1 00:00:09,840 --> 00:00:12,500 PROFESSOR: I think you will get going. 2 00:00:12,500 --> 00:00:15,600 A lot of people are not here, but I think if the MRS meeting 3 00:00:15,600 --> 00:00:19,490 were going on across the river I would be there too if I 4 00:00:19,490 --> 00:00:23,120 didn't have a class to go to. 5 00:00:23,120 --> 00:00:28,560 We had last time finished a discussion strain. 6 00:00:28,560 --> 00:00:34,910 And we introduced strain by defining it as a displacement, 7 00:00:34,910 --> 00:00:38,170 each component of a vector displacement U as being given 8 00:00:38,170 --> 00:00:45,520 by a set of coefficients, eij times x of j, which says that 9 00:00:45,520 --> 00:00:49,000 the displacement in a deformed body varies linearly with the 10 00:00:49,000 --> 00:00:52,170 position of the particular point in the body. 11 00:00:52,170 --> 00:00:55,360 We could also express the same thing in terms 12 00:00:55,360 --> 00:00:57,330 of change of length. 13 00:00:57,330 --> 00:01:01,790 The change in the vector U would be given by-- we show 14 00:01:01,790 --> 00:01:03,320 the same set of coefficients-- 15 00:01:03,320 --> 00:01:07,150 eij times delta x of j. 16 00:01:07,150 --> 00:01:09,350 So whether you want the displacement vector or you 17 00:01:09,350 --> 00:01:13,450 want the fractional change of length, you get this by an 18 00:01:13,450 --> 00:01:17,680 expression of the same sort. 19 00:01:17,680 --> 00:01:23,540 We saw though that unless we define the coefficients 20 00:01:23,540 --> 00:01:28,800 correctly we could have a situation where a body is not 21 00:01:28,800 --> 00:01:32,900 only deformed but is rotated as well. 22 00:01:32,900 --> 00:01:37,020 And we saw that in general unless we define these 23 00:01:37,020 --> 00:01:43,420 coefficients carefully we would include in this tensor a 24 00:01:43,420 --> 00:01:46,740 component of pure rotation, rigid body rotation. 25 00:01:46,740 --> 00:01:50,070 And this would not measure deformation. 26 00:01:50,070 --> 00:01:53,470 However, regardless of its suitability or unsuitability, 27 00:01:53,470 --> 00:01:57,060 the positional vector is a vector. 28 00:01:57,060 --> 00:01:59,280 It transforms like a vector. 29 00:01:59,280 --> 00:02:03,660 The displacement U is a vector, and it will transform 30 00:02:03,660 --> 00:02:09,810 like a vector; ergo, the 3 by 3 array eij is a second-rank 31 00:02:09,810 --> 00:02:13,140 tensor and we'll transform like a tensor. 32 00:02:13,140 --> 00:02:20,310 We showed though that we could take eij and get rid of the 33 00:02:20,310 --> 00:02:24,000 pure rotation that might be contained in those 34 00:02:24,000 --> 00:02:30,840 coefficients by writing it as a tensor epsilon ij plus 35 00:02:30,840 --> 00:02:34,210 another tensor omega ij. 36 00:02:34,210 --> 00:02:37,890 And that was a new wrinkle, the concept of adding two 37 00:02:37,890 --> 00:02:41,460 tensors together to get a third tensor. 38 00:02:41,460 --> 00:02:43,980 We did run into that before. 39 00:02:43,980 --> 00:02:48,500 And we define the elements of the two tensors epsilon and 40 00:02:48,500 --> 00:03:00,580 omega; epsilon ij was given by one half of eij plus eji, and 41 00:03:00,580 --> 00:03:07,750 omega ij is given by one half of omega ij. 42 00:03:07,750 --> 00:03:10,810 And notice the order of the subscripts here in this 43 00:03:10,810 --> 00:03:11,420 difference. 44 00:03:11,420 --> 00:03:15,580 That is a matter of definition, but it defines 45 00:03:15,580 --> 00:03:19,250 which term is omega ij and which is omega 46 00:03:19,250 --> 00:03:21,670 ji minus omega ji. 47 00:03:24,220 --> 00:03:24,590 I'm sorry. 48 00:03:24,590 --> 00:03:28,970 This should be eij minus eji. 49 00:03:28,970 --> 00:03:35,430 And we showed that at a tensor of this form which would have 50 00:03:35,430 --> 00:03:50,220 all of the diagonal terms zero and the off-diagonal terms 51 00:03:50,220 --> 00:03:56,540 equal to the negative of one another that a tensor of this 52 00:03:56,540 --> 00:04:07,210 form does indeed correspond to pure body rotation. 53 00:04:07,210 --> 00:04:10,230 We showed that by looking at a point at the end of a 54 00:04:10,230 --> 00:04:17,320 positional vector U and looked at a displacement which would 55 00:04:17,320 --> 00:04:19,610 be exactly at right angles to it. 56 00:04:19,610 --> 00:04:28,030 And the U would be given by the tensor relation. 57 00:04:28,030 --> 00:04:36,070 We showed that U dot r is identically zero when the 58 00:04:36,070 --> 00:04:43,940 displacement vector is given by this tensor with diagonal 59 00:04:43,940 --> 00:04:47,080 term zero and the off-diagonal terms equal to the negative of 60 00:04:47,080 --> 00:04:48,540 one another. 61 00:04:48,540 --> 00:04:54,170 So we have then from a general tensor of the form eij 62 00:04:54,170 --> 00:04:59,080 constructed in the terms epsilon ij something that is a 63 00:04:59,080 --> 00:05:02,230 measure of true deformation. 64 00:05:02,230 --> 00:05:05,750 It is a symmetric tensor by definition. 65 00:05:05,750 --> 00:05:08,950 And therefore in addition to all of the other properties of 66 00:05:08,950 --> 00:05:15,770 second-rank tensor, namely having a law of transformation 67 00:05:15,770 --> 00:05:19,140 and staying symmetric for any arbitrary change of coordinate 68 00:05:19,140 --> 00:05:28,480 systems, the radius normal property also works. 69 00:05:28,480 --> 00:05:32,670 Hence, the representation surface that we construct from 70 00:05:32,670 --> 00:05:40,980 the tensor epsilon ij xi xj equals 1, gives 71 00:05:40,980 --> 00:05:42,345 us the strain quadric. 72 00:05:47,430 --> 00:05:53,906 And it has the property that the radius gives us the value 73 00:05:53,906 --> 00:06:01,825 of strain in a given direction. 74 00:06:07,550 --> 00:06:12,430 And the value of the radius literally is going to be the 75 00:06:12,430 --> 00:06:18,390 vector that's on the left hand side, the magnitude of U times 76 00:06:18,390 --> 00:06:23,500 the component of U that is parallel to the radial vector 77 00:06:23,500 --> 00:06:25,300 per unit radial vector. 78 00:06:28,820 --> 00:06:33,170 And this then is the tensile strain, the fractional change 79 00:06:33,170 --> 00:06:40,050 of length as we look in this direction. 80 00:06:40,050 --> 00:06:46,530 The radius normal is going to give us the direction of the 81 00:06:46,530 --> 00:06:53,780 total displacement U. And the value of the property in that 82 00:06:53,780 --> 00:06:58,490 direction is going to be the product of that displacement 83 00:06:58,490 --> 00:07:01,940 with a unit vector in the direction of interest. 84 00:07:01,940 --> 00:07:04,850 And therefore that is the tensile strain as we said 85 00:07:04,850 --> 00:07:07,220 before from the general properties of the quadric. 86 00:07:11,110 --> 00:07:14,600 So everything we said about second-rank tensor and in 87 00:07:14,600 --> 00:07:21,920 particular symmetric tensor holds for the strain tensor. 88 00:07:21,920 --> 00:07:31,990 We can talk about diagonalizing the strain 89 00:07:31,990 --> 00:07:36,290 tensor into a form epsilon 1, 1, epsilon 22, zero zero, 90 00:07:36,290 --> 00:07:37,920 epsilon 33. 91 00:07:37,920 --> 00:07:40,950 And I'll remind you that we know how to do this to set up 92 00:07:40,950 --> 00:07:45,120 the normal equations and solve for the eigenvalues and then 93 00:07:45,120 --> 00:07:49,100 look for the principal axes of a tensor. 94 00:07:49,100 --> 00:07:54,180 There are special forms of the strain tensor. 95 00:07:54,180 --> 00:07:58,890 Something that 1 diagonal goes into this form is 96 00:07:58,890 --> 00:08:00,245 called plane strain. 97 00:08:04,989 --> 00:08:09,610 And I love the melodic nature of that, the plane strain. 98 00:08:09,610 --> 00:08:13,640 If this were strain induced by a Modernistic transformation, 99 00:08:13,640 --> 00:08:16,990 we could call it the Bain plane strain. 100 00:08:16,990 --> 00:08:19,120 If we diagonalized it, it would be the 101 00:08:19,120 --> 00:08:21,830 main Bain plane strains. 102 00:08:21,830 --> 00:08:26,150 And if that deformation took place in an aircraft window, 103 00:08:26,150 --> 00:08:31,740 we could call it the plane-pane main Bain strains. 104 00:08:31,740 --> 00:08:34,630 And we could continue on indefinitely, but I think 105 00:08:34,630 --> 00:08:37,270 you're finding this tiresome. 106 00:08:37,270 --> 00:08:42,590 Another form of the strain tensor that doesn't look what 107 00:08:42,590 --> 00:08:48,190 it actually represents is epsilon 1 zero minus epsilon 1 108 00:08:48,190 --> 00:08:50,170 zero zero zero zero. 109 00:08:53,920 --> 00:08:56,990 That looks peculiar, very special. 110 00:08:56,990 --> 00:09:00,870 But, again, this is analogous to what we found for stress. 111 00:09:00,870 --> 00:09:08,860 If we rotate this tensor 45 degrees, we would find that 112 00:09:08,860 --> 00:09:12,070 treat tensor was diagonally, had all the 113 00:09:12,070 --> 00:09:13,700 diagonal elements zero. 114 00:09:13,700 --> 00:09:21,600 And we would have the epsilon 11 prime-- 115 00:09:21,600 --> 00:09:22,940 let's just call it epsilon prime-- 116 00:09:22,940 --> 00:09:25,940 and this epsilon prime zero zero zero. 117 00:09:25,940 --> 00:09:36,100 And this is pure shear 118 00:09:36,100 --> 00:09:42,000 Something that we'll show shortly, and in fact I left it 119 00:09:42,000 --> 00:09:45,680 to you as a problem on a problems set. 120 00:09:45,680 --> 00:09:49,910 The sum of the diagonal elements, epsilon 11 plus 121 00:09:49,910 --> 00:09:57,070 epsilon 22 plus epsilon 33 is something that's defined as 122 00:09:57,070 --> 00:09:58,550 the trace of the tensor. 123 00:10:03,660 --> 00:10:09,950 And for the strain tensor, the trace turns out to be equal to 124 00:10:09,950 --> 00:10:14,730 delta V over the fractional change in volume. 125 00:10:14,730 --> 00:10:18,070 And we'll show that in just a little bit as well. 126 00:10:18,070 --> 00:10:23,960 So a characteristic of pure shear is that it does not 127 00:10:23,960 --> 00:10:26,660 result in any change of volume of the solid. 128 00:10:26,660 --> 00:10:29,710 There's deformation, but the volume stays exactly the same. 129 00:10:57,030 --> 00:11:00,370 Let me then proceed to show that the trace of the strain 130 00:11:00,370 --> 00:11:04,180 tensor is indeed the change in volume. 131 00:11:04,180 --> 00:11:19,530 And want I'll do is to look at an element of volume that is 132 00:11:19,530 --> 00:11:24,810 oriented along the principal elements of strain. 133 00:11:24,810 --> 00:11:30,970 So if this as x1, this is x2, and this is x3, the strain 134 00:11:30,970 --> 00:11:32,690 tensor I'll assume is oriented. 135 00:11:32,690 --> 00:11:36,716 So we have a term epsilon 11 zero zero zero zero; epsilon 136 00:11:36,716 --> 00:11:42,560 22 zero zero zero; epsilon 33. 137 00:11:42,560 --> 00:11:50,500 So let's suppose that the solid originally has edges L1, 138 00:11:50,500 --> 00:11:54,590 L2, and L3 along x1, x2, and x3, respectively. 139 00:11:54,590 --> 00:11:58,040 So the initial volume before any deformation will be simply 140 00:11:58,040 --> 00:12:01,020 L1 times L2 times L3. 141 00:12:06,157 --> 00:12:10,260 Let's suppose now we impose the strain, and the volume 142 00:12:10,260 --> 00:12:14,090 then we'll increase to some value of V plus delta V. And 143 00:12:14,090 --> 00:12:19,270 the length L1 will change to a value L1 times 1 144 00:12:19,270 --> 00:12:21,870 plus epsilon 11. 145 00:12:21,870 --> 00:12:25,480 That is to say it'll be L1 plus epsilon 11 times L1, 146 00:12:25,480 --> 00:12:28,000 which can be factored out in this fashion. 147 00:12:28,000 --> 00:12:33,240 L2 changes to L2 times 1 plus epsilon 22. 148 00:12:33,240 --> 00:12:41,660 And L3 changes to L3 plus epsilon 33 times L3, which 149 00:12:41,660 --> 00:12:44,250 I'll write in this fashion. 150 00:12:44,250 --> 00:12:50,480 So expanding this product, it'll be L1, L2, L3. 151 00:12:50,480 --> 00:12:54,930 And then if I simply multiply out these terms, I'll have 1, 152 00:12:54,930 --> 00:13:02,650 and then I'll have epsilon 11 plus epsilon 22 plus epsilon 153 00:13:02,650 --> 00:13:06,140 11 times epsilon 22. 154 00:13:06,140 --> 00:13:10,370 And this will be times 1 plus epsilon 33. 155 00:13:10,370 --> 00:13:15,330 And if I carry out that multiplication, 156 00:13:15,330 --> 00:13:18,140 that will be 1 plus-- 157 00:13:18,140 --> 00:13:25,470 and I'll get exactly these terms again-- epsilon 11 plus 158 00:13:25,470 --> 00:13:34,110 epsilon 22 plus epsilon 11 times epsilon 22. 159 00:13:34,110 --> 00:13:41,700 And then a term epsilon 33 plus epsilon 11 epsilon 33 160 00:13:41,700 --> 00:13:46,440 plus epsilon 22 times epsilon 33. 161 00:13:46,440 --> 00:13:51,685 And then a third-order term, epsilon 11 times epsilon 22 162 00:13:51,685 --> 00:13:54,860 times epsilon 33. 163 00:13:54,860 --> 00:13:58,360 Now having gone through the painful process of expanding 164 00:13:58,360 --> 00:14:03,160 that, I will proceed to say that the epsilons are always 165 00:14:03,160 --> 00:14:05,470 going to be very small. 166 00:14:05,470 --> 00:14:08,180 Elastic strains are typically on the order of 10 167 00:14:08,180 --> 00:14:09,780 to the minus 6. 168 00:14:09,780 --> 00:14:14,020 So a product of two of those strains is 10 to the minus 12. 169 00:14:14,020 --> 00:14:16,360 And a product of three of those terms is going to be on 170 00:14:16,360 --> 00:14:21,260 the order of 10 to the minus 18. 171 00:14:21,260 --> 00:14:23,770 So this last term is going to be minuscule. 172 00:14:23,770 --> 00:14:30,120 And now I use a higher order term intentionally rather than 173 00:14:30,120 --> 00:14:31,140 a higher rank term. 174 00:14:31,140 --> 00:14:33,250 This is really a term of higher order. 175 00:14:33,250 --> 00:14:38,170 The same is true of all the peer-wise projects of this 176 00:14:38,170 --> 00:14:39,180 [INAUDIBLE]. 177 00:14:39,180 --> 00:14:43,850 So I now have left if I turn out these higher order terms 178 00:14:43,850 --> 00:14:49,430 to pasture, I'll have 1 plus epsilon 11 plus epsilon 22 179 00:14:49,430 --> 00:14:51,260 plus epsilon 33. 180 00:14:51,260 --> 00:14:53,210 And the rest is negligible. 181 00:14:53,210 --> 00:15:01,260 And this will be the original volume V times 1 plus epsilon 182 00:15:01,260 --> 00:15:07,020 11 plus epsilon 22 plus epsilon 33. 183 00:15:07,020 --> 00:15:13,390 So delta V over V, if I subtract off V on the left, 184 00:15:13,390 --> 00:15:17,360 delta V over V then is simply going to be equal to epsilon 185 00:15:17,360 --> 00:15:23,600 11 plus epsilon 22 plus epsilon 33. 186 00:15:23,600 --> 00:15:26,690 So this is indeed the fractional change of volume, 187 00:15:26,690 --> 00:15:31,095 and it's the trace of epsilon ij. 188 00:15:35,250 --> 00:15:39,090 Now, physically, you wouldn't expect the volume to change if 189 00:15:39,090 --> 00:15:41,360 I change my reference axes because 190 00:15:41,360 --> 00:15:43,410 that's a scalar quantity. 191 00:15:43,410 --> 00:15:49,000 So I've shown by a hand-waving, backdoor argument 192 00:15:49,000 --> 00:15:52,880 that in fact the trace of a tensor does stay invariant for 193 00:15:52,880 --> 00:15:54,690 at least this particular tensor. 194 00:15:54,690 --> 00:15:57,220 But it's fairly straightforward to show that 195 00:15:57,220 --> 00:16:01,890 for any tensor aij the trace of the tensor is invariant 196 00:16:01,890 --> 00:16:04,830 when you change the coordinate system. 197 00:16:04,830 --> 00:16:06,260 And that's not hard to do. 198 00:16:06,260 --> 00:16:08,440 If you're clever, it takes about three lines. 199 00:16:08,440 --> 00:16:11,910 So I'll invite you to the exhilaration of discovering 200 00:16:11,910 --> 00:16:13,807 that for yourself on a problem set. 201 00:16:21,770 --> 00:16:23,080 Any questions or comments? 202 00:16:36,820 --> 00:16:37,680 All right. 203 00:16:37,680 --> 00:16:45,030 Having defined strain as a symmetric second-rank tensor 204 00:16:45,030 --> 00:16:56,740 when we factor out rigid body rotation, we can view strain 205 00:16:56,740 --> 00:17:00,730 as a physical property. 206 00:17:00,730 --> 00:17:03,230 And I think I tried to hoodwink you last time by 207 00:17:03,230 --> 00:17:10,530 saying this is a property of material. 208 00:17:10,530 --> 00:17:14,339 So therefore strain tensor has to also conform to all of the 209 00:17:14,339 --> 00:17:17,490 symmetry restrains that we have derived very 210 00:17:17,490 --> 00:17:18,839 exhaustively. 211 00:17:18,839 --> 00:17:22,839 And you said, no, no, no, no, that can't be; I don't believe 212 00:17:22,839 --> 00:17:29,810 that I can only give a uniform uniaxial deformation to an 213 00:17:29,810 --> 00:17:30,920 orthorhombic crystal. 214 00:17:30,920 --> 00:17:33,190 And I say, yeah, but to get the deformation, you have to 215 00:17:33,190 --> 00:17:34,410 squeeze the crystal. 216 00:17:34,410 --> 00:17:38,080 And that is going to be linked to the deformation by the 217 00:17:38,080 --> 00:17:41,400 elastic constants, and they are physical properties. 218 00:17:41,400 --> 00:17:43,510 And that maybe makes you think a little bit. 219 00:17:43,510 --> 00:17:48,520 But actually even though operationally you impose a 220 00:17:48,520 --> 00:17:52,600 strain by imposing a stress or by doing something else to the 221 00:17:52,600 --> 00:17:56,490 crystal that results in a strain, you can nevertheless 222 00:17:56,490 --> 00:18:00,060 pick any stimulus you wish to achieve a 223 00:18:00,060 --> 00:18:02,910 desired state of strain. 224 00:18:02,910 --> 00:18:06,400 So there is no symmetry constraint on the form of the 225 00:18:06,400 --> 00:18:08,690 strain tensor that can result. 226 00:18:08,690 --> 00:18:12,560 So the stain tensor then is not a property tensor. 227 00:18:12,560 --> 00:18:15,990 It is also a field tensor just like stress. 228 00:18:15,990 --> 00:18:19,600 So it can have any form we choose to create in it. 229 00:18:22,540 --> 00:18:28,080 So we now have the potential of having 230 00:18:28,080 --> 00:18:30,770 stress as a field tensor. 231 00:18:36,930 --> 00:18:40,710 And we have strain as a field tensor. 232 00:18:45,200 --> 00:18:50,540 And within limits, we can create strains of these two 233 00:18:50,540 --> 00:18:53,750 field tensors of any form that we wish. 234 00:18:53,750 --> 00:18:58,600 And we can now regard these as things that we do to crystals 235 00:18:58,600 --> 00:19:03,500 and look at a whole range of properties that result when 236 00:19:03,500 --> 00:19:09,510 the generalized force is not a vector, a tensor of first 237 00:19:09,510 --> 00:19:13,170 rank, but the generalized force is a-- 238 00:19:13,170 --> 00:19:15,300 let's say-- a stress tensor. 239 00:19:15,300 --> 00:19:17,320 It's a generalized force which itself is not 240 00:19:17,320 --> 00:19:20,950 a first rank tensor. 241 00:19:20,950 --> 00:19:22,460 It's a second rank tensor. 242 00:19:22,460 --> 00:19:24,530 So we'll see a whole collection of interesting 243 00:19:24,530 --> 00:19:27,850 properties that fall into this category. 244 00:19:27,850 --> 00:19:31,620 But first I'd like to look at one remaining second-rank 245 00:19:31,620 --> 00:19:34,860 tensor property, and that is thermal expansion. 246 00:19:42,870 --> 00:19:47,360 And this is a curious second-rank tensor property, 247 00:19:47,360 --> 00:19:49,480 and we couldn't discuss it until we had 248 00:19:49,480 --> 00:19:50,800 talked about strain. 249 00:19:50,800 --> 00:19:54,390 And let me introduce it by one-dimensional example. 250 00:19:54,390 --> 00:20:02,020 Let's suppose we have a rodlike specimen of length L. 251 00:20:02,020 --> 00:20:05,850 And let's suppose it is in equilibrium at some 252 00:20:05,850 --> 00:20:11,850 temperature T. And then we heat it up to some temperature 253 00:20:11,850 --> 00:20:14,910 T plus delta T. 254 00:20:14,910 --> 00:20:19,430 In response to that particular change in temperature, the 255 00:20:19,430 --> 00:20:25,743 length of the rod will increase to a length L plus 256 00:20:25,743 --> 00:20:28,710 delta L. It expands. 257 00:20:28,710 --> 00:20:32,390 And we would find experimentally that delta L, 258 00:20:32,390 --> 00:20:36,680 first of all, is proportional to L. If you make the rod 259 00:20:36,680 --> 00:20:40,010 twice as long, the amount of expansion in an absolute sense 260 00:20:40,010 --> 00:20:41,810 is twice as large. 261 00:20:41,810 --> 00:20:48,110 And secondly, provided you're well above very, very low 262 00:20:48,110 --> 00:20:51,210 temperatures, close to absolute zero, delta L will 263 00:20:51,210 --> 00:20:54,935 also be proportional to the change in temperature. 264 00:21:00,830 --> 00:21:04,200 Thermal expansion is one of those properties which 265 00:21:04,200 --> 00:21:05,600 thermodynamically must go to zero as 266 00:21:05,600 --> 00:21:07,330 temperature goes to zero. 267 00:21:07,330 --> 00:21:10,740 So the thermal expansion coefficient that we've yet to 268 00:21:10,740 --> 00:21:15,450 define will be a function of temperature as we get down to 269 00:21:15,450 --> 00:21:20,400 very low temperatures on the order of a few degrees Kelvin. 270 00:21:20,400 --> 00:21:24,140 So we can define then the linear thermal expansion 271 00:21:24,140 --> 00:21:37,570 coefficient for this rod as 1 over L, delta L over delta T. 272 00:21:37,570 --> 00:21:43,180 So this is something that will give us the fractional change 273 00:21:43,180 --> 00:21:45,470 of lengths. 274 00:21:45,470 --> 00:21:53,410 And what we can do now in general is to say that if we 275 00:21:53,410 --> 00:21:57,140 create a general strain, not a tensile strain, we have a 276 00:21:57,140 --> 00:22:06,580 general strain epsilon 11 plus epsilon 12, epsilon 13, and so 277 00:22:06,580 --> 00:22:10,740 on, has to be symmetric. 278 00:22:10,740 --> 00:22:13,840 So epsilon 21 is equal to epsilon 12. 279 00:22:21,500 --> 00:22:28,690 We'll say that this tensor epsilon ij will be written in 280 00:22:28,690 --> 00:22:34,140 general for the three-dimensional case as a 281 00:22:34,140 --> 00:22:45,850 second-rank tensor aij times a scalar quantity delta T. 282 00:22:45,850 --> 00:22:47,700 The strain is the field tensor. 283 00:22:47,700 --> 00:22:48,980 That's the response. 284 00:22:48,980 --> 00:22:52,460 The stimulus that creates this response is an incremental 285 00:22:52,460 --> 00:22:53,930 change in temperature. 286 00:22:53,930 --> 00:22:59,380 But as this is a second-rank tensor and this is a scalar, 287 00:22:59,380 --> 00:23:02,370 an array of coefficients in which we multiply every 288 00:23:02,370 --> 00:23:06,200 element of a tensor by a scalar is also a tensor. 289 00:23:06,200 --> 00:23:07,950 So this is called the linear thermal 290 00:23:07,950 --> 00:23:09,530 expansion coefficient tensor. 291 00:23:23,600 --> 00:23:36,100 And it is a symmetric tensor by definition since a measure 292 00:23:36,100 --> 00:23:42,160 of true strain is by definition a symmetric tensor. 293 00:23:42,160 --> 00:23:45,490 So this is the linear thermal expansion coefficient tensor. 294 00:23:53,346 --> 00:23:54,645 AUDIENCE: Quick question. 295 00:23:54,645 --> 00:23:55,346 PROFESSOR: Yes. 296 00:23:55,346 --> 00:23:58,068 AUDIENCE: [INAUDIBLE] the temperature gradient 297 00:23:58,068 --> 00:23:58,791 [INAUDIBLE] 298 00:23:58,791 --> 00:24:00,960 and talk about the directionality say between 299 00:24:00,960 --> 00:24:04,350 here and here because of the temperature [INAUDIBLE]? 300 00:24:04,350 --> 00:24:05,180 PROFESSOR: Oh, yeah. 301 00:24:05,180 --> 00:24:05,310 Sure. 302 00:24:05,310 --> 00:24:08,470 Sure if you wanted to do that. 303 00:24:08,470 --> 00:24:11,210 That assumes that the thermal conductivity would be very 304 00:24:11,210 --> 00:24:15,560 small so that the temperature inside the body would not 305 00:24:15,560 --> 00:24:17,130 attempt to [INAUDIBLE]. 306 00:24:17,130 --> 00:24:17,606 Yeah. 307 00:24:17,606 --> 00:24:18,082 Yeah. 308 00:24:18,082 --> 00:24:20,840 In the same way that you could create strains that are in 309 00:24:20,840 --> 00:24:22,090 homogeneous as well. 310 00:24:31,935 --> 00:24:34,640 We'll now for the first time start talking not in terms of 311 00:24:34,640 --> 00:24:39,590 abstractions but in terms of some real physical properties. 312 00:24:39,590 --> 00:24:44,140 What is the range of linear thermal 313 00:24:44,140 --> 00:24:46,780 expansion coefficient tensors? 314 00:24:46,780 --> 00:24:49,380 And unlike many physical properties, that's an easy one 315 00:24:49,380 --> 00:24:58,350 to answer, 10 to the minus 6 degree C per degree C. 316 00:24:58,350 --> 00:25:02,290 Materials have a very, very small range linear thermal 317 00:25:02,290 --> 00:25:03,510 expansion coefficients. 318 00:25:03,510 --> 00:25:06,910 And let me give you some examples for real materials. 319 00:25:06,910 --> 00:25:12,930 They go up to maybe 10 to the minus 5 for a very soft weakly 320 00:25:12,930 --> 00:25:15,460 bonded materials. 321 00:25:15,460 --> 00:25:17,630 But the for metals-- 322 00:25:22,590 --> 00:25:25,450 and I'm giving you a single number now because these are 323 00:25:25,450 --> 00:25:28,530 averages for polycrystalline materials-- 324 00:25:28,530 --> 00:25:31,150 the value of A times 10 to the 6. 325 00:25:31,150 --> 00:25:37,240 For lead, a low- melting metal, is 28, 2.8 times 10 to 326 00:25:37,240 --> 00:25:39,480 the minus 5. 327 00:25:39,480 --> 00:25:42,560 For copper, 18. 328 00:25:42,560 --> 00:25:45,350 For iron, 12. 329 00:25:45,350 --> 00:25:49,500 Four tungsten, a very refractory metal, 5. 330 00:25:49,500 --> 00:25:54,810 And for diamond, a very strongly bonded materials, 331 00:25:54,810 --> 00:26:04,810 it's 0.89 times 10 minus 6. 332 00:26:04,810 --> 00:26:22,720 If we look at compounds, again, the average linear 333 00:26:22,720 --> 00:26:27,180 thermal expansion coefficient times 10 to the 6 for 334 00:26:27,180 --> 00:26:29,050 polycrystalline material. 335 00:26:29,050 --> 00:26:34,930 For aluminum bromide, hardly a technology material of 336 00:26:34,930 --> 00:26:39,370 commerce, but in light of this because this has one of the 337 00:26:39,370 --> 00:26:43,590 largest linear thermal expansion coefficients of any 338 00:26:43,590 --> 00:26:47,250 material, very weakly bonded to compound. 339 00:26:47,250 --> 00:26:53,600 For a more typical ionic compound, NaCL, 40 times 10 to 340 00:26:53,600 --> 00:26:56,530 the minus 6. 341 00:26:56,530 --> 00:27:02,200 For calcium fluoride, this has a bivalent cation in it, so 342 00:27:02,200 --> 00:27:06,750 the material is stronger, more strongly bonded, 20. 343 00:27:06,750 --> 00:27:12,860 For MgO, both a divalent cation and the divalent anion, 344 00:27:12,860 --> 00:27:15,580 as you would expect, the thermal expansion coefficient 345 00:27:15,580 --> 00:27:17,250 drop still more. 346 00:27:17,250 --> 00:27:23,990 For Al2O3, a trivalent cation that goes down to 8.8, and 347 00:27:23,990 --> 00:27:25,840 that's really an average because this 348 00:27:25,840 --> 00:27:27,720 is a hexagonal material. 349 00:27:27,720 --> 00:27:33,670 And a second-rank property tensor for a hexagonal crystal 350 00:27:33,670 --> 00:27:39,380 has to have two equal diagonal elements and one independent 351 00:27:39,380 --> 00:27:41,695 diagonal element when you diagonalize the tensor. 352 00:27:45,420 --> 00:27:54,870 Spinal, MgAl2O4 this is representative of all of the 353 00:27:54,870 --> 00:27:58,350 ferrites for example, 7.6. 354 00:27:58,350 --> 00:28:01,550 Silicon carbide, a very refractory 355 00:28:01,550 --> 00:28:07,635 covalent compound, 4.7. 356 00:28:07,635 --> 00:28:12,980 For glasses, here's a surprise. 357 00:28:12,980 --> 00:28:16,120 You might think that a glass might have about the same 358 00:28:16,120 --> 00:28:20,010 linear expansion coefficient as the crystalline form of the 359 00:28:20,010 --> 00:28:21,610 same composition. 360 00:28:21,610 --> 00:28:30,520 But a sodium calcium silicon oxide glass, so called 361 00:28:30,520 --> 00:28:34,840 soda-lime silicate glass, is 9, fairly low. 362 00:28:34,840 --> 00:28:44,640 A borosilicate glass, like the Vicor that we love to make 363 00:28:44,640 --> 00:28:48,620 glass apparatus out of, 4.5. 364 00:28:48,620 --> 00:28:56,210 And one of the near record holder is fused silica, 365 00:28:56,210 --> 00:29:01,920 silicio glass, and here a measly value of 0.5. 366 00:29:01,920 --> 00:29:06,630 So glasses have very low linear thermal expansion 367 00:29:06,630 --> 00:29:10,970 coefficients compared to crystallize materials that are 368 00:29:10,970 --> 00:29:12,790 predominantly ionic. 369 00:29:12,790 --> 00:29:15,830 And the reason lies in the nature of the structure. 370 00:29:15,830 --> 00:29:19,790 Something like MgO has ordered ions, and when you heat it up, 371 00:29:19,790 --> 00:29:22,280 it expands isotropically. 372 00:29:22,280 --> 00:29:26,770 A glass is this random network of edge-shared polyhedra. 373 00:29:26,770 --> 00:29:30,210 So when you heat it up, yes, the dimensions of the 374 00:29:30,210 --> 00:29:33,290 polyhedra increase in proportion to the change of 375 00:29:33,290 --> 00:29:34,150 temperature. 376 00:29:34,150 --> 00:29:38,450 But the framework because it's so meandering and open can 377 00:29:38,450 --> 00:29:42,080 buckle to accommodate the increased size of the 378 00:29:42,080 --> 00:29:44,240 tetrahedra that are in the linkage. 379 00:29:44,240 --> 00:29:47,060 So the result is the network buckles, but the overall 380 00:29:47,060 --> 00:29:52,050 microscopic thermal expansion is very, very slight even 381 00:29:52,050 --> 00:29:56,030 though the tetrahedra are changing dimensions in the 382 00:29:56,030 --> 00:30:00,190 same degree that they are in these crystalline inorganic 383 00:30:00,190 --> 00:30:01,440 nonmetallic materials. 384 00:30:06,310 --> 00:30:10,360 Let me give you a handout not that it's up to date, but 385 00:30:10,360 --> 00:30:15,030 there was a very nice, convenient one-page article in 386 00:30:15,030 --> 00:30:18,610 the Journal of the American Ceramic Society a number of 387 00:30:18,610 --> 00:30:22,330 years ago that lists thermal expansion coefficients for low 388 00:30:22,330 --> 00:30:24,650 expansion oxides. 389 00:30:24,650 --> 00:30:27,260 So let me pass that around and let you take off 390 00:30:27,260 --> 00:30:28,460 for copy with that. 391 00:30:28,460 --> 00:30:30,975 And that gives some examples of the value of this property 392 00:30:30,975 --> 00:30:35,450 for, again, for polycrystalline materials, 393 00:30:35,450 --> 00:30:38,130 which if you looked at a single crystal would show 394 00:30:38,130 --> 00:30:40,350 anisotropy. 395 00:30:40,350 --> 00:30:44,100 And here you'll find a few materials in the list near the 396 00:30:44,100 --> 00:30:47,470 top they're arranged in order of increasing linear thermal 397 00:30:47,470 --> 00:30:53,090 expansion, you'll find some that are just about zero when 398 00:30:53,090 --> 00:30:55,830 you have a polycrystalline form of the material. 399 00:30:55,830 --> 00:31:01,160 And you'll have some that have this very unusual behavior 400 00:31:01,160 --> 00:31:03,920 where the linear thermal expansion coefficient is 401 00:31:03,920 --> 00:31:05,200 actually negative. 402 00:31:05,200 --> 00:31:06,950 So let me pass this around for reference. 403 00:31:12,570 --> 00:31:17,980 Now this may seem to be a curiosity that you can fine 404 00:31:17,980 --> 00:31:21,330 for a polycrystalline material for which the individual 405 00:31:21,330 --> 00:31:27,410 grains expand anisotropically a bulk linear thermal 406 00:31:27,410 --> 00:31:31,290 expansion coefficient of zero. 407 00:31:31,290 --> 00:31:34,920 Why should one care other than that being a curious result? 408 00:31:34,920 --> 00:31:36,170 Can anybody guess? 409 00:31:39,272 --> 00:31:40,700 AUDIENCE: Can you say that again? 410 00:31:40,700 --> 00:31:45,090 PROFESSOR: That these are materials which in a 411 00:31:45,090 --> 00:31:48,650 polycrystalline form with random grain orientation have 412 00:31:48,650 --> 00:31:51,840 a linear thermal expansion coefficient this is 413 00:31:51,840 --> 00:31:53,090 essentially zero? 414 00:31:55,960 --> 00:31:58,410 AUDIENCE: [INAUDIBLE]? 415 00:31:58,410 --> 00:31:59,390 PROFESSOR: Yes. 416 00:31:59,390 --> 00:32:00,380 Exactly. 417 00:32:00,380 --> 00:32:07,120 The materials out of which you build furnace linings and 418 00:32:07,120 --> 00:32:11,110 tanks for melting glass or smelting steel have to be made 419 00:32:11,110 --> 00:32:12,690 out of polycrystalline materials. 420 00:32:12,690 --> 00:32:15,680 You can't have a monolithic single crystal that's large 421 00:32:15,680 --> 00:32:17,330 enough to melt a significant amount of 422 00:32:17,330 --> 00:32:19,070 glass in for example. 423 00:32:19,070 --> 00:32:23,110 And being polycrystalline, when the material expands, 424 00:32:23,110 --> 00:32:27,820 there is a great deal of stress between neighboring 425 00:32:27,820 --> 00:32:30,530 grains because they're randomly oriented. 426 00:32:30,530 --> 00:32:35,400 If the bulk microscopic thermal expansion coefficient 427 00:32:35,400 --> 00:32:38,760 is zero, that is going to be refractory that is very, very 428 00:32:38,760 --> 00:32:40,630 resistant to thermal shock. 429 00:32:40,630 --> 00:32:45,430 When you heat it up, you don't find intergranular cracking. 430 00:32:45,430 --> 00:32:55,040 So the materials that have essentially zero spatially 431 00:32:55,040 --> 00:32:57,910 average thermal expansion coefficients are very 432 00:32:57,910 --> 00:32:59,620 attractive for refractories. 433 00:32:59,620 --> 00:33:03,260 You can pick up any issue of the Journal of 434 00:33:03,260 --> 00:33:04,580 the American Ceramics-- 435 00:33:04,580 --> 00:33:06,360 not the journal, but the Bulletin of the American 436 00:33:06,360 --> 00:33:12,120 Ceramic Society and you will find refractory companies 437 00:33:12,120 --> 00:33:16,700 hustling materials like Cordierite, which is a 438 00:33:16,700 --> 00:33:21,710 silicate material that on average has one positive 439 00:33:21,710 --> 00:33:24,490 thermal expansion coefficient, one negative thermal expansion 440 00:33:24,490 --> 00:33:25,350 coefficient. 441 00:33:25,350 --> 00:33:27,810 And the volume change of the individual grains is 442 00:33:27,810 --> 00:33:29,380 essentially zero. 443 00:33:29,380 --> 00:33:36,385 So that makes a dandy material for refractories. 444 00:33:42,030 --> 00:33:47,210 Cordierite has another interesting property that is a 445 00:33:47,210 --> 00:33:49,825 really nice example of anisotropy 446 00:33:49,825 --> 00:33:53,340 of a physical property. 447 00:33:53,340 --> 00:33:55,995 If I ask you to rattle off some properties which you know 448 00:33:55,995 --> 00:33:59,230 to be very anisotropic, one of them that you wouldn't think 449 00:33:59,230 --> 00:34:02,600 of is color. 450 00:34:02,600 --> 00:34:05,470 Your t-shirt is pink, so how can that be 451 00:34:05,470 --> 00:34:06,790 a function of direction? 452 00:34:06,790 --> 00:34:09,460 Well your t-shirt is not a single crystal. 453 00:34:09,460 --> 00:34:11,920 And their single crystals which if you hold them up to 454 00:34:11,920 --> 00:34:17,760 light and pass through a plain polarized beam of light the 455 00:34:17,760 --> 00:34:21,210 crystal has a very different color for one direction of 456 00:34:21,210 --> 00:34:23,360 polarization then another. 457 00:34:23,360 --> 00:34:26,580 And these colors, interestingly, very often are 458 00:34:26,580 --> 00:34:28,630 strikingly different. 459 00:34:28,630 --> 00:34:32,690 They're crystals that are green in light polarized in 460 00:34:32,690 --> 00:34:36,670 one direction and red for light polarized in the 461 00:34:36,670 --> 00:34:38,679 opposite direction. 462 00:34:38,679 --> 00:34:44,969 And there is in the old Icelandic sagas-- 463 00:34:44,969 --> 00:34:46,310 few people have heard of them. 464 00:34:46,310 --> 00:34:52,460 The Icelandic sagas which were written in about the year 500 465 00:34:52,460 --> 00:34:56,550 are one of the early forms of Western literature that is 466 00:34:56,550 --> 00:35:00,730 really great, enduring world-class literature. 467 00:35:00,730 --> 00:35:05,310 In the sagas, there's one episode where Thor is sleeping 468 00:35:05,310 --> 00:35:09,820 and Olaf comes along and steals his sunstone. 469 00:35:09,820 --> 00:35:13,690 And Thor is upset he takes his battle as and cut 470 00:35:13,690 --> 00:35:15,880 Olaf's head in twain. 471 00:35:15,880 --> 00:35:18,240 And people puzzled for hundreds of 472 00:35:18,240 --> 00:35:20,010 years, what is sunstone. 473 00:35:23,190 --> 00:35:26,080 This doesn't compute in today's age. 474 00:35:26,080 --> 00:35:29,950 And finally a Norwegian archaeologist came up with a 475 00:35:29,950 --> 00:35:36,240 hypothesis that in Iceland there are remarkably large and 476 00:35:36,240 --> 00:35:41,920 remarkably transparent and perfect single crystals of 477 00:35:41,920 --> 00:35:43,420 Cordierite. 478 00:35:43,420 --> 00:35:46,920 And Cordierite is strikingly pleochroic. 479 00:35:46,920 --> 00:35:49,800 That's color that is a function of direction. 480 00:35:49,800 --> 00:35:54,770 And what this archaeologist theorized is that the Vikings 481 00:35:54,770 --> 00:35:59,670 would take a sunstone when they went sailing on the North 482 00:35:59,670 --> 00:36:04,120 Sea, which is notoriously damp and overcast and gray, and 483 00:36:04,120 --> 00:36:07,050 your principal means of navigation was the sun. 484 00:36:07,050 --> 00:36:09,890 But when the sky was overcast, you couldn't see the sun. 485 00:36:09,890 --> 00:36:13,290 But the sun as it comes through the cloud gives you 486 00:36:13,290 --> 00:36:15,890 light that's very strongly polarized. 487 00:36:15,890 --> 00:36:20,460 So the theory is that the Vikings would have very 488 00:36:20,460 --> 00:36:23,560 perfect crystals of transparent Cordierite, would 489 00:36:23,560 --> 00:36:28,340 hold them up to the sky, and turn them around until the 490 00:36:28,340 --> 00:36:33,100 Cordierite crystal lit up a bright, golden sunny yellow. 491 00:36:33,100 --> 00:36:35,020 What better name for the sunstone. 492 00:36:35,020 --> 00:36:38,310 You found the sun from the direction of polarization of 493 00:36:38,310 --> 00:36:40,950 the light scattered from the cloud when you've got the 494 00:36:40,950 --> 00:36:42,800 crystal oriented just so. 495 00:36:42,800 --> 00:36:46,180 In a different orientation, it would look murkier, look sort 496 00:36:46,180 --> 00:36:49,940 of a bluish purple for the other principal direction of 497 00:36:49,940 --> 00:36:51,560 the birefringence. 498 00:36:51,560 --> 00:36:57,180 So there is putting crystal anisotropy hundreds of years 499 00:36:57,180 --> 00:37:02,390 ago to a very useful purpose to navigate on cloudy days. 500 00:37:02,390 --> 00:37:04,810 Unfortunately, the Vikings used it to sail down the 501 00:37:04,810 --> 00:37:07,750 British coast and sacked the next village. 502 00:37:07,750 --> 00:37:12,590 So it shows you that even the most simple of science can be 503 00:37:12,590 --> 00:37:16,460 put to application in war research. 504 00:37:16,460 --> 00:37:19,300 Anyway, won't go there anymore. 505 00:37:19,300 --> 00:37:23,870 So index of refraction and color can also be a strong 506 00:37:23,870 --> 00:37:32,040 function of direction 507 00:37:32,040 --> 00:37:38,540 The numbers that I put on the blackboard suggest that the 508 00:37:38,540 --> 00:37:43,390 magnitude of thermal expansion coefficients are influenced 509 00:37:43,390 --> 00:37:47,060 very strongly by the strength of the interatomic bonding. 510 00:37:47,060 --> 00:37:50,420 And this shows up in a very, very striking way if you group 511 00:37:50,420 --> 00:37:56,810 together classes of comparable materials. 512 00:37:56,810 --> 00:38:01,190 And if you plot the value of the linear thermal expansion 513 00:38:01,190 --> 00:38:08,100 coefficient in units of 10 to the minus 6 per degree C as a 514 00:38:08,100 --> 00:38:14,510 function of the melting point in degrees Kelvin, the numbers 515 00:38:14,510 --> 00:38:20,390 range from about 400 K for materials like sulfur up to 516 00:38:20,390 --> 00:38:24,160 about 3,700 for tungsten. 517 00:38:24,160 --> 00:38:28,120 And the variation is so beautiful it 518 00:38:28,120 --> 00:38:30,690 could make you cry. 519 00:38:30,690 --> 00:38:34,330 It goes as the inverse of the melting point. 520 00:38:34,330 --> 00:38:40,500 And way up here are materials like for sulfur and lithium. 521 00:38:40,500 --> 00:38:44,420 And the way out here are materials like tungsten. 522 00:38:44,420 --> 00:38:50,500 And their relation is given by a equals 0.020 over the 523 00:38:50,500 --> 00:38:52,950 melting point in degrees Kelvin. 524 00:38:57,150 --> 00:39:02,420 So very strong correlation between melting point and 525 00:39:02,420 --> 00:39:05,510 linear expansion for simple elements. 526 00:39:05,510 --> 00:39:09,790 If you do the same thing for compounds, you find a similar 527 00:39:09,790 --> 00:39:15,555 parabolic relation except it is offset. 528 00:39:22,450 --> 00:39:33,190 For oxides and halides, many of them with the rock salt 529 00:39:33,190 --> 00:39:35,420 structure, so they are isotropic. 530 00:39:35,420 --> 00:39:40,190 And again, if you plot here the melting point in degrees 531 00:39:40,190 --> 00:39:44,520 Kelvin, the numbers here range up to 60 times 10 532 00:39:44,520 --> 00:39:46,320 to the minus 6. 533 00:39:46,320 --> 00:39:53,110 And again, a variation with 1 over T, but 534 00:39:53,110 --> 00:39:55,030 offset from the origin. 535 00:39:55,030 --> 00:40:02,570 And the curve here is that a equals 0.038 over T, the 536 00:40:02,570 --> 00:40:09,280 melting point T sub m in Kelvin, minus 7.0 times 10 to 537 00:40:09,280 --> 00:40:10,530 the minus 6. 538 00:40:17,730 --> 00:40:21,690 Finally, let me finish with some data for single crystals 539 00:40:21,690 --> 00:40:26,200 that provide examples of and anisotropy. 540 00:40:26,200 --> 00:40:30,130 These are all hexagonal materials. 541 00:40:30,130 --> 00:40:33,680 And I will give you the-- 542 00:40:33,680 --> 00:40:34,600 not all hexagonal. 543 00:40:34,600 --> 00:40:35,230 I take that back. 544 00:40:35,230 --> 00:40:37,240 But they're all uniaxial crystals. 545 00:40:37,240 --> 00:40:42,870 The value of the thermal expansion coefficient along C 546 00:40:42,870 --> 00:40:46,270 and the thermal expansion coefficient that is parallel 547 00:40:46,270 --> 00:40:49,100 to C, so these are the elements that would be in the 548 00:40:49,100 --> 00:40:53,850 diagonal isothermal expansion tensor. 549 00:40:53,850 --> 00:41:00,240 So for Al2O3, which is a hexagonal rhombohedral oxide, 550 00:41:00,240 --> 00:41:06,330 the two expansion coefficients are 8.39 and 9.0 times 10 to 551 00:41:06,330 --> 00:41:09,840 the minus 6, not terribly anisotropic. 552 00:41:09,840 --> 00:41:13,550 But the structure of alumina is a close-packed arrangement 553 00:41:13,550 --> 00:41:18,890 of oxygen with aluminum filling 2/3 of the available 554 00:41:18,890 --> 00:41:20,700 octahedral holes. 555 00:41:20,700 --> 00:41:26,460 So it's hexagonal only because of the sites that are filled 556 00:41:26,460 --> 00:41:28,490 in the array. 557 00:41:28,490 --> 00:41:31,540 For TiO2, which is [INAUDIBLE] 558 00:41:31,540 --> 00:41:39,270 and therefore also uniaxial, 6.8 perpendicular to C; 8.3 559 00:41:39,270 --> 00:41:49,730 parallel to C. For zirconium silicate, 3.7 and 6.2, almost 560 00:41:49,730 --> 00:41:51,500 a 2:1 difference. 561 00:41:51,500 --> 00:42:01,510 For the quartz form of silica, which is hexagonal, 14 562 00:42:01,510 --> 00:42:07,550 perpendicular to C; 9 parallel to C. 563 00:42:07,550 --> 00:42:10,700 For carbon, the graphite form, a layer structure-- 564 00:42:15,030 --> 00:42:17,220 you can almost guess how this is going to turn out. 565 00:42:17,220 --> 00:42:21,100 It's going to be humongous perpendicular to the lawyers 566 00:42:21,100 --> 00:42:24,660 and very small within the plane of these tightly bonded 567 00:42:24,660 --> 00:42:26,370 hexagonal nets. 568 00:42:26,370 --> 00:42:32,630 And that is indeed the case, 1 perpendicular to C; 27 569 00:42:32,630 --> 00:42:38,090 parallel to C. So there is an anisotropy of a factor of 27. 570 00:42:38,090 --> 00:42:41,480 A few more. 571 00:42:41,480 --> 00:42:55,220 Aluminum titanate, perpendicular to C minus 2.6; 572 00:42:55,220 --> 00:43:01,636 parallel to C 11.5. 573 00:43:01,636 --> 00:43:06,180 A very well known example of anisotropy, extreme 574 00:43:06,180 --> 00:43:09,450 anisotropy, where one principal coefficient is 575 00:43:09,450 --> 00:43:14,840 positive and two are negative is the calcite forms calcium 576 00:43:14,840 --> 00:43:22,100 carbonate, minus 6 and 25. 577 00:43:22,100 --> 00:43:26,250 For zinc metal, a hexagonal close-packed metal, surprising 578 00:43:26,250 --> 00:43:30,930 degree of anisotropy, 14 and 64. 579 00:43:30,930 --> 00:43:37,210 And for tellurium, which is a chain structure like selenium, 580 00:43:37,210 --> 00:43:44,040 27 perpendicular to C and minus 1.6 in a direction 581 00:43:44,040 --> 00:43:47,640 parallel to C. 582 00:43:47,640 --> 00:43:51,080 Even simple hexagonal close-packed metals can do 583 00:43:51,080 --> 00:43:53,430 strange things at low temperatures. 584 00:43:53,430 --> 00:43:57,200 For zinc, the expansion coefficients perpendicular to 585 00:43:57,200 --> 00:44:04,100 C and parallel to C as a function of temperature go at 586 00:44:04,100 --> 00:44:09,850 300 degrees C from 13 and 64. 587 00:44:09,850 --> 00:44:17,046 At 150, the values have dropped to 8 and 65. 588 00:44:20,470 --> 00:44:26,230 At 60 degrees K perpendicular to C, the thermal expansion is 589 00:44:26,230 --> 00:44:30,080 negative and has not dropped terribly much of at all 590 00:44:30,080 --> 00:44:34,000 parallel to C. So this is a hexagonal close-packed 591 00:44:34,000 --> 00:44:34,730 structure lot. 592 00:44:34,730 --> 00:44:37,720 A lot of action and strange things going on in the plane 593 00:44:37,720 --> 00:44:42,290 at the close-packed layers, but not very much change in a 594 00:44:42,290 --> 00:44:44,140 direction perpendicular to the layers. 595 00:45:02,570 --> 00:45:05,000 Let me raise one more issue. 596 00:45:07,620 --> 00:45:11,600 If we were to look at some of these strangely anisotropic 597 00:45:11,600 --> 00:45:15,220 materials which had negative thermal expansion coefficients 598 00:45:15,220 --> 00:45:19,720 in one direction and positive in another, what would the 599 00:45:19,720 --> 00:45:22,810 representation quadric look like? 600 00:45:22,810 --> 00:45:26,430 Suppose we looked at calcite, for example, which has a large 601 00:45:26,430 --> 00:45:29,120 negative thermal expansion coefficient. 602 00:45:29,120 --> 00:45:32,630 And if we look at the thermal expansion quadric when the 603 00:45:32,630 --> 00:45:34,840 tensor was referred to the principal 604 00:45:34,840 --> 00:45:40,415 axes, parallel to C-- 605 00:45:40,415 --> 00:45:47,220 well, the thermal expansion tensor aij along its principal 606 00:45:47,220 --> 00:45:53,470 axes would be along A1 in a direction perpendicular to C, 607 00:45:53,470 --> 00:45:59,520 it would be for the data given here, minus 6, zero, zero, 608 00:45:59,520 --> 00:46:03,850 zero, minus 6, zero, zero, zero. 609 00:46:03,850 --> 00:46:08,135 And A33 is 25. 610 00:46:10,970 --> 00:46:13,100 So that's very, very anisotropic. 611 00:46:15,850 --> 00:46:17,470 But what is this going to be? 612 00:46:17,470 --> 00:46:21,640 This is going to be a quadric that has the shape of an 613 00:46:21,640 --> 00:46:24,160 hyperboloid of two sheets. 614 00:46:24,160 --> 00:46:26,330 So it's going to look like this. 615 00:46:32,630 --> 00:46:35,650 And there'll be an asymptote like this and then 616 00:46:35,650 --> 00:46:37,145 asymptote like this. 617 00:46:44,020 --> 00:46:47,330 Along the asymptote, the radius of 618 00:46:47,330 --> 00:46:50,500 the quadric is infinite. 619 00:46:50,500 --> 00:46:55,300 So therefore along these directions the 620 00:46:55,300 --> 00:47:00,670 strain is equal to zero. 621 00:47:00,670 --> 00:47:05,140 Strain goes as 1 over the radius squared. 622 00:47:10,460 --> 00:47:15,350 In this range of directions here, the radius is imaginary. 623 00:47:20,300 --> 00:47:23,680 But remember that the strain is given by 1 624 00:47:23,680 --> 00:47:26,220 over the radius squared. 625 00:47:26,220 --> 00:47:33,040 So this says that the strain is negative, so the material 626 00:47:33,040 --> 00:47:35,190 has contracted. 627 00:47:35,190 --> 00:47:41,710 And finally, in this range of directions where the radius is 628 00:47:41,710 --> 00:47:46,320 finite and positive, if the value of strain is 1 over 629 00:47:46,320 --> 00:47:50,270 radius squared, this says that the maximum deformation is 630 00:47:50,270 --> 00:47:54,380 along the direction of the C axis, and it's positive. 631 00:47:54,380 --> 00:47:57,870 The material expands along C; the minimum radius of the 632 00:47:57,870 --> 00:48:02,540 quadric corresponds to the maximum thermal expansion. 633 00:48:06,660 --> 00:48:09,980 So let me close with a question. 634 00:48:09,980 --> 00:48:14,410 Along the asymptote of the quadric, the strain is zero. 635 00:48:16,940 --> 00:48:20,450 Does this mean that this direction in a crystal of 636 00:48:20,450 --> 00:48:24,730 calcite does not move, no deformation at all when you 637 00:48:24,730 --> 00:48:25,980 heat it up? 638 00:48:31,681 --> 00:48:33,669 I see faint shakes of the head. 639 00:48:33,669 --> 00:48:37,148 I don't know if that's just in awe of what's going on here or 640 00:48:37,148 --> 00:48:39,633 an opinion on the question. 641 00:48:39,633 --> 00:48:42,615 Do you think it'll be no strain? 642 00:48:42,615 --> 00:48:45,100 AUDIENCE: It would be [INAUDIBLE]. 643 00:48:45,100 --> 00:48:46,591 PROFESSOR: Good answer. 644 00:48:46,591 --> 00:48:51,780 Let's remember a property of the representation quadric. 645 00:48:51,780 --> 00:48:56,650 The direction of what happens is normal to the surface of 646 00:48:56,650 --> 00:48:57,770 the quadric. 647 00:48:57,770 --> 00:49:01,420 The thing that is happening here is the displacement; ui 648 00:49:01,420 --> 00:49:06,440 is given by epsilon ij times U sub j. 649 00:49:06,440 --> 00:49:10,260 So when we look in this direction, the displacement, 650 00:49:10,260 --> 00:49:16,010 which would be normal to the surface of the quadric, gives 651 00:49:16,010 --> 00:49:19,790 us this as a displacement. 652 00:49:19,790 --> 00:49:23,640 So along the asymptotes of the quadric, you can't say that 653 00:49:23,640 --> 00:49:27,730 there's no deformation, but that the deformation 654 00:49:27,730 --> 00:49:31,110 corresponds to pure rotation and not any 655 00:49:31,110 --> 00:49:34,260 fractional change of length. 656 00:49:34,260 --> 00:49:37,850 In these directions, the length changes, but it is a 657 00:49:37,850 --> 00:49:40,790 decrease in length within the asymptotes. 658 00:49:40,790 --> 00:49:48,270 For those directions, the strain is an extension. 659 00:49:48,270 --> 00:49:50,910 And again, if you want to know the direction of the 660 00:49:50,910 --> 00:49:54,810 displacement, you find the normal to the surface of the 661 00:49:54,810 --> 00:49:58,770 quadric in the direction of interest, and that's the 662 00:49:58,770 --> 00:50:01,040 direction in which the radius vector points. 663 00:50:08,800 --> 00:50:09,040 All right. 664 00:50:09,040 --> 00:50:11,060 That's I think a good place to quit. 665 00:50:15,630 --> 00:50:21,460 I think I'll say a little bit more after the break about the 666 00:50:21,460 --> 00:50:25,940 atomistic reason for increases in interionic separations, 667 00:50:25,940 --> 00:50:29,140 which we can get some insight into from a simple model. 668 00:50:29,140 --> 00:50:34,150 But before we disperse, I don't know if everybody got a 669 00:50:34,150 --> 00:50:38,230 copy of problem set 13. 670 00:50:38,230 --> 00:50:41,200 If anybody didn't, there's some extra copies up here. 671 00:50:41,200 --> 00:50:45,640 But I will with great pleasure pass out to you problems set 672 00:50:45,640 --> 00:50:49,730 number 14, which has served two purposes. 673 00:50:49,730 --> 00:50:52,780 One is to have you show for yourself that the trace of a 674 00:50:52,780 --> 00:50:54,620 second-rank tensor-- 675 00:50:54,620 --> 00:50:58,210 I called it second-order tensor-- 676 00:50:58,210 --> 00:51:01,880 second-rank tensor the sum of the diagonal elements is 677 00:51:01,880 --> 00:51:05,440 invariant for a change of reference axes for a general 678 00:51:05,440 --> 00:51:08,700 direction cosine scheme for the change of axes. 679 00:51:08,700 --> 00:51:12,510 And then the other two questions are to give you some 680 00:51:12,510 --> 00:51:17,490 practice in diagonalizing a second-rank property tensor 681 00:51:17,490 --> 00:51:19,920 which is not in diagonal form. 682 00:51:19,920 --> 00:51:21,920 And I asked you to do this in two ways. 683 00:51:21,920 --> 00:51:26,160 One is by direct solution of the secular equation and 684 00:51:26,160 --> 00:51:28,000 finding the eigenvectors. 685 00:51:28,000 --> 00:51:30,660 And then the third example, which is for a completely 686 00:51:30,660 --> 00:51:33,990 general tensor, to do this by the method of successive 687 00:51:33,990 --> 00:51:35,580 approximations. 688 00:51:35,580 --> 00:51:38,240 And you'll find after a couple of iterations you get fairly 689 00:51:38,240 --> 00:51:39,630 close to convergence. 690 00:51:39,630 --> 00:51:42,480 So I shall pass this around. 691 00:51:42,480 --> 00:51:46,470 And again, I think there's something to be 692 00:51:46,470 --> 00:51:47,720 learned from that. 693 00:51:51,370 --> 00:51:53,340 Let's take our stretch then. 694 00:51:57,190 --> 00:52:01,936 I'm sure that lecture involved a lot of stress and strain. 695 00:52:01,936 --> 00:52:04,880 Ha, ha. 696 00:52:04,880 --> 00:52:06,340 Let's resume in 10 minutes. 697 00:52:06,340 --> 00:52:12,590 And we'll then move on to some additional higher rank tensor 698 00:52:12,590 --> 00:52:17,150 properties, which will be very, very interesting. 699 00:52:17,150 --> 00:52:18,400 Go for it.