbed, and it's simply C1 4 times 303 00:20:22,310 --> 00:20:25,900 epsilon 4, and so on. 304 00:20:25,900 --> 00:20:30,410 If we look at another line, one that involves some of the 305 00:20:30,410 --> 00:20:37,740 off-diagonal terms and strain, if we, for example, look at 306 00:20:37,740 --> 00:20:46,330 sigma 2 3 and we write this down as sigma 2 3 times 1 1 307 00:20:46,330 --> 00:20:58,180 times epsilon 1 1 plus C2 3 2 2 times epsilon 2 2 plus C2 3 308 00:20:58,180 --> 00:21:07,670 3 3 times epsilon 3 3 plus C-- 309 00:21:07,670 --> 00:21:11,040 and I'll just write down a few of these additional terms. 310 00:21:11,040 --> 00:21:22,330 This would be C2 3 3 2 times epsilon 3 2 plus C2 3 2 3, 311 00:21:22,330 --> 00:21:26,390 epsilon 2 3, and then other terms for additional terms, 312 00:21:26,390 --> 00:21:29,560 and epsilon ij with i not equal to j, which are going to 313 00:21:29,560 --> 00:21:31,490 behave the same way. 314 00:21:31,490 --> 00:21:35,870 So if we convert this, we go to sigma, and we would call 315 00:21:35,870 --> 00:21:37,120 this sigma 4. 316 00:21:41,110 --> 00:21:46,560 This would be C4 1 in matrix notation times epsilon 1 plus 317 00:21:46,560 --> 00:21:55,390 C4 2 time epsilon 2 2 plus C4 3 times epsilon 3. 318 00:21:55,390 --> 00:21:59,620 And now this matrix element would be C4 4. 319 00:22:04,890 --> 00:22:09,810 And in place of the tensor element of strain epsilon 3 2, 320 00:22:09,810 --> 00:22:13,500 we would write 1/2 of epsilon 4. 321 00:22:13,500 --> 00:22:18,780 And the next term is again a C4 4 times a 1/2 epsilon 4 and 322 00:22:18,780 --> 00:22:22,370 the terms in epsilon 5 and epsilon 6 would 323 00:22:22,370 --> 00:22:23,830 behave the same way. 324 00:22:23,830 --> 00:22:27,200 So you can see that the factor 2 is absorbed, and this 325 00:22:27,200 --> 00:22:32,030 becomes simply C4 4 times epsilon 4. 326 00:22:32,030 --> 00:22:35,460 So all the factors of 2 have disappeared. 327 00:22:35,460 --> 00:22:40,070 And we can say, provided we remember the way we have 328 00:22:40,070 --> 00:22:45,580 condensed the elements of tensor strain, we can say with 329 00:22:45,580 --> 00:22:50,750 complete confidence that in our matrix notation, sigma i 330 00:22:50,750 --> 00:23:01,530 is Cij times epsilon j where i goes 1, 2, and 3, and j goes 331 00:23:01,530 --> 00:23:03,965 1, 2, 3 all the way up to six. 332 00:23:07,176 --> 00:23:09,640 AUDIENCE: [INAUDIBLE] 333 00:23:09,640 --> 00:23:10,240 PROFESSOR: No. 334 00:23:10,240 --> 00:23:13,140 No, no, these are the-- 335 00:23:13,140 --> 00:23:13,740 yeah, I'm sorry. 336 00:23:13,740 --> 00:23:14,990 Yeah, inj. 337 00:23:20,750 --> 00:23:22,660 So it's a six by six matrix, right. 338 00:23:22,660 --> 00:23:24,380 Right you are. 339 00:23:24,380 --> 00:23:27,400 And this is a six by six. 340 00:23:27,400 --> 00:23:31,192 And in here are 36 businesses stiffnesses. 341 00:23:42,520 --> 00:23:48,040 OK, any comments other than, yuck? 342 00:23:48,040 --> 00:23:49,430 And again, if you stay in one coordinate 343 00:23:49,430 --> 00:23:50,530 system it's not so bad. 344 00:23:50,530 --> 00:23:54,950 In fact, instead of having nine by nine 81 terms you have 345 00:23:54,950 --> 00:23:58,000 36 stiffnesses. 346 00:23:58,000 --> 00:24:03,455 And that's a great convenience. 347 00:24:11,290 --> 00:24:18,150 Unfortunately, things are not quite so simple if we work 348 00:24:18,150 --> 00:24:22,720 with the compliances represented by the symbol S. 349 00:24:22,720 --> 00:24:30,490 And if we attempt to write strain in terms of stress, 350 00:24:30,490 --> 00:24:38,245 we'll have an S1 1 1 1 times sigma 1. 351 00:24:41,440 --> 00:24:52,550 We'll have an S1 1 2 2 times sigma 2 2 plus an S1 1 3 3 352 00:24:52,550 --> 00:25:02,444 times a sigma 3 3 plus an S1 1 2 3 times a sigma 2 3 plus an 353 00:25:02,444 --> 00:25:15,590 S1 1 3 2 times a sigma 3 2, and an S1 1 2 3 3 1 times 354 00:25:15,590 --> 00:25:23,520 sigma 3 1 plus an S1 1 3 times a sigma 1 3, and then other 355 00:25:23,520 --> 00:25:27,790 off-diagonal terms which I won't bother to mention. 356 00:25:27,790 --> 00:25:32,200 So if we convert this to matrix form, epsilon 1 1 would 357 00:25:32,200 --> 00:25:37,550 be replaced by the matrix term epsilon 1. 358 00:25:37,550 --> 00:25:44,980 S1 1 1 1 would become S1 1, and this would be sigma 1, and 359 00:25:44,980 --> 00:25:55,325 an S1 2 times a sigma 2, plus and S1 3 times sigma 3. 360 00:25:57,900 --> 00:26:03,370 And now we have a problem, Houston. 361 00:26:03,370 --> 00:26:12,270 Because we would have an S1 4 times a sigma 4 plus an S1 4 362 00:26:12,270 --> 00:26:15,380 times sigma 4. 363 00:26:15,380 --> 00:26:21,950 And now in the sigmas there is no factor 1/2 in the matrix 364 00:26:21,950 --> 00:26:26,520 representation of stress as there was with strain. 365 00:26:26,520 --> 00:26:30,770 Because we ate the factor of 2 in defining strain. 366 00:26:30,770 --> 00:26:33,580 But now there is no factor of 2 and stress. 367 00:26:33,580 --> 00:26:36,020 So we're going to have to do something with that. 368 00:26:36,020 --> 00:26:41,790 So this would give us some additional terms, S1 5 times 369 00:26:41,790 --> 00:26:47,210 sigma 5, and then an S1 5 times a sigma 5 again. 370 00:26:50,130 --> 00:26:51,990 So what are we going to do? 371 00:26:51,990 --> 00:26:52,860 Again, we're stuck. 372 00:26:52,860 --> 00:27:01,530 We can either say that epsilon i is equal to Sij times sigma 373 00:27:01,530 --> 00:27:11,350 j if j is not equal to 4, 5, or 6. 374 00:27:11,350 --> 00:27:15,170 Or we can absorb, again, a factor of 2 in the definition 375 00:27:15,170 --> 00:27:18,510 of the compliances. 376 00:27:18,510 --> 00:27:22,060 And that, again, since we will usually be working in one, and 377 00:27:22,060 --> 00:27:25,650 the same coordinate system, is the convenient thing to do. 378 00:27:25,650 --> 00:27:30,600 So what we will do is to define this as S1 379 00:27:30,600 --> 00:27:34,480 4 times sigma 4. 380 00:27:34,480 --> 00:27:42,450 And in so doing, we have to combine the factor of 2 into 381 00:27:42,450 --> 00:27:44,600 the definition of S1 4. 382 00:27:44,600 --> 00:27:53,250 So S1 4 would be equal to 1/2 of S1 1 2 3 plus S1 1 3 2. 383 00:27:57,000 --> 00:28:02,840 So it's going to be 1/2 of one of these equal terms that 384 00:28:02,840 --> 00:28:06,880 involve one on-diagonal subscript and one 385 00:28:06,880 --> 00:28:08,930 off-diagonal subscript. 386 00:28:08,930 --> 00:28:14,730 So this is the only way we're going to be able to write a 387 00:28:14,730 --> 00:28:17,185 nice, clean matrix representation. 388 00:28:20,820 --> 00:28:22,536 Things get worse. 389 00:28:22,536 --> 00:28:23,390 AUDIENCE: Sir? 390 00:28:23,390 --> 00:28:25,180 PROFESSOR: Yes? 391 00:28:25,180 --> 00:28:26,590 AUDIENCE: You're sure of your [INAUDIBLE]? 392 00:28:29,410 --> 00:28:34,050 PROFESSOR: Yeah, this would be equal to 2S1 4. 393 00:28:34,050 --> 00:28:35,160 I'm sure it's 1/2. 394 00:28:35,160 --> 00:28:37,240 I'm not sure of what term would go in front. 395 00:28:37,240 --> 00:28:40,370 So I would have 2S1 4 times sigma 4. 396 00:28:40,370 --> 00:28:43,710 And if I want to write this-- 397 00:28:43,710 --> 00:28:45,340 I'm sorry. 398 00:28:45,340 --> 00:28:50,480 Let me go back to the full matrix expression. 399 00:28:50,480 --> 00:29:00,580 I would have S1 1 2 3 times sigma 2 3 plus S1 1 3 2 400 00:29:00,580 --> 00:29:02,640 times sigma 3 2. 401 00:29:02,640 --> 00:29:07,920 I know that sigma 2 3 is equal to sigma 3 2, so I could write 402 00:29:07,920 --> 00:29:18,490 this as S1 1 2 3 plus S1 1 3 2 just simply times sigma 2 3, 403 00:29:18,490 --> 00:29:19,810 one of them. 404 00:29:19,810 --> 00:29:25,860 Now what I would really like to do is to write this as S1. 405 00:29:25,860 --> 00:29:29,250 Let's change this to sigma 4. 406 00:29:29,250 --> 00:29:31,590 I would like to write this as S1 4. 407 00:29:31,590 --> 00:29:34,800 So it follows then that S1 4 is equal to S1 2 408 00:29:34,800 --> 00:29:39,570 3 plus S1 1 3 2. 409 00:29:39,570 --> 00:29:45,060 And that is the other way around, isn't it? 410 00:29:45,060 --> 00:29:45,930 Thank you. 411 00:29:45,930 --> 00:29:48,593 I knew there was a factor of 1/2, but it goes in here. 412 00:29:48,593 --> 00:29:52,204 AUDIENCE: And then there's no 2 at all. 413 00:29:52,204 --> 00:29:56,578 If you wanted to say that S1 1 2 3 is equal to S, 1 1 3 2, 414 00:29:56,578 --> 00:30:00,480 then you would have S1 4 equal to 2S1 1 3 2. 415 00:30:00,480 --> 00:30:03,140 PROFESSOR: OK, you're right, you're right. 416 00:30:03,140 --> 00:30:04,090 So it's this. 417 00:30:04,090 --> 00:30:05,090 Let's leave it at that. 418 00:30:05,090 --> 00:30:09,740 But if these are equal, we could say-- and we did show 419 00:30:09,740 --> 00:30:12,950 that the compliance tensor is symmetric, so we could say 420 00:30:12,950 --> 00:30:25,405 that this is equal to 2S1 1 2 3 or 2S1 1 3 421 00:30:25,405 --> 00:30:27,560 2 as they're equal. 422 00:30:36,020 --> 00:30:38,370 I told you it was going to get bad, but I didn't think I 423 00:30:38,370 --> 00:30:42,380 would be contributing to it the extent that I am. 424 00:30:42,380 --> 00:30:46,450 I don't know if you really want to see this, but this 425 00:30:46,450 --> 00:30:50,708 gets even worse when we deal with something like epsilon 2 426 00:30:50,708 --> 00:30:57,450 3, where this is a term that would be 427 00:30:57,450 --> 00:30:59,015 replaced by epsilon 4. 428 00:31:01,580 --> 00:31:05,260 And then all of our absorbed factors come back to haunt us. 429 00:31:05,260 --> 00:31:06,540 Let me go through this quickly. 430 00:31:06,540 --> 00:31:19,640 This would be S2 3 1 1 times sigma 1 1 plus S2 3 2 2 times 431 00:31:19,640 --> 00:31:38,445 sigma 2 2 plus S2 3 3 3 times sigma 3 3. 432 00:31:41,430 --> 00:31:47,590 And then we'll have these terms, off-diagonal terms S2 3 433 00:31:47,590 --> 00:31:59,720 3 1, sigma 3 1 plus S2 3 1 3 times sigma 1 3, and so on, 434 00:31:59,720 --> 00:32:00,970 other terms. 435 00:32:04,970 --> 00:32:11,900 OK, and going from epsilon 2 3 to epsilon 4, we have to put 436 00:32:11,900 --> 00:32:13,220 on a factor 1/2. 437 00:32:16,180 --> 00:32:26,520 And that says that that 1/2 is going to create problems in 438 00:32:26,520 --> 00:32:28,090 the first term in this expansion. 439 00:32:28,090 --> 00:32:30,050 We'd call this sigma 1 1. 440 00:32:30,050 --> 00:32:33,870 We'd like to call this S4 1. 441 00:32:33,870 --> 00:32:39,290 But what is S4 1 from what we have defined here? 442 00:32:39,290 --> 00:32:43,460 It's the sum of two tensor elements. 443 00:32:43,460 --> 00:32:53,110 So we will have to define, now, again, S4 1 equals the 444 00:32:53,110 --> 00:32:55,220 same as we did before. 445 00:32:55,220 --> 00:33:04,180 It's going to be equal to the term S2 3 1 1 plus S3 2 1 1 1. 446 00:33:06,800 --> 00:33:13,060 And therefore if I put a 1/2 in this definition, then that 447 00:33:13,060 --> 00:33:17,000 will cancel, so far. 448 00:33:17,000 --> 00:33:20,040 But then in the terms that come down in the lower 449 00:33:20,040 --> 00:33:21,370 quadrant of the matrix-- 450 00:33:21,370 --> 00:33:23,320 and I will just right in two terms. 451 00:33:23,320 --> 00:33:29,852 We've got an S2 3 31 that we would write as S4 452 00:33:29,852 --> 00:33:39,920 4 times sigma 4. 453 00:33:39,920 --> 00:33:47,370 And then I would have another term, S4 4 times sigma 4. 454 00:33:47,370 --> 00:33:57,090 But the S4 4 really is a sum of two tensor elements. 455 00:33:57,090 --> 00:34:05,490 So my definition of an S4 4 is that it should be-- 456 00:34:09,395 --> 00:34:12,639 AUDIENCE: Isn't that sigma 5 [INAUDIBLE]? 457 00:34:12,639 --> 00:34:14,879 PROFESSOR: You're right, you're right. 458 00:34:14,879 --> 00:34:17,420 Yeah, this is 5. 459 00:34:17,420 --> 00:34:19,900 So this was right, 3 1, and this is sigma 5. 460 00:34:19,900 --> 00:34:22,130 You're right. 461 00:34:22,130 --> 00:34:27,755 And I go to S4 5, yeah. 462 00:34:33,090 --> 00:34:34,920 No, sigma-- 463 00:34:34,920 --> 00:34:37,668 OK, yeah. 464 00:34:37,668 --> 00:34:40,760 OK, let me cut to the chase. 465 00:34:40,760 --> 00:34:44,960 What we're going to have to do is to put in not the factor 466 00:34:44,960 --> 00:34:47,940 1/2 that we had here, but there's going to 467 00:34:47,940 --> 00:34:50,350 be a factor 4 that-- 468 00:34:50,350 --> 00:34:51,600 something like S-- 469 00:34:55,192 --> 00:34:56,620 which one are we dealing with? 470 00:35:05,420 --> 00:35:17,950 Things like S2 3 3 1, something like S4-- 471 00:35:17,950 --> 00:35:19,440 and we're dealing with 5. 472 00:35:19,440 --> 00:35:34,840 S4 5 is going to be defined as S1, S2 3 1 3 plus S2 3 3 1 473 00:35:34,840 --> 00:35:41,040 plus S3 2 1 3 plus S3 2 3 1. 474 00:35:46,420 --> 00:35:52,090 So we're going to have to put in a factor of 1/4 in front of 475 00:35:52,090 --> 00:35:57,930 the S4 5 in order to accommodate for those terms. 476 00:35:57,930 --> 00:36:05,090 So our definition, then, in going from tensor compliances 477 00:36:05,090 --> 00:36:10,060 to matrix compliances is much more complicated. 478 00:36:10,060 --> 00:36:12,820 And the rule-- and just to summarize this, I gave you a 479 00:36:12,820 --> 00:36:15,510 handout last time. 480 00:36:15,510 --> 00:36:18,810 The necessary conventions to absorb these factors of 2 and 481 00:36:18,810 --> 00:36:20,450 4 is that-- 482 00:36:30,630 --> 00:36:32,105 and I didn't bring it with me. 483 00:36:45,020 --> 00:36:46,480 OK, I just simply did not bring it with me. 484 00:36:46,480 --> 00:36:48,110 So I wouldn't attempt to summarize it. 485 00:36:53,560 --> 00:36:53,980 All right. 486 00:36:53,980 --> 00:36:57,370 So hopefully I've convinced you of nothing other than that 487 00:36:57,370 --> 00:36:59,320 these conversions are messy. 488 00:36:59,320 --> 00:37:15,200 But the summary is that for the compliances, the Sijkl, 489 00:37:15,200 --> 00:37:26,700 you take this equal to Slm if l and m are 490 00:37:26,700 --> 00:37:32,590 equal to 1, 2, or 3. 491 00:37:32,590 --> 00:37:54,550 Then you have to replace Sijkl by 1/2 of Slm if i, j, or kl 492 00:37:54,550 --> 00:37:58,710 is 4, 5, or 6. 493 00:37:58,710 --> 00:38:07,620 And then you have to write Sijkl as 1/4 of Slm if i, j, 494 00:38:07,620 --> 00:38:15,870 and kl are 4, 5, or 6. 495 00:38:15,870 --> 00:38:19,450 But once you're into the coordinate system, which will 496 00:38:19,450 --> 00:38:26,630 be fixed in matrix notation, things are simple, since you 497 00:38:26,630 --> 00:38:29,030 don't have to worry about these factors of 2 or 3. 498 00:38:33,140 --> 00:38:35,650 The reason I did this is I would like to now talk a 499 00:38:35,650 --> 00:38:42,070 little bit about how one can define important scalar moduli 500 00:38:42,070 --> 00:38:44,430 that describe a particular phenomenon. 501 00:38:44,430 --> 00:38:47,840 If you ask the question, how do mechanical properties vary 502 00:38:47,840 --> 00:38:54,500 with direction, we've got six unique elements of stress. 503 00:38:54,500 --> 00:38:58,080 And we've got six independent elements of 504 00:38:58,080 --> 00:38:59,980 strain that can work. 505 00:38:59,980 --> 00:39:03,340 And if you wanted to show how each one of those elements 506 00:39:03,340 --> 00:39:06,160 varied with the other one, you'd need 36-- 507 00:39:06,160 --> 00:39:07,640 6 times 6, 36-- 508 00:39:07,640 --> 00:39:09,530 representation surfaces and this is 509 00:39:09,530 --> 00:39:11,590 not going to be fruitful. 510 00:39:11,590 --> 00:39:15,010 So just as we did for piezoelectricity where we 511 00:39:15,010 --> 00:39:20,230 could define a scalar modulus which represented the 512 00:39:20,230 --> 00:39:24,190 component of polarization normal to a very thin plate-- 513 00:39:24,190 --> 00:39:27,540 making it a thin plate means you're going to measure 514 00:39:27,540 --> 00:39:30,610 primarily the charge on the large surface, because 515 00:39:30,610 --> 00:39:33,200 polarization is charge per unit area. 516 00:39:33,200 --> 00:39:37,740 So if you looked at the surface of a plate with x1 517 00:39:37,740 --> 00:39:41,960 normal to the plate, you are going to have a specimen that 518 00:39:41,960 --> 00:39:45,840 primarily gives you a measure of P1, or the charge on a 519 00:39:45,840 --> 00:39:48,050 surface normal to x1. 520 00:39:48,050 --> 00:39:53,060 And then you could apply any of six different states of 521 00:39:53,060 --> 00:39:54,390 simple stress. 522 00:39:54,390 --> 00:39:58,250 And one thing that you could do is squeeze it with the 523 00:39:58,250 --> 00:40:00,350 tensile stress in the same direction as the 524 00:40:00,350 --> 00:40:01,390 normal to the surface. 525 00:40:01,390 --> 00:40:04,775 And that was the longitudinal piezoelectric modulus. 526 00:40:04,775 --> 00:40:09,530 Then on the quiz you looked at another modulus that related a 527 00:40:09,530 --> 00:40:16,580 component of polarization on another surface in response to 528 00:40:16,580 --> 00:40:22,290 a tensile stress that was not parallel to it. 529 00:40:22,290 --> 00:40:25,630 There are a number of such moduli and mechanical 530 00:40:25,630 --> 00:40:26,760 properties. 531 00:40:26,760 --> 00:40:32,210 Probably the most important one is something called 532 00:40:32,210 --> 00:40:35,100 Young's modulus, which I'm sure you've all heard of. 533 00:40:41,890 --> 00:40:48,490 And Young's modulus involves taking a very long rod of the 534 00:40:48,490 --> 00:40:52,560 material and hanging a weight on it. 535 00:40:52,560 --> 00:40:56,390 And that weight will induce a strain. 536 00:40:56,390 --> 00:41:03,940 And if we take this as the direction of x1, we will, with 537 00:41:03,940 --> 00:41:09,350 Young's modulus, relate the change of length to the 538 00:41:09,350 --> 00:41:11,220 initial length of the rod. 539 00:41:18,300 --> 00:41:22,130 And my question now is, rhetorically, do we want to do 540 00:41:22,130 --> 00:41:24,895 this in terms of stiffnesses or compliances? 541 00:41:28,470 --> 00:41:30,780 Does it make any difference? 542 00:41:30,780 --> 00:41:32,650 Yeah, makes a lot of difference in terms of the 543 00:41:32,650 --> 00:41:34,670 simplicity of the result that you get. 544 00:41:34,670 --> 00:41:41,460 Suppose we wanted to do this in terms of the compliances. 545 00:41:41,460 --> 00:41:45,775 What we'd be then applying would be a sigma 1. 546 00:41:50,360 --> 00:41:54,990 And this would be given by the compliance C1 1 547 00:41:54,990 --> 00:41:57,990 times the strain E1. 548 00:41:57,990 --> 00:41:59,750 That looks like exactly what we want. 549 00:41:59,750 --> 00:42:04,590 This delta l over l for this one-dimensional specimen with 550 00:42:04,590 --> 00:42:07,990 a one-dimensional directional applied stress, this would be 551 00:42:07,990 --> 00:42:10,710 simply epsilon 1. 552 00:42:10,710 --> 00:42:12,700 And that's force per unit area here. 553 00:42:12,700 --> 00:42:14,680 This would be sigma 1 1. 554 00:42:14,680 --> 00:42:20,070 And it looks as though what we want is a C1 1 1 that relates 555 00:42:20,070 --> 00:42:21,510 a sigma 1 1 to an epsilon 1 1. 556 00:42:26,890 --> 00:42:29,070 Is this going to be a definition that I would want 557 00:42:29,070 --> 00:42:30,650 to make for describing this? 558 00:42:33,220 --> 00:42:35,600 My colleague here shakes his head very seriously. 559 00:42:35,600 --> 00:42:36,920 No. 560 00:42:36,920 --> 00:42:40,069 Do you want to share your reservation? 561 00:42:40,069 --> 00:42:44,999 AUDIENCE: Because as I recall, Young's modulus, once strain 562 00:42:44,999 --> 00:42:48,203 is weaker output [INAUDIBLE] in terms of stress you're 563 00:42:48,203 --> 00:42:52,080 going to want a compliance. 564 00:42:52,080 --> 00:42:53,940 PROFESSOR: That's one answer. 565 00:42:53,940 --> 00:42:55,826 AUDIENCE: You could theoretically get either one 566 00:42:55,826 --> 00:42:57,076 [INAUDIBLE]. 567 00:42:59,140 --> 00:43:02,420 PROFESSOR: If I could restate your objection, we can't 568 00:43:02,420 --> 00:43:05,650 impose a strain to get a stress. 569 00:43:05,650 --> 00:43:09,630 We really stresses the independent variable, 570 00:43:09,630 --> 00:43:10,880 practically speaking. 571 00:43:10,880 --> 00:43:13,640 We can stress it, but we can't instantly say, [INAUDIBLE] 572 00:43:13,640 --> 00:43:15,400 develop a epsilon 1. 573 00:43:15,400 --> 00:43:16,292 Yeah? 574 00:43:16,292 --> 00:43:17,630 AUDIENCE: [INAUDIBLE] 575 00:43:17,630 --> 00:43:18,970 PROFESSOR: You betcha. 576 00:43:18,970 --> 00:43:20,660 That's the reason. 577 00:43:20,660 --> 00:43:24,490 Added onto this is not only this term, but there will be a 578 00:43:24,490 --> 00:43:27,930 C1 2 times epsilon 2 plus a C1 3 times an 579 00:43:27,930 --> 00:43:29,780 epsilon 3, and so on. 580 00:43:29,780 --> 00:43:33,860 So we're not going to be able to get a nice, tidy relation 581 00:43:33,860 --> 00:43:37,120 between the tensile strain that we're measuring and the 582 00:43:37,120 --> 00:43:41,580 tensile uniaxial stress that is produced there. 583 00:43:41,580 --> 00:43:47,270 So the relation of stress in terms of strain that involves 584 00:43:47,270 --> 00:43:51,020 the stiffnesses is just not going to work. 585 00:43:51,020 --> 00:43:59,550 But if we would instead express epsilon1 in terms of a 586 00:43:59,550 --> 00:44:06,844 compliance, a matrix compliance S1 1 times sigma 1, 587 00:44:06,844 --> 00:44:09,100 that's all that she wrote. 588 00:44:09,100 --> 00:44:13,010 We're measuring this by design by selecting an elongated 589 00:44:13,010 --> 00:44:18,220 specimen for which we'll primarily be seeing epsilon 1. 590 00:44:18,220 --> 00:44:21,230 This is not to say these other strains exist. 591 00:44:21,230 --> 00:44:28,460 There will be lateral strains here that will be epsilon 2 592 00:44:28,460 --> 00:44:30,500 and epsilon 3. 593 00:44:30,500 --> 00:44:33,510 There will be shear strains. 594 00:44:33,510 --> 00:44:34,080 Sounds mind-boggling. 595 00:44:34,080 --> 00:44:36,803 You pull the sample in this direction, and 596 00:44:36,803 --> 00:44:38,200 what it does is shear. 597 00:44:38,200 --> 00:44:40,430 Well, if this were a single crystal, that would happen. 598 00:44:40,430 --> 00:44:43,000 And these would be other components of deformation. 599 00:44:43,000 --> 00:44:48,740 But if we measure a pi sigma 1 and measure epsilon 1, then 600 00:44:48,740 --> 00:44:50,990 this is the way we have to define it. 601 00:44:50,990 --> 00:44:58,470 And the definition of Young's modulus is that 1 over S1 1 is 602 00:44:58,470 --> 00:45:01,620 sigma 1 over epsilon 1. 603 00:45:01,620 --> 00:45:04,265 And this is Young's modulus. 604 00:45:12,550 --> 00:45:17,660 I would hasten to observe that it is unfortunate that this 605 00:45:17,660 --> 00:45:22,900 very practical modulus involves the S's, which have 606 00:45:22,900 --> 00:45:27,230 all these complicated factors of 2 and 4. 607 00:45:27,230 --> 00:45:33,760 And therefore, if we try to go to a particular symmetry and 608 00:45:33,760 --> 00:45:38,360 ask the question, how does Young's modulus change as we 609 00:45:38,360 --> 00:45:41,700 cut this rod in different orientations 610 00:45:41,700 --> 00:45:42,950 from the single crystal. 611 00:45:47,290 --> 00:45:52,690 So I would like to illustrate for you how we would do this. 612 00:45:52,690 --> 00:45:55,030 I'm going to take a very, very simple example. 613 00:45:59,610 --> 00:46:10,370 If you look in the table of symmetry restrictions that I 614 00:46:10,370 --> 00:46:17,570 passed out a couple of times ago, prior to the quiz, for an 615 00:46:17,570 --> 00:46:26,740 isotropic the tensor has a different form than it does 616 00:46:26,740 --> 00:46:27,960 for a cubic crystal. 617 00:46:27,960 --> 00:46:30,940 Cubic crystals are elastically anisotropic. 618 00:46:30,940 --> 00:46:35,520 But the form of the stiffness tensor for an isotropic 619 00:46:35,520 --> 00:46:44,255 material was S1 1, S1 2, S1 2, 0, 0, 0, S1 2, S1 620 00:46:44,255 --> 00:46:48,730 1, S1 2, 0, 0, 0. 621 00:46:48,730 --> 00:46:52,980 Then the diagonal terms, if the material was isotropic, 622 00:46:52,980 --> 00:47:00,460 was 2s1 1 minus S1 2 for this term, 0, 0, 0, 0, 0, 0, and 623 00:47:00,460 --> 00:47:09,035 again, a 2S1 1 minus S1 2 plus a 0, and then 0, 0, 0, 0, 2S1 624 00:47:09,035 --> 00:47:11,060 1 minus S1 2. 625 00:47:15,190 --> 00:47:15,870 OK. 626 00:47:15,870 --> 00:47:19,840 Let me now derive Young's modulus as a function of 627 00:47:19,840 --> 00:47:25,330 direction and show that, in fact, this says that 628 00:47:25,330 --> 00:47:29,730 regardless how you orient the rod, the value of Young's 629 00:47:29,730 --> 00:47:31,110 modulus will stay the same. 630 00:47:34,660 --> 00:47:38,230 Let me also do something else first, though, which is 631 00:47:38,230 --> 00:47:43,940 something we should have scratched our head over when 632 00:47:43,940 --> 00:47:45,200 we first encountered it. 633 00:47:53,410 --> 00:47:55,420 Here's something curious. 634 00:47:55,420 --> 00:47:58,480 When we looked at symmetry constraints on single 635 00:47:58,480 --> 00:48:08,720 crystals, we found that for a cubic crystal, C6 6 had to be 636 00:48:08,720 --> 00:48:17,420 equal to 1/2 of C1 1 minus C1 2 if the stiffnesses were to 637 00:48:17,420 --> 00:48:24,830 be independent for all symmetry transformations. 638 00:48:24,830 --> 00:48:33,580 For the compliances, however, we found that S6 6 had to be 639 00:48:33,580 --> 00:48:40,170 equal to 2 times S1 1 minus S1 2. 640 00:48:40,170 --> 00:48:44,010 Two very different equalities between tensor elements to 641 00:48:44,010 --> 00:48:49,800 make the elastic behavior be invariant to the symmetry 642 00:48:49,800 --> 00:48:52,520 transformations of a cubic crystal. 643 00:48:52,520 --> 00:48:53,160 How come? 644 00:48:53,160 --> 00:48:55,680 How come these are so different? 645 00:48:55,680 --> 00:48:58,920 Well, they have to be, in terms of the tensor, the same 646 00:48:58,920 --> 00:49:00,640 sort of a quality. 647 00:49:00,640 --> 00:49:04,850 And the reason they don't look the same is because of our 648 00:49:04,850 --> 00:49:09,350 different absorption of the factors of 2 and 4 in defining 649 00:49:09,350 --> 00:49:15,810 the stiffnesses an in defining the compliances. 650 00:49:15,810 --> 00:49:21,650 Remember that for the Cijkl's there was no factor of 2 or 4 651 00:49:21,650 --> 00:49:24,830 introduced in defining the matrix elements. 652 00:49:24,830 --> 00:49:26,820 So this one is OK. 653 00:49:26,820 --> 00:49:30,790 This is a true equality between tensor elements, for 654 00:49:30,790 --> 00:49:33,045 any tensor whatsoever of fourth 655 00:49:33,045 --> 00:49:35,770 rank for a cubic crystal. 656 00:49:35,770 --> 00:49:39,380 For these terms, these are off-diagonal. 657 00:49:39,380 --> 00:49:41,730 This is an off-diagonal compliance. 658 00:49:41,730 --> 00:49:52,550 And our definition is that Sijkl is equal to Smn, for m 659 00:49:52,550 --> 00:49:56,070 and n not equal to 4, 5, or 6. 660 00:49:56,070 --> 00:49:57,920 Here's this crazy thing again. 661 00:49:57,920 --> 00:50:08,560 It's equal to 1/2 of Smn for m or n equal to 4, 5, or 6. 662 00:50:08,560 --> 00:50:16,470 And it's equal to 1/4 of Smn for m and n 663 00:50:16,470 --> 00:50:20,330 equal to 4, 5, or 6. 664 00:50:20,330 --> 00:50:29,075 So S6 6 here is actually 4S1 2 1 2. 665 00:50:34,640 --> 00:50:41,080 And that's supposedly equal to 2S1 1, which is really S1 1 1, 666 00:50:41,080 --> 00:50:42,330 minus S1 1 2 2. 667 00:50:45,430 --> 00:50:53,550 And this says that 1/2 of S1 2 1 2 is equal to S1 1 1 1 668 00:50:53,550 --> 00:50:56,170 minus S1 1 2 2. 669 00:50:56,170 --> 00:51:02,850 So this is exactly the same constraint on tensor elements, 670 00:51:02,850 --> 00:51:05,900 when we take out these factors of 2 and 4 671 00:51:05,900 --> 00:51:07,130 that have been absorbed. 672 00:51:07,130 --> 00:51:09,760 So this, in fact, although it looks very different with 673 00:51:09,760 --> 00:51:13,230 those factors absorbed, is exactly this same relation. 674 00:51:13,230 --> 00:51:17,360 So the symmetry constraint is the same. 675 00:51:17,360 --> 00:51:19,208 AUDIENCE: How'd you get 1/2 there? 676 00:51:19,208 --> 00:51:20,140 PROFESSOR: Hmm? 677 00:51:20,140 --> 00:51:21,860 How did I get 1/2 here? 678 00:51:21,860 --> 00:51:29,770 It's 4S1 2 2 had to be equal to 2S1 1 minus S1 2. 679 00:51:29,770 --> 00:51:31,220 And this was the equality. 680 00:51:31,220 --> 00:51:32,190 I'm working up this way. 681 00:51:32,190 --> 00:51:36,780 So this was the statement in the matrix forms of the 682 00:51:36,780 --> 00:51:39,010 compliances that I handed out. 683 00:51:39,010 --> 00:51:40,770 Looks different from the term on the left. 684 00:51:40,770 --> 00:51:45,400 So I wrote this now in terms of the matrix elements. 685 00:51:45,400 --> 00:51:47,270 And the factor of 4 comes in here. 686 00:51:47,270 --> 00:51:53,300 The 2 was here in the equality that had to be held that the 687 00:51:53,300 --> 00:51:54,640 tensors stay invariant. 688 00:51:54,640 --> 00:51:57,340 And if I bring this 2 over on the left-hand side, it's 689 00:51:57,340 --> 00:51:58,896 exactly the same as when-- 690 00:51:58,896 --> 00:51:59,888 AUDIENCE: Shouldn't it be 2 now? 691 00:51:59,888 --> 00:52:01,138 [INAUDIBLE]. 692 00:52:03,856 --> 00:52:05,344 [INAUDIBLE] the right side. 693 00:52:09,312 --> 00:52:12,220 PROFESSOR: No, this is just replacement of the matrix 694 00:52:12,220 --> 00:52:15,352 terms with the definition of the tensor. 695 00:52:15,352 --> 00:52:16,828 AUDIENCE: Right, but [INAUDIBLE]. 696 00:52:16,828 --> 00:52:18,796 AUDIENCE: You're just dividing 4 by 2, right? 697 00:52:18,796 --> 00:52:20,272 PROFESSOR: Yeah, exactly. 698 00:52:20,272 --> 00:52:22,732 AUDIENCE: So wouldn't that make [INAUDIBLE]? 699 00:52:22,732 --> 00:52:24,208 PROFESSOR: Oh, ha, OK. 700 00:52:24,208 --> 00:52:26,504 AUDIENCE: You want to be able to [INAUDIBLE] by 4, so it'd 701 00:52:26,504 --> 00:52:30,112 be S12 1 2 equals 1/2 the difference 702 00:52:30,112 --> 00:52:31,600 to be equal to that. 703 00:52:31,600 --> 00:52:32,865 PROFESSOR: OK, OK. 704 00:52:38,030 --> 00:52:39,280 Thank you. 705 00:52:44,280 --> 00:52:46,860 OK, let me then go back to what I started out to do. 706 00:52:46,860 --> 00:52:53,630 And I will write just one line of how we would go about 707 00:52:53,630 --> 00:53:00,630 expanding this just to indicate how intricate it is, 708 00:53:00,630 --> 00:53:03,690 and then I'll give a few examples for two point groups 709 00:53:03,690 --> 00:53:05,575 of how Young's modulus varies with direction. 710 00:53:08,650 --> 00:53:09,130 OK. 711 00:53:09,130 --> 00:53:17,060 What we're going to say, then, is that one of these 712 00:53:17,060 --> 00:53:27,990 compliances, S1 1 1 1 prime is going to be equal to l 1/4 713 00:53:27,990 --> 00:53:40,980 times S1 1 1 1 plus l 2/4 times S2 2 2 2 plus l 3/4 714 00:53:40,980 --> 00:53:42,230 times S3 3 3 3 . 715 00:53:45,100 --> 00:53:54,340 And just now doing what we know, that any Sijkl prime is 716 00:53:54,340 --> 00:54:07,740 going to be Cii, Cj capital J, Ck capital K, Cl capital L-- 717 00:54:07,740 --> 00:54:11,110 these are direction cosines now, not stiffnesses-- 718 00:54:11,110 --> 00:54:12,360 times Sijkl. 719 00:54:15,620 --> 00:54:21,290 So it's a quadrupole summation over all of the non-zero 720 00:54:21,290 --> 00:54:22,230 tensor elements. 721 00:54:22,230 --> 00:54:24,640 So what I'm doing now is taking these terms and 722 00:54:24,640 --> 00:54:27,480 expanding them to the full tensor form. 723 00:54:27,480 --> 00:54:33,670 So S1 1 1 1 prime, that's the first term in 724 00:54:33,670 --> 00:54:35,950 the transformed tensor. 725 00:54:35,950 --> 00:54:46,860 That's going to be these terms plus l1l2 squared times S1 1 2 726 00:54:46,860 --> 00:54:54,558 2 plus S2 2 1 1 plus l1 squared l3 squared times S1 1 727 00:54:54,558 --> 00:55:06,920 3 3 plus S3 3 1 1, and then still another term, l2 squared 728 00:55:06,920 --> 00:55:12,350 l3 squared times S2 2 3 3 plus S3 3 2 2. 729 00:55:16,250 --> 00:55:25,330 And then there will be terms of the form l2 squared l3 730 00:55:25,330 --> 00:55:38,100 squared times, again, four terms, S3 2 3 2 plus S1 2 3 3 731 00:55:38,100 --> 00:55:48,280 plus S2 3 3 2 plus S2 3 2 3, and then similarly, terms in 732 00:55:48,280 --> 00:55:54,260 the squares of l1 and l2, and these would involve four terms 733 00:55:54,260 --> 00:56:00,340 of the form S1 2 1 2 plus permutation 0, and then l1 734 00:56:00,340 --> 00:56:09,080 squared l3 squared times, again, four terms, S1 3 1 3 735 00:56:09,080 --> 00:56:11,520 and permutation 0 for four terms. 736 00:56:15,050 --> 00:56:15,420 OK. 737 00:56:15,420 --> 00:56:16,510 So what do we have now? 738 00:56:16,510 --> 00:56:23,580 We have an expression for S1 1 1 1 prime. 739 00:56:23,580 --> 00:56:33,210 And this is, in fact, 1 over Young's modulus E. And we've 740 00:56:33,210 --> 00:56:38,560 done this summation over the supposed non-zero tensor 741 00:56:38,560 --> 00:56:43,565 elements in the matrix for an isotropic material. 742 00:56:46,340 --> 00:56:50,390 And if I simplify this, and this will be just one more 743 00:56:50,390 --> 00:56:58,600 tedious and then we can see, that is for these equalities, 744 00:56:58,600 --> 00:57:04,110 we ought to, for l1, l2, l3, get the same value, S1 1. 745 00:57:04,110 --> 00:57:06,020 And that is indeed what happens. 746 00:57:06,020 --> 00:57:11,070 So the summation, if I simplify, is l 1/4 times S1 1 747 00:57:11,070 --> 00:57:19,850 plus l 2.4 times S2 2 plus l 3/4 times S3 3. 748 00:57:19,850 --> 00:57:26,240 And then a collection of terms plus l1 squared l2 squared, S1 749 00:57:26,240 --> 00:57:27,860 2 plus S2 1. 750 00:57:31,960 --> 00:57:38,570 And then similar terms in l1 and l3, l1 squared l3 squared, 751 00:57:38,570 --> 00:57:47,160 S1 3 plus S 3 1 plus l2 squared l3 squared times 752 00:57:47,160 --> 00:57:50,050 S2 3 plus S3 2. 753 00:57:54,400 --> 00:58:00,231 And then some terms that stand by themselves, l2 squared l3 754 00:58:00,231 --> 00:58:11,410 squared times S4 4 plus l1 squared l2 squared S6 6 plus 755 00:58:11,410 --> 00:58:14,620 l1 squared plus l3 squared times S5 5. 756 00:58:18,580 --> 00:58:23,620 And consolidating this, this is going to be equal to l 1/4 757 00:58:23,620 --> 00:58:32,700 plus l 2/4 plus l 3/4 times S1 1. 758 00:58:32,700 --> 00:58:37,760 And then combining terms in the second power of direction 759 00:58:37,760 --> 00:58:43,110 cosines, l1 squared l2 squared plus l1 squared l3 squared 760 00:58:43,110 --> 00:58:44,960 plus l2 squared l3 squared. 761 00:58:47,900 --> 00:58:57,190 And this is all times 2S1 2 plus 2S1 1 minus 2S1 2. 762 00:58:57,190 --> 00:59:00,260 So S1 2 drops out. 763 00:59:00,260 --> 00:59:10,170 And all this will be simply l 1/4 plus l 2/4 plus l 3/4. 764 00:59:10,170 --> 00:59:13,570 And then I'll add in the other terms here that involve 765 00:59:13,570 --> 00:59:16,085 products of squares of direction cosines. 766 00:59:25,280 --> 00:59:28,400 And all this is times S1 1. 767 00:59:28,400 --> 00:59:32,660 But this turns out to be equal to simply l1 squared plus l2 768 00:59:32,660 --> 00:59:36,480 squared plus l3 squared, the sum of the squares of the 769 00:59:36,480 --> 00:59:42,900 direction cosines of our rod, quantity squared times S1 1. 770 00:59:42,900 --> 00:59:46,060 And this term inside the parentheses is 1. 771 00:59:46,060 --> 00:59:54,830 So indeed, S1 1 prime is equal to S1 1. 772 00:59:54,830 --> 00:59:59,600 So the value of Young's modulus has not changed with 773 00:59:59,600 --> 01:00:05,260 direction if the form of the compliance tensor is like so. 774 01:00:08,930 --> 01:00:13,600 And I think you would probably have taken my word for that, 775 01:00:13,600 --> 01:00:19,370 but it was intended primarily as an example of how, when S1 776 01:00:19,370 --> 01:00:22,630 1 prime is the reciprocal of Young's modulus, how you would 777 01:00:22,630 --> 01:00:26,260 set up a transformation for S1 1 prime. 778 01:00:26,260 --> 01:00:30,660 And the form of this polynomial would be exactly 779 01:00:30,660 --> 01:00:34,090 the same, even for a triclinic crystal, if you included all 780 01:00:34,090 --> 01:00:38,220 the non-zero terms in this fashion. 781 01:00:38,220 --> 01:00:41,880 All right, so let me wrap things up in another couple 782 01:00:41,880 --> 01:00:49,530 minutes for some cases that are real symmetries, where 783 01:00:49,530 --> 01:00:51,060 Young's modulus is anisotropic. 784 01:00:53,560 --> 01:01:01,360 And working in exactly the same way for the tensor that 785 01:01:01,360 --> 01:01:08,390 is the appropriate one for a cubic crystal, we would lift 786 01:01:08,390 --> 01:01:13,700 out the form of the stiffness matrix. 787 01:01:13,700 --> 01:01:17,540 The reciprocal of Young's modulus is S1 1 prime, so what 788 01:01:17,540 --> 01:01:20,310 we're saying is we have a long, skinny rod. 789 01:01:20,310 --> 01:01:22,200 This is x1. 790 01:01:22,200 --> 01:01:25,740 And what we're doing, if this is a single crystal, is 791 01:01:25,740 --> 01:01:30,730 examining how Young's modulus changes as we change the 792 01:01:30,730 --> 01:01:35,030 direction of the rod to a new orientation, x1 prime, that's 793 01:01:35,030 --> 01:01:41,220 described by direction cosines l1, l2, L3, relative to x1. 794 01:01:43,780 --> 01:01:47,670 The expression that results by exactly the process that we 795 01:01:47,670 --> 01:01:53,400 muddled through a moment ago is that for a cubic crystal, 796 01:01:53,400 --> 01:02:05,470 the form of S1 1 prime as a function of direction does 797 01:02:05,470 --> 01:02:07,160 indeed give anisotropy. 798 01:02:07,160 --> 01:02:16,420 Isaiah It's S1 1 minus 2 times S1 1 minus S1 2 799 01:02:16,420 --> 01:02:20,450 minus 1/2 of S4 4. 800 01:02:20,450 --> 01:02:24,090 Remember, for a cubic crystal, 1 1, 2 2, and 1 4 are the only 801 01:02:24,090 --> 01:02:25,700 non-zero terms. 802 01:02:25,700 --> 01:02:29,285 And then this is times a polynomial l1 squared l2 803 01:02:29,285 --> 01:02:36,520 squared plus l2 squared l3 squared plus 804 01:02:36,520 --> 01:02:38,150 l3 squared l1 squared. 805 01:02:41,420 --> 01:02:44,120 So this is, again the reciprocal of Young's modulus. 806 01:02:44,120 --> 01:02:50,460 And it turns out to have a constant term, S1 1 , which is 807 01:02:50,460 --> 01:02:53,620 what we found for the isotropic material. 808 01:02:53,620 --> 01:02:59,070 But from that, as the direction cosines change, we 809 01:02:59,070 --> 01:03:04,475 subtract off a term that is a linear combination of 1 1, 1 810 01:03:04,475 --> 01:03:07,930 2, and 4 4. 811 01:03:07,930 --> 01:03:15,100 So the question is, is this positive or negative? 812 01:03:15,100 --> 01:03:18,360 The direction cosines are all squared, so this term here is 813 01:03:18,360 --> 01:03:19,960 always going to be positive. 814 01:03:19,960 --> 01:03:23,860 So are we going to take the thing that we found for an 815 01:03:23,860 --> 01:03:29,150 isotropic material, which was a constant, Young's modulus, 816 01:03:29,150 --> 01:03:34,570 which was 1 over S1 1 prime, and that's equal to the 817 01:03:34,570 --> 01:03:37,670 Young's modulus E. 818 01:03:37,670 --> 01:03:42,020 Are we going to add or subtract something to it? 819 01:03:42,020 --> 01:03:46,830 Well, it turns out that if you look at real materials, it can 820 01:03:46,830 --> 01:03:49,960 be either positive or negative. 821 01:03:49,960 --> 01:03:51,355 It's usually positive. 822 01:03:56,680 --> 01:04:02,350 So that says we are going to subtract off a term which goes 823 01:04:02,350 --> 01:04:05,620 as products of squares of direction cosines. 824 01:04:05,620 --> 01:04:08,970 And this is something that's going to be zero along the 825 01:04:08,970 --> 01:04:14,790 direction 1 0 0. 826 01:04:14,790 --> 01:04:22,640 So along the reference axes x1, x2, x3, which if you 827 01:04:22,640 --> 01:04:25,900 remember, where this form of the matrix came from was 828 01:04:25,900 --> 01:04:30,570 taking the axes along the edges of the cubic crystal. 829 01:04:30,570 --> 01:04:35,730 It turns out that this is going to be equal to 1/3 along 830 01:04:35,730 --> 01:04:37,040 the direction 1 1 1. 831 01:04:42,770 --> 01:04:51,300 So if this term is plus, nothing gets added onto the 832 01:04:51,300 --> 01:04:55,410 surface in the directions that correspond to the four-fold 833 01:04:55,410 --> 01:04:59,900 axes or the twofold axes of a cubic crystal. 834 01:04:59,900 --> 01:05:02,440 Nothing gets added on if it's plus. 835 01:05:02,440 --> 01:05:07,690 But along the body diagonals, this takes on a value of 1 3, 836 01:05:07,690 --> 01:05:10,360 so a positive thing gets added on. 837 01:05:10,360 --> 01:05:13,420 And the best way I can describe this surface-- it's 838 01:05:13,420 --> 01:05:14,910 not a simple surface-- 839 01:05:14,910 --> 01:05:20,470 it looks like a cube with fuzzy edges. 840 01:05:20,470 --> 01:05:23,330 So we had a cube and started to dissolve it. 841 01:05:23,330 --> 01:05:25,930 So this is how the reciprocal of Young's 842 01:05:25,930 --> 01:05:27,760 modulus varies with direction. 843 01:05:34,960 --> 01:05:38,280 If, on the other hand, this term is negative-- 844 01:05:38,280 --> 01:05:41,690 so this is what you get if it's positive. 845 01:05:41,690 --> 01:05:47,150 If it's negative, again, you start with the basic isotropic 846 01:05:47,150 --> 01:05:51,650 variation of 1 over S1 1 1. 847 01:05:51,650 --> 01:06:00,490 If it is negative, and there's one metal, that's molybdenum-- 848 01:06:00,490 --> 01:06:04,480 molybdenum has a negative value of these compliances. 849 01:06:04,480 --> 01:06:15,002 It looks like a surface that has indentations along the 1 1 850 01:06:15,002 --> 01:06:16,440 1 direction. 851 01:06:16,440 --> 01:06:19,760 So it looks like something with a dimple in it. 852 01:06:22,460 --> 01:06:28,820 There is a final case, which is not realized for any 853 01:06:28,820 --> 01:06:33,070 material that I know of, if that term is 854 01:06:33,070 --> 01:06:34,320 0, then it's isotropic. 855 01:06:43,840 --> 01:06:46,800 Let me give you one other variation that comes straight 856 01:06:46,800 --> 01:06:49,270 out of [INAUDIBLE]. 857 01:06:49,270 --> 01:06:53,320 And this is for a hexagonal crystal that might be a 858 01:06:53,320 --> 01:06:57,010 hexagonal close-packed crystal. 859 01:06:57,010 --> 01:07:01,810 For zinc specifically, which is a hexagonal close-packed 860 01:07:01,810 --> 01:07:12,810 metal, the form of the matrix is S1 1, S1 2, S1 3, 0, 0, 0, 861 01:07:12,810 --> 01:07:24,880 S1 1, S1 2, 0, 0, 0, S3 3, 0, 0, 0, S4 4, 0, 0, S4 862 01:07:24,880 --> 01:07:29,260 4, 0, and S4 4. 863 01:07:29,260 --> 01:07:35,470 In the values of these specific compliances are 8.4 864 01:07:35,470 --> 01:07:52,651 for S1 1, for S1 2 1.1, for S1 3, minus 7.8, for S3 3, 28.7, 865 01:07:52,651 --> 01:07:56,940 for S4 4, 26.4. 866 01:07:56,940 --> 01:08:03,190 And these are all in units of 10 to the minus 12 meters 867 01:08:03,190 --> 01:08:04,440 squared per Newton. 868 01:08:07,060 --> 01:08:09,720 That's good old MKS units. 869 01:08:09,720 --> 01:08:14,090 It turns out that there are 10 to the 2 meters squared per 870 01:08:14,090 --> 01:08:23,014 Newton per 1 centimeter squared per dyne, if 871 01:08:23,014 --> 01:08:24,430 you like CGS units. 872 01:08:27,090 --> 01:08:38,729 The form of the reciprocal of Young's modulus, it turns out 873 01:08:38,729 --> 01:08:40,369 to be a surface of revolution. 874 01:08:44,370 --> 01:08:48,290 This is x3, which is the direction of C. And that's the 875 01:08:48,290 --> 01:08:54,240 surface of revolution and the value of S1 1 prime as a 876 01:08:54,240 --> 01:08:57,930 function of the angle theta. 877 01:08:57,930 --> 01:09:00,950 That's the only parameter we need examine the variation 878 01:09:00,950 --> 01:09:03,660 with since it's a surface of revolution. 879 01:09:03,660 --> 01:09:09,930 S1 1 prime is equal to S1 1 times the sine of theta to the 880 01:09:09,930 --> 01:09:16,170 fourth power plus S3 3 times the cosine of theta to the 881 01:09:16,170 --> 01:09:26,520 fourth power plus S4 4 plus 2S1 3 times sine squared 882 01:09:26,520 --> 01:09:32,040 theta, cosine squared theta. 883 01:09:32,040 --> 01:09:35,439 So it's a fairly exotic surface, even as a surface of 884 01:09:35,439 --> 01:09:36,750 revolution. 885 01:09:36,750 --> 01:09:40,680 And what this looks like, it's something that peaks out at 3 886 01:09:40,680 --> 01:09:47,810 times 10 to the minus 12 centimeters squared per dyne 887 01:09:47,810 --> 01:09:51,630 along the direction of C. It's something that comes down very 888 01:09:51,630 --> 01:09:55,740 sharply as you approach the normal to the C axis. 889 01:09:55,740 --> 01:10:01,320 And then there's a cute little wiggle just near the axis. 890 01:10:01,320 --> 01:10:03,680 So it looks something like that. 891 01:10:03,680 --> 01:10:06,950 Not a terribly isotropic surface, and a variation of 892 01:10:06,950 --> 01:10:12,060 Young's modulus of 3 to 1 in the direction parallel to C 893 01:10:12,060 --> 01:10:15,590 and perpendicular to C. So there are lots of exotic 894 01:10:15,590 --> 01:10:16,840 surfaces of this sort. 895 01:10:19,970 --> 01:10:20,330 All right. 896 01:10:20,330 --> 01:10:21,850 I did run a little bit over. 897 01:10:21,850 --> 01:10:26,350 And I will absent myself and let you express yourself 898 01:10:26,350 --> 01:10:29,420 candidly on the questionnaire. 899 01:10:32,050 --> 01:10:34,140 Except for these numbers that I just put up on the 900 01:10:34,140 --> 01:10:38,410 blackboard, I haven't really said anything about, much 901 01:10:38,410 --> 01:10:39,240 about, numbers. 902 01:10:39,240 --> 01:10:43,900 So let me pass around some examples, not for metals, 903 01:10:43,900 --> 01:10:46,513 which we just looked at, but for oxides. 904 01:10:51,340 --> 01:10:58,360 And this is interesting, because you'll remember there 905 01:10:58,360 --> 01:11:03,330 was an additional equality between the stiffnesses called 906 01:11:03,330 --> 01:11:04,020 the [? Kowshi ?] 907 01:11:04,020 --> 01:11:08,270 equality, which was supposed to hold for physical reasons, 908 01:11:08,270 --> 01:11:11,840 not for reasons of symmetry, if the forces were central, if 909 01:11:11,840 --> 01:11:15,240 the crystal was under a state of no stress, and if all of 910 01:11:15,240 --> 01:11:19,690 the atoms were situated at a center of symmetry. 911 01:11:19,690 --> 01:11:24,040 The first set of data that you have on the top sheet is for 912 01:11:24,040 --> 01:11:26,080 MGO, which is cubic. 913 01:11:26,080 --> 01:11:29,550 It's predominantly ionic compounds. 914 01:11:29,550 --> 01:11:33,180 And the atoms are all octahedrally coordinated. 915 01:11:33,180 --> 01:11:36,950 So all of the requirements for the [? Kowshi ?] 916 01:11:36,950 --> 01:11:38,630 equality should hold. 917 01:11:38,630 --> 01:11:42,465 And you can see they're not really exact. 918 01:11:42,465 --> 01:11:46,070 And that's probably due to the covalent character. 919 01:11:46,070 --> 01:11:51,860 The two different sets of data here are for measurements by 920 01:11:51,860 --> 01:11:54,000 two separate observers. 921 01:11:54,000 --> 01:11:58,290 The second page is for aluminum oxide. 922 01:11:58,290 --> 01:12:02,300 And again, you see the order of magnitudes of the numbers 923 01:12:02,300 --> 01:12:05,710 on the order of 10 to the 12 dynes per centimeter squared 924 01:12:05,710 --> 01:12:09,875 in general, and variation from one to another for a lumen of 925 01:12:09,875 --> 01:12:11,285 a factor of about four. 926 01:12:16,700 --> 01:12:19,230 I meant to bring the book in, but I left it behind in my 927 01:12:19,230 --> 01:12:19,950 rush to come off. 928 01:12:19,950 --> 01:12:21,440 Where do you get these numbers? 929 01:12:21,440 --> 01:12:22,940 And it's hard to really find. 930 01:12:26,630 --> 01:12:31,120 And the book from which I took these data are by two workers, 931 01:12:31,120 --> 01:12:33,430 Simmons and Wang. 932 01:12:33,430 --> 01:12:41,980 And it's called Handbook of Elastic Constants, 933 01:12:41,980 --> 01:12:45,340 I think that is. 934 01:12:45,340 --> 01:12:48,000 And this is a book that was published by MIT Press. 935 01:12:51,500 --> 01:12:55,370 Valuable repository of data, but it's all numbers. 936 01:12:55,370 --> 01:12:58,860 And there are many, many materials in here. 937 01:12:58,860 --> 01:13:00,660 Simmons and Wang, interestingly, are 938 01:13:00,660 --> 01:13:02,180 geophysicists. 939 01:13:02,180 --> 01:13:05,410 Why should geophysicists care about stiffnesses? 940 01:13:05,410 --> 01:13:05,640 Why? 941 01:13:05,640 --> 01:13:08,130 Because these guys are going all over the face of the earth 942 01:13:08,130 --> 01:13:11,710 setting off little bits of explosive that send off 943 01:13:11,710 --> 01:13:12,730 seismic waves. 944 01:13:12,730 --> 01:13:16,160 And they probed the interior of the earth by the 945 01:13:16,160 --> 01:13:18,760 propagation of elastic waves. 946 01:13:18,760 --> 01:13:22,930 And the elastic waves depend on the square root of 947 01:13:22,930 --> 01:13:26,490 stiffnesses over the density of the material. 948 01:13:26,490 --> 01:13:29,910 So they're very, very interested in the elastic 949 01:13:29,910 --> 01:13:33,120 properties of particularly rock-forming minerals. 950 01:13:33,120 --> 01:13:37,400 Another interesting comment on this book is that there is no 951 01:13:37,400 --> 01:13:43,140 set of data for a single triclinic crystal. 952 01:13:43,140 --> 01:13:46,430 Too many stiffnesses that are independent, I guess, and too 953 01:13:46,430 --> 01:13:50,560 few materials that, fortunately, are anisotropic 954 01:13:50,560 --> 01:13:51,810 and triclinic. 955 01:13:54,210 --> 01:13:55,660 OK, I am going to quit. 956 01:13:55,660 --> 01:13:58,800 And thank you for your attention. 957 01:13:58,800 --> 01:14:01,620 You've been a good class. 958 01:14:01,620 --> 01:14:05,480 And we've covered all sorts of exotic aspects of the behavior 959 01:14:05,480 --> 01:14:07,190 of crystalline materials. 960 01:14:07,190 --> 01:14:11,245 And you might think with some justification that you know 961 01:14:11,245 --> 01:14:14,210 all there is to know about symmetry and tensor properties 962 01:14:14,210 --> 01:14:14,940 of materials. 963 01:14:14,940 --> 01:14:17,100 But nevertheless, I would caution you that you don't 964 01:14:17,100 --> 01:14:18,630 know everything. 965 01:14:18,630 --> 01:14:23,710 So I have a final handout fill in the chinks, the cracks that 966 01:14:23,710 --> 01:14:24,890 we've not been able to fill. 967 01:14:24,890 --> 01:14:31,720 And this is a set of data that is titled "Think You Know 968 01:14:31,720 --> 01:14:36,050 Everything?" And this will fill in some things that you 969 01:14:36,050 --> 01:14:38,366 really don't know. 970 01:14:38,366 --> 01:14:41,110 And it has a lot of very useful items on here. 971 01:14:41,110 --> 01:14:45,280 For example, there are 293 different ways to 972 01:14:45,280 --> 01:14:48,036 make change for $1. 973 01:14:48,036 --> 01:14:50,570 You didn't know that. 974 01:14:50,570 --> 01:14:55,000 2/3 of the world's eggplant is grown in New Jersey. 975 01:14:55,000 --> 01:14:58,630 I'm a native of New Jersey, and even I didn't know that. 976 01:14:58,630 --> 01:15:01,790 The longest one-syllable word in the English language is 977 01:15:01,790 --> 01:15:05,950 "screeched." OK. 978 01:15:09,040 --> 01:15:10,290 And so it goes. 979 01:15:14,200 --> 01:15:15,640 In most advertisements-- 980 01:15:15,640 --> 01:15:18,230 this is one I never checked out-- in most advertisements 981 01:15:18,230 --> 01:15:23,502 the time displayed on a watch is 10 minutes after 10:00. 982 01:15:23,502 --> 01:15:26,090 Don't know why. 983 01:15:26,090 --> 01:15:31,740 How many ridges are around the edge of a dime? 984 01:15:31,740 --> 01:15:35,840 118. 985 01:15:35,840 --> 01:15:38,600 And how many little dimples are there on a 986 01:15:38,600 --> 01:15:40,836 regulation golf ball? 987 01:15:40,836 --> 01:15:47,370 If you count them up, you'll find that there are 336. 988 01:15:47,370 --> 01:15:52,310 So I'll end with one for the benefit of some of our 989 01:15:52,310 --> 01:15:53,310 visiting students. 990 01:15:53,310 --> 01:15:58,640 In England, the Speaker of the House is not allowed to speak. 991 01:15:58,640 --> 01:16:00,870 There is my oxymoron of the day. 992 01:16:00,870 --> 01:16:02,560 And with that I shall leave you, and I 993 01:16:02,560 --> 01:16:03,630 hope you enjoy these. 994 01:16:03,630 --> 01:16:05,430 It's one of these things that circulates 995 01:16:05,430 --> 01:16:09,750 around on the internet. 996 01:16:09,750 --> 01:16:16,560 So as the sheet finishes, now you do know everything. 997 01:16:16,560 --> 01:16:19,080 So I'll leave you, and again I thank you for your faithful 998 01:16:19,080 --> 01:16:19,750 attendance. 999 01:16:19,750 --> 01:16:29,910 And you shall see me when you come to call for your quizzes. 1000 01:16:29,910 --> 01:16:33,990 And I will, as most of you requested, send out an email 1001 01:16:33,990 --> 01:16:36,410 to let you know when you can come and pick those up. 1002 01:16:36,410 --> 01:16:37,740 So again, thank you. 1003 01:16:37,740 --> 01:16:41,230 Enjoy the mid-term break. 1004 01:16:41,230 --> 01:16:44,610 And take advantage of some of the really interesting things 1005 01:16:44,610 --> 01:16:48,150 that go on extracurricularly around the institute. 1006 01:16:48,150 --> 01:16:49,100 So that's it. 1007 01:16:49,100 --> 01:16:52,340 Thank you, and au revoir. 1008 01:16:52,340 --> 01:16:58,542 [APPLAUSE]