1 00:00:07,110 --> 00:00:09,380 PROFESSOR: OK, its always disappointing when, just as 2 00:00:09,380 --> 00:00:12,930 you're ready to tie everything up in a nice, neat little 3 00:00:12,930 --> 00:00:16,950 package and leave the room to a round of applause, you screw 4 00:00:16,950 --> 00:00:18,330 up the last little detail. 5 00:00:18,330 --> 00:00:21,010 Anyway, what's wrong here is, I said for an arbitrary 6 00:00:21,010 --> 00:00:24,750 rotation around X3, and I'm comforted only by the fact 7 00:00:24,750 --> 00:00:28,120 that nobody else caught this little glitch either, the 8 00:00:28,120 --> 00:00:37,970 answer elements are a11, a11, a33, a12, not 0, and 9 00:00:37,970 --> 00:00:43,810 -a12, 0, 0, 0, 0. 10 00:00:43,810 --> 00:00:47,000 So there's a skew-symmetric form when there's just a 11 00:00:47,000 --> 00:00:48,390 single rotation axis. 12 00:00:48,390 --> 00:00:56,870 So this would be asymmetry n and this would be one of a 13 00:00:56,870 --> 00:00:59,480 family of two different symmetries, the dihedral 14 00:00:59,480 --> 00:01:03,810 groups n22, the two-fold axes would require that these be 15 00:01:03,810 --> 00:01:10,420 equal, so, 0, 0, 0, a11, 0, 0, 0, a33. 16 00:01:10,420 --> 00:01:14,270 So this is not the form of the tensor for a four-fold axis, 17 00:01:14,270 --> 00:01:18,390 it's the form for a crystal with symmetry 422. 18 00:01:18,390 --> 00:01:20,860 And same for a three-fold axis. 19 00:01:20,860 --> 00:01:24,210 This is the form for 32 and 622. 20 00:01:24,210 --> 00:01:27,760 For just the single rotation axis of any sort other than 2, 21 00:01:27,760 --> 00:01:32,330 these two terms are equal in magnitude, but non-0. 22 00:01:32,330 --> 00:01:37,700 Now if you look at the equation for the tensor, the 23 00:01:37,700 --> 00:01:46,180 term in a12, x1, x2, and a21, x1, x2 are opposite in sign 24 00:01:46,180 --> 00:01:48,990 and disappear. 25 00:01:48,990 --> 00:02:01,520 So the quadric is, in fact a surface of revolution 26 00:02:01,520 --> 00:02:04,390 So the question is why are there two degrees of freedom 27 00:02:04,390 --> 00:02:06,840 rather than three? 28 00:02:06,840 --> 00:02:13,280 And I think the answer to that is going to lie in what is 29 00:02:13,280 --> 00:02:17,030 required of the eigenvectors when the off diagonal turns 30 00:02:17,030 --> 00:02:19,890 are skew-symmetric, and that is something I've not thought 31 00:02:19,890 --> 00:02:22,955 about before. 32 00:02:22,955 --> 00:02:30,630 All right, so that one minor loose end sticking out of the 33 00:02:30,630 --> 00:02:35,640 lid as we try to close the box on this segment of our 34 00:02:35,640 --> 00:02:36,290 discussion. 35 00:02:36,290 --> 00:02:41,680 I'd like t move on to a different set of 36 00:02:41,680 --> 00:02:42,930 considerations. 37 00:02:45,120 --> 00:02:50,280 One of the things that we can subject a crystal to as a 38 00:02:50,280 --> 00:02:55,960 generalized force is not merely a vector force, but 39 00:02:55,960 --> 00:02:59,480 stimulations that are more complicated. 40 00:02:59,480 --> 00:03:03,220 And I would like to in particular now, as a prelude 41 00:03:03,220 --> 00:03:07,290 to get into getting into property 42 00:03:07,290 --> 00:03:08,900 tensors of higher rank. 43 00:03:08,900 --> 00:03:12,670 I'd like to . first discuss stress, and then define 44 00:03:12,670 --> 00:03:17,620 strain, and you've all heard of these 45 00:03:17,620 --> 00:03:19,690 questions in other contexts. 46 00:03:19,690 --> 00:03:22,990 I'm going to introduce it, though, in terms of the tensor 47 00:03:22,990 --> 00:03:25,830 notation that we've used for other sorts of tensors, just 48 00:03:25,830 --> 00:03:28,750 so that we have everything in terms of the same language. 49 00:03:28,750 --> 00:03:34,000 So let me introduce stress by supposing I have a volume 50 00:03:34,000 --> 00:03:38,070 element in a solid, and it will not surprise you that I'm 51 00:03:38,070 --> 00:03:43,830 going to define the coordinate system in the solid by three 52 00:03:43,830 --> 00:03:46,430 axes, X1, X2, X3 in a Cartesian. 53 00:03:46,430 --> 00:03:54,910 system, and I will look at a surface on the exterior of the 54 00:03:54,910 --> 00:04:03,880 solid that cuts axis X1 at A, axis X2 at B, and axis X3 at 55 00:04:03,880 --> 00:04:06,070 C. 56 00:04:06,070 --> 00:04:10,880 Then I'll assume that I have a force density, force per unit 57 00:04:10,880 --> 00:04:14,280 area applied to the surface. 58 00:04:14,280 --> 00:04:17,890 And if this is force per unit area, why 59 00:04:17,890 --> 00:04:21,209 don't I call it a pressure? 60 00:04:21,209 --> 00:04:24,430 Well, a special case of what we normally think of as a 61 00:04:24,430 --> 00:04:27,320 pressure is a hydrostatic pressure, and that is 62 00:04:27,320 --> 00:04:30,720 something that's always exactly normal to the surface. 63 00:04:30,720 --> 00:04:34,800 So I don't want to call it a pressure to mislead you into 64 00:04:34,800 --> 00:04:39,350 the assumption that this force per unit area, which is a 65 00:04:39,350 --> 00:04:41,035 vector, has to be normal to the surface. 66 00:04:41,035 --> 00:04:47,210 So I'll call it a force density, which becomes a 67 00:04:47,210 --> 00:04:49,610 hydrostatic pressure when that force vector is 68 00:04:49,610 --> 00:04:51,370 normal to the surface. 69 00:04:51,370 --> 00:04:55,990 And I'll call that force density vector K, and it will 70 00:04:55,990 --> 00:04:57,240 have three components, KI. 71 00:05:01,880 --> 00:05:06,340 Now, the solid is in equilibrium, I will assume. 72 00:05:06,340 --> 00:05:14,460 And if that external force is not balance on the other side 73 00:05:14,460 --> 00:05:19,280 by balancing forced, the volume element in the solid 74 00:05:19,280 --> 00:05:22,520 would undergo a linear acceleration, or an angular 75 00:05:22,520 --> 00:05:24,570 acceleration at the very least. 76 00:05:24,570 --> 00:05:30,000 So I will assume that this force on the outside, KI, 77 00:05:30,000 --> 00:05:32,350 which is a force per unit area. 78 00:05:32,350 --> 00:05:36,040 So to make that a force, I will have to multiply that 79 00:05:36,040 --> 00:05:39,040 force density by the area of the surface 80 00:05:39,040 --> 00:05:41,460 ABC, on which it acts. 81 00:05:44,320 --> 00:05:47,780 And I'll say that this has to be balanced for each of the 82 00:05:47,780 --> 00:05:53,910 three directions, X1 and X2 and X3 by a force per unit 83 00:05:53,910 --> 00:05:57,380 area acting on these internal surfaces. 84 00:05:57,380 --> 00:06:01,240 So I'll assume that there is a force acting in the x1 85 00:06:01,240 --> 00:06:10,140 direction, and I'll call those a sigma IJ. 86 00:06:10,140 --> 00:06:14,160 And I'll say that there's a first force acting in the X1 87 00:06:14,160 --> 00:06:15,860 direction on-- 88 00:06:15,860 --> 00:06:19,910 now I need to somehow specify the orientation to the 89 00:06:19,910 --> 00:06:23,160 surface, because on this internal surface there's also 90 00:06:23,160 --> 00:06:25,320 a force acting in this direction. 91 00:06:25,320 --> 00:06:28,000 Now on this back surface there's also a force acting in 92 00:06:28,000 --> 00:06:28,970 this direction. 93 00:06:28,970 --> 00:06:32,440 So the forces acting on all three internal surfaces in the 94 00:06:32,440 --> 00:06:37,380 X1 direction have to balance exactly the X1 direction, the 95 00:06:37,380 --> 00:06:40,230 X1 component of the force density. 96 00:06:40,230 --> 00:06:44,710 So let me turn this to K1, and I'll say this has to be a 97 00:06:44,710 --> 00:06:48,800 force per unit area acting on area. 98 00:06:48,800 --> 00:06:55,850 First of all, I'll call the origin O on area AOC. 99 00:06:58,720 --> 00:07:04,726 Plus a force acting in the X1 direction on area A-- 100 00:07:04,726 --> 00:07:07,260 whoops, what did I do? 101 00:07:07,260 --> 00:07:14,800 I wanted to do this first on area BOC, excuse me. 102 00:07:14,800 --> 00:07:18,520 And a force in the X1 direction acting on area AOC. 103 00:07:26,080 --> 00:07:32,996 And finally a force acting on area AOB. 104 00:07:32,996 --> 00:07:40,190 And these are all acting in the X1 direction, and I now 105 00:07:40,190 --> 00:07:43,860 would like to introduce some notation that indicates the 106 00:07:43,860 --> 00:07:49,460 direction of the direction orientation of the surface on 107 00:07:49,460 --> 00:07:51,560 which that force acts. 108 00:07:51,560 --> 00:07:55,770 And I will use the normal to these internal surfaces to 109 00:07:55,770 --> 00:07:56,850 define that. 110 00:07:56,850 --> 00:08:02,460 So the force per unit area on area AOB, that is a surface 111 00:08:02,460 --> 00:08:05,360 whose normal is X3. 112 00:08:05,360 --> 00:08:10,320 And I will call this component sigma 13. 113 00:08:10,320 --> 00:08:17,940 And I'll call this one area AOC is normal to X2. 114 00:08:17,940 --> 00:08:20,700 That's why I did that little sleight of hand as I began 115 00:08:20,700 --> 00:08:28,820 this, and finally area BOC has as its normal X1. 116 00:08:28,820 --> 00:08:37,030 So my use of subscripts here is that this first subscript 117 00:08:37,030 --> 00:08:39,430 is the direction in which the force acts. 118 00:08:47,300 --> 00:08:50,580 And the second subscript will be the normal to 119 00:08:50,580 --> 00:08:53,505 the internal surface. 120 00:09:14,730 --> 00:09:19,490 So I've set up algebraically a system of 121 00:09:19,490 --> 00:09:20,830 subscripts that works. 122 00:09:20,830 --> 00:09:22,470 We'll see that these have some physical 123 00:09:22,470 --> 00:09:25,210 significance in just a moment. 124 00:09:25,210 --> 00:09:27,890 So what will I call these? 125 00:09:27,890 --> 00:09:30,080 Well, let me get this in a slightly different form. 126 00:09:30,080 --> 00:09:35,750 Let me take the left hand side and divide area ABC into all 127 00:09:35,750 --> 00:09:36,770 of these other areas. 128 00:09:36,770 --> 00:09:42,120 So this then will have sigma 11 times area 129 00:09:42,120 --> 00:09:49,110 BOC over area ABC. 130 00:09:49,110 --> 00:09:54,290 And not surprisingly, sigma 12 will then have multiplying it 131 00:09:54,290 --> 00:10:03,210 a term area AOC divided by area ABC. 132 00:10:03,210 --> 00:10:06,460 And this is all looking very cumbersome, but you'll be 133 00:10:06,460 --> 00:10:09,600 amazed at how this cleans up in just a moment. 134 00:10:09,600 --> 00:10:13,890 And this will be area AOB over finally area AOC. 135 00:10:13,890 --> 00:10:15,140 ABC again. 136 00:10:19,420 --> 00:10:21,520 OK, so how can we tidy this up? 137 00:10:21,520 --> 00:10:25,880 Well, let me imagine that I have some surface, a planar 138 00:10:25,880 --> 00:10:30,300 surface, and its normal points in this direction. 139 00:10:30,300 --> 00:10:34,600 And I'm going to take that area and project it onto 140 00:10:34,600 --> 00:10:38,165 another surface whose normal points in this direction. 141 00:10:41,280 --> 00:10:46,380 And let's suppose that the angle between these two 142 00:10:46,380 --> 00:10:51,400 directions is phi, and the angle between the 143 00:10:51,400 --> 00:10:54,910 two surfaces is phi. 144 00:10:54,910 --> 00:11:01,680 The area of this original surface is A, and its normal 145 00:11:01,680 --> 00:11:03,820 points in this direction. 146 00:11:03,820 --> 00:11:07,150 And I project it onto a surface whose normal is an 147 00:11:07,150 --> 00:11:08,700 angle phi away. 148 00:11:08,700 --> 00:11:12,640 This new area, A prime is equal to A times 149 00:11:12,640 --> 00:11:13,890 the cosine of phi. 150 00:11:19,260 --> 00:11:24,990 So if we accept that, then the first line of this equation, 151 00:11:24,990 --> 00:11:28,310 which will be eventually a set of three equations, this is 152 00:11:28,310 --> 00:11:32,030 the ratio of area BOC over area ABC, and what is that? 153 00:11:32,030 --> 00:11:38,830 That is the angle between the direction of the force density 154 00:11:38,830 --> 00:11:45,930 and area ABC. 155 00:11:45,930 --> 00:11:47,510 And here is area ABC. 156 00:11:50,800 --> 00:11:51,820 That's the normal here. 157 00:11:51,820 --> 00:11:56,960 And the first term involves area BOC, and that is the 158 00:11:56,960 --> 00:12:05,460 direction of X, the normal to that is X1, and therefore the 159 00:12:05,460 --> 00:12:11,330 ratio of these two areas is the same thing as the cosine 160 00:12:11,330 --> 00:12:16,780 of the angle between K and X1. 161 00:12:19,620 --> 00:12:24,000 And this second ratio of areas is going to be sigma 12 times 162 00:12:24,000 --> 00:12:27,520 the cosine of the angle between the force density 163 00:12:27,520 --> 00:12:32,690 vector K and X2. 164 00:12:32,690 --> 00:12:36,710 And the third term is going to be sigma 13 times the cosine 165 00:12:36,710 --> 00:12:42,010 of the angle between K and X3. 166 00:12:44,550 --> 00:12:48,860 What these cosines are are just the cosines of the angle 167 00:12:48,860 --> 00:12:50,720 between K and the three reference 168 00:12:50,720 --> 00:12:53,740 axes, X1, X2, and X3. 169 00:12:53,740 --> 00:13:05,800 So this is simply the direction cosine L1 of K. And 170 00:13:05,800 --> 00:13:13,860 this is the direction cosine L2 of K. And this is the 171 00:13:13,860 --> 00:13:17,790 direction cosine L3. 172 00:13:17,790 --> 00:13:23,430 So the X1 component of the force density vector is going 173 00:13:23,430 --> 00:13:29,955 to be equal to a term sigma 11 times L1, plus the term sigma 174 00:13:29,955 --> 00:13:33,860 12 times L2, the term sigma 13 L3. 175 00:13:33,860 --> 00:13:37,390 And if I would write down similar balances, forces for 176 00:13:37,390 --> 00:13:41,260 the X2 component of the force density vector, not 177 00:13:41,260 --> 00:13:47,870 surprisingly these things are going to give me coefficients 178 00:13:47,870 --> 00:13:53,855 that I'll label sigma 121, sigma 22 times L2, plus sigma 179 00:13:53,855 --> 00:13:56,810 23 times L3. 180 00:13:56,810 --> 00:14:01,590 And in general then the requirement that this body be 181 00:14:01,590 --> 00:14:06,170 in equilibrium states that each component of the force 182 00:14:06,170 --> 00:14:13,160 density vector should be given by a coefficient sigma IJ 183 00:14:13,160 --> 00:14:18,160 times L sub J. These are the components of the force 184 00:14:18,160 --> 00:14:19,410 density vector. 185 00:14:23,580 --> 00:14:33,680 This is the direction cosines of the direction of K. 186 00:14:33,680 --> 00:14:40,780 This is a unit vector in the direction of K. This is the 187 00:14:40,780 --> 00:14:41,840 direction of a vector. 188 00:14:41,840 --> 00:14:50,620 These are the components of K. And therefore these 189 00:14:50,620 --> 00:14:56,700 coefficients sigma IJ relate to vectors, and they must be a 190 00:14:56,700 --> 00:14:58,970 second rank tensor. 191 00:15:04,540 --> 00:15:06,870 And these are called, as you all know, it comes as no 192 00:15:06,870 --> 00:15:09,910 surprise as the components of stress. 193 00:15:16,890 --> 00:15:22,280 So the reason that these are really tensors, and not just a 194 00:15:22,280 --> 00:15:25,840 vector force per unit area essentially comes down to the 195 00:15:25,840 --> 00:15:29,850 fact that the behavior of the body, dynamically and 196 00:15:29,850 --> 00:15:32,490 mechanically, is going to depend not only on the 197 00:15:32,490 --> 00:15:36,850 direction of the force, but on the direction of the surface 198 00:15:36,850 --> 00:15:38,070 on which it acts. 199 00:15:38,070 --> 00:15:41,870 Quite clear here, if I shove, is a force density, 200 00:15:41,870 --> 00:15:44,040 and that's a vector. 201 00:15:44,040 --> 00:15:48,560 But the way in which this desk responds to my pushing on it 202 00:15:48,560 --> 00:15:52,590 with a force vector of that magnitude is going to depend 203 00:15:52,590 --> 00:15:55,990 whether I do this, which is going to compress it, or 204 00:15:55,990 --> 00:15:58,850 whether it takes the same force and direct it this way, 205 00:15:58,850 --> 00:16:00,820 which if I don't move the thing is going 206 00:16:00,820 --> 00:16:03,120 to result in a shear. 207 00:16:03,120 --> 00:16:05,830 So the behavior of the body will depend on the direction 208 00:16:05,830 --> 00:16:09,040 of the force per unit area that I apply, and the 209 00:16:09,040 --> 00:16:11,340 direction of the surface on which I apply it. 210 00:16:11,340 --> 00:16:15,080 In this case, that would be the surface ABC. 211 00:16:15,080 --> 00:16:15,962 Yes, sir? 212 00:16:15,962 --> 00:16:20,390 AUDIENCE: You say that the force density is not parallel 213 00:16:20,390 --> 00:16:22,850 to the normal of the surface? 214 00:16:22,850 --> 00:16:23,834 PROFESSOR: That's correct. 215 00:16:23,834 --> 00:16:28,262 AUDIENCE: So when you do the ratio of [INAUDIBLE] 216 00:16:28,262 --> 00:16:31,214 isn't it supposed to be [INAUDIBLE] 217 00:16:31,214 --> 00:16:35,642 of the normal to the [INAUDIBLE] and X1? 218 00:16:35,642 --> 00:16:36,626 PROFESSOR: Another way of defining it. 219 00:16:36,626 --> 00:16:41,030 This is a vector, and what I'm doing is splitting that vector 220 00:16:41,030 --> 00:16:46,210 up into three internal forces that act on surfaces whose 221 00:16:46,210 --> 00:16:47,360 normal are here. 222 00:16:47,360 --> 00:16:52,745 So actually, what I'm going to do is to take the way in which 223 00:16:52,745 --> 00:16:55,390 the internal force pushes back. 224 00:16:55,390 --> 00:16:57,710 I see, that's what's confusing. 225 00:16:57,710 --> 00:17:03,790 If the force that I'm putting on the body, I didn't mean to 226 00:17:03,790 --> 00:17:06,230 indicate any particular direction, but if this 227 00:17:06,230 --> 00:17:10,390 component of K points in this direction, this force that 228 00:17:10,390 --> 00:17:13,776 balances it has to go in the opposite direction. 229 00:17:13,776 --> 00:17:17,881 AUDIENCE: So you're saying that there were surface ABC is 230 00:17:17,881 --> 00:17:20,769 defined as normal to K? 231 00:17:20,769 --> 00:17:23,474 PROFESSOR: ABC is not normal. 232 00:17:23,474 --> 00:17:23,930 No. 233 00:17:23,930 --> 00:17:31,700 ABC is not normal to K. All I'm saying is that it if it 234 00:17:31,700 --> 00:17:35,920 were normal to K, then the balancing force would be just 235 00:17:35,920 --> 00:17:41,620 a function of this orientation of the surface. 236 00:17:41,620 --> 00:17:45,160 But if I allow this to be a general force per unit area, 237 00:17:45,160 --> 00:17:50,050 then the balancing force on the inside depends on the 238 00:17:50,050 --> 00:17:57,892 angle between K and the angle between this internal surface. 239 00:17:57,892 --> 00:18:04,794 AUDIENCE: I don't see the relation between K and the 240 00:18:04,794 --> 00:18:06,766 ratio between other ends. 241 00:18:06,766 --> 00:18:08,738 BOC and ABC [INAUDIBLE]. 242 00:18:19,091 --> 00:18:19,945 PROFESSOR: OK. 243 00:18:19,945 --> 00:18:22,600 The reason that there are three areas, is that what's 244 00:18:22,600 --> 00:18:23,080 troubling you? 245 00:18:23,080 --> 00:18:26,500 Because I'm really now splitting the balancing force 246 00:18:26,500 --> 00:18:29,830 up into a force per unit area on this surface, and a force 247 00:18:29,830 --> 00:18:33,160 per unit area on this surface, and a force per unit area on 248 00:18:33,160 --> 00:18:34,580 this surface. 249 00:18:34,580 --> 00:18:38,310 And that's so that I can look at these different components 250 00:18:38,310 --> 00:18:44,400 of resistance in terms of a Cartesian system. 251 00:18:44,400 --> 00:18:49,130 Clear that the ratio of these two areas does go as the-- 252 00:18:49,130 --> 00:18:52,080 I erased it now, that is going to go as the-- 253 00:18:56,060 --> 00:18:59,280 This is A prime, and this is the original A, the relation 254 00:18:59,280 --> 00:19:03,260 between those areas is given by the cosine 255 00:19:03,260 --> 00:19:05,222 of this angle phi. 256 00:19:05,222 --> 00:19:09,120 If I [INAUDIBLE] this down in this projection, and this is 257 00:19:09,120 --> 00:19:11,200 the area of A. 258 00:19:11,200 --> 00:19:15,110 This is the direction of the normal to the surface ABC. 259 00:19:15,110 --> 00:19:17,940 And I'm going to let this be the direction of the normal to 260 00:19:17,940 --> 00:19:21,670 one of the internal surfaces, and these internal surfaces 261 00:19:21,670 --> 00:19:26,908 are going to have as their normal either X1, X2, or X3. 262 00:19:26,908 --> 00:19:28,158 AUDIENCE: So [INAUDIBLE] 263 00:19:30,812 --> 00:19:33,740 between the normal to ABC and X1. 264 00:19:33,740 --> 00:19:35,010 PROFESSOR: And X1, yes. 265 00:19:35,010 --> 00:19:37,650 And that's what I'm saying its, and that is going to be 266 00:19:37,650 --> 00:19:39,530 the same thing. 267 00:19:39,530 --> 00:19:42,910 As the cosine of that angle is just going to be the direction 268 00:19:42,910 --> 00:19:48,105 cosine of the normal to ABC relative to X1, X2, X3. 269 00:19:48,105 --> 00:19:49,560 OK? 270 00:19:49,560 --> 00:19:53,925 AUDIENCE: So in the end, you're saying that K is 271 00:19:53,925 --> 00:19:58,290 parallel to the normal of ABC? 272 00:19:58,290 --> 00:19:59,260 PROFESSOR: No, I'm not. 273 00:19:59,260 --> 00:20:00,510 No, I'm not. 274 00:20:04,950 --> 00:20:08,020 This is a relation between the areas. 275 00:20:08,020 --> 00:20:10,920 And these are the forces acting on 276 00:20:10,920 --> 00:20:13,740 those internal surfaces. 277 00:20:13,740 --> 00:20:16,550 And I'm saying there are three of them, and those three 278 00:20:16,550 --> 00:20:20,100 components that are all acting in the X1 direction will 279 00:20:20,100 --> 00:20:23,360 depend on the ratio of these areas. 280 00:20:23,360 --> 00:20:27,220 The ratio of this area is the cosine of the angle 281 00:20:27,220 --> 00:20:39,910 between X1 and K. OK. 282 00:20:39,910 --> 00:20:42,294 This you're OK with? 283 00:20:42,294 --> 00:20:43,208 AUDIENCE: Yeah. 284 00:20:43,208 --> 00:20:44,580 PROFESSOR: OK. 285 00:20:44,580 --> 00:20:52,020 And what I'm saying now is that there is acting on this 286 00:20:52,020 --> 00:21:00,380 internal surface something that has to balance the 287 00:21:00,380 --> 00:21:07,380 component of K, which points in the direction of X1. 288 00:21:07,380 --> 00:21:10,950 And the component of K before we divided out the area, the 289 00:21:10,950 --> 00:21:15,860 component of K1 times the area gave me the net force. 290 00:21:15,860 --> 00:21:20,620 So this K1 now is a force per unit area, the component of K 291 00:21:20,620 --> 00:21:24,740 per unit area that points along the X1 direction. 292 00:21:24,740 --> 00:21:28,010 Now as a force per unit area I multiply it by an area that 293 00:21:28,010 --> 00:21:29,550 gave me force. 294 00:21:29,550 --> 00:21:32,110 There are three forces internally that balance this 295 00:21:32,110 --> 00:21:34,650 force, and they all act in the X1 direction. 296 00:21:37,578 --> 00:21:39,042 AUDIENCE: You're just [INAUDIBLE]. 297 00:21:43,434 --> 00:21:47,826 Shouldn't it be the cosine of the angle between the normals 298 00:21:47,826 --> 00:21:49,790 of [INAUDIBLE]? 299 00:21:49,790 --> 00:21:50,160 PROFESSOR: Yeah. 300 00:21:50,160 --> 00:21:51,710 That's what I'm saying here. 301 00:21:51,710 --> 00:21:59,040 That for X, for this surface, yeah, that's what I'm saying. 302 00:21:59,040 --> 00:22:02,060 One of these surfaces here, if we look at the first term, 303 00:22:02,060 --> 00:22:07,540 this is the direction of X1, and the force here will be the 304 00:22:07,540 --> 00:22:15,100 force along K. And if I take this force and put it down in 305 00:22:15,100 --> 00:22:19,450 X1, this is going to be the force times the 306 00:22:19,450 --> 00:22:21,520 cosine of the angle. 307 00:22:21,520 --> 00:22:25,245 And the two areas are going to go as the cosine of the angle. 308 00:22:28,090 --> 00:22:31,042 AUDIENCE: [INAUDIBLE] 309 00:22:31,042 --> 00:22:33,010 cosine of the normal [INAUDIBLE]? 310 00:22:43,350 --> 00:22:43,970 PROFESSOR: OK. 311 00:22:43,970 --> 00:22:47,180 This is indeed the ratio between the internal area and 312 00:22:47,180 --> 00:22:49,430 the external area. 313 00:22:49,430 --> 00:22:53,790 Since the internal area has a normal that's either X1, X2, 314 00:22:53,790 --> 00:22:57,640 or X3 the angle between the normal to these internal 315 00:22:57,640 --> 00:23:03,920 surfaces and K is going to be the direction cosine of K. 316 00:23:03,920 --> 00:23:09,210 I suggest since we're getting out of time quickly we either 317 00:23:09,210 --> 00:23:13,530 resolve this after the end of class or leave 318 00:23:13,530 --> 00:23:16,050 it until next time. 319 00:23:16,050 --> 00:23:19,790 That's a bad note on which to end, too. 320 00:23:19,790 --> 00:23:23,115 The point I want to leave, still subject to resolution 321 00:23:23,115 --> 00:23:30,390 and debate is that the coefficient sigma IJ I would 322 00:23:30,390 --> 00:23:36,750 like to claim relate to vectors. 323 00:23:36,750 --> 00:23:43,730 One is the direction of the external surface, and if it 324 00:23:43,730 --> 00:23:49,400 relates to vectors, then this set of coefficients qualifies 325 00:23:49,400 --> 00:23:50,912 as a second rank tensor. 326 00:23:55,250 --> 00:23:59,890 And if you accept that, however grudgingly, everything 327 00:23:59,890 --> 00:24:04,400 that we've said about second rank tensors holds for the 328 00:24:04,400 --> 00:24:07,985 quantity sigma IJ, which are called the elements of stress. 329 00:24:16,620 --> 00:24:21,100 In particular, we can talk about a stress quadrant. 330 00:24:27,060 --> 00:24:29,920 We can talk about the value of a stress in 331 00:24:29,920 --> 00:24:31,170 a particular direction. 332 00:24:40,480 --> 00:24:44,480 If we have a stress tensor in which show all the terms are 333 00:24:44,480 --> 00:24:46,425 non-0, it can be diagonalized. 334 00:25:00,020 --> 00:25:09,890 We can also say that if the crystal has symmetry, for 335 00:25:09,890 --> 00:25:16,550 example, if the crystal is cubic, you can say that the 336 00:25:16,550 --> 00:25:19,000 form of a second rank tensor for a cubic 337 00:25:19,000 --> 00:25:21,610 crystal has to be diagonal. 338 00:25:25,920 --> 00:25:28,770 And these diagonal elements represent compressive 339 00:25:28,770 --> 00:25:33,240 stresses, so you can say that you can only subject a cubic 340 00:25:33,240 --> 00:25:36,648 crystal to compressive stress. 341 00:25:36,648 --> 00:25:39,340 On the other hand, if the crystal is a triclinic 342 00:25:39,340 --> 00:25:46,020 crystal, you can say that there's a sigma 11, there's a 343 00:25:46,020 --> 00:25:53,000 sigma 12, a sigma 13, a sigma 21, a sigma 22, sigma 23, 344 00:25:53,000 --> 00:25:56,980 sigma 31, sigma 32, sigma 33. 345 00:25:56,980 --> 00:26:01,820 So for a triclinic crystal, there appear to be nine 346 00:26:01,820 --> 00:26:04,940 elements, but next time we will show that the nature of 347 00:26:04,940 --> 00:26:08,660 stress is the body has to be in equilibrium requires that 348 00:26:08,660 --> 00:26:12,360 the stress tensor be by definition symmetric if 349 00:26:12,360 --> 00:26:14,350 there's to be no net couple to force on it. 350 00:26:18,920 --> 00:26:21,904 Going to buy this? 351 00:26:21,904 --> 00:26:24,640 You didn't buy the things that I said that perhaps were 352 00:26:24,640 --> 00:26:26,020 slightly incorrect. 353 00:26:26,020 --> 00:26:28,040 This is massively incorrect. 354 00:26:28,040 --> 00:26:28,787 Jason? 355 00:26:28,787 --> 00:26:33,657 AUDIENCE: You can rotate a cubic crystal [INAUDIBLE]. 356 00:26:33,657 --> 00:26:34,144 PROFESSOR: No. 357 00:26:34,144 --> 00:26:37,066 Because say a tensor of this form acts 358 00:26:37,066 --> 00:26:39,014 just like the scalar. 359 00:26:39,014 --> 00:26:42,790 Just like a cubic crystal, you don't suddenly get off 360 00:26:42,790 --> 00:26:44,575 diagonal properties if you transfer it 361 00:26:44,575 --> 00:26:47,580 to a different axis. 362 00:26:47,580 --> 00:26:50,480 So unless you tell me what's wrong here, I'm going to leave 363 00:26:50,480 --> 00:26:53,630 this on your plate until next week. 364 00:26:53,630 --> 00:26:56,450 And I don't want you fretting over the weekend. 365 00:26:56,450 --> 00:27:04,820 This is clearly nonsense, and it is a contradiction that 366 00:27:04,820 --> 00:27:08,130 arises only if you carry too much of what we've been doing 367 00:27:08,130 --> 00:27:11,800 for the last month with you into a discussion of stress 368 00:27:11,800 --> 00:27:12,950 and strain. 369 00:27:12,950 --> 00:27:17,282 Stress is something that you impose on a crystal. 370 00:27:17,282 --> 00:27:20,600 It has nothing to do with what kind of crystal it is and 371 00:27:20,600 --> 00:27:23,270 what's going on in the interior of the crystal. 372 00:27:23,270 --> 00:27:27,010 There may be constraints on the coefficients that relate 373 00:27:27,010 --> 00:27:29,740 stress and strains, and there surely are. 374 00:27:29,740 --> 00:27:32,470 But I can take a cubic crystal, and I can squeeze it, 375 00:27:32,470 --> 00:27:35,810 and I can twist it and share it anyway I like. 376 00:27:35,810 --> 00:27:38,370 So this is an important distinction between the 377 00:27:38,370 --> 00:27:40,900 tensors that represent physical properties, which 378 00:27:40,900 --> 00:27:43,950 we've been discussing up to this point. 379 00:27:43,950 --> 00:27:51,090 And the properties such as conductivity, diffusivity, 380 00:27:51,090 --> 00:27:54,310 susceptibility and so on and all their many varieties, 381 00:27:54,310 --> 00:27:56,530 these are something which we'll refer 382 00:27:56,530 --> 00:27:57,780 to as property tensors. 383 00:28:03,200 --> 00:28:06,290 And they are tensors demonstrably, but they are 384 00:28:06,290 --> 00:28:08,250 tensors which describe the physical 385 00:28:08,250 --> 00:28:11,080 behavior of a crystal. 386 00:28:11,080 --> 00:28:17,905 Nye uses a term that I don't think is self explanatory, so 387 00:28:17,905 --> 00:28:18,700 I don't care for it. 388 00:28:18,700 --> 00:28:20,580 He calls these field tensors. 389 00:28:26,280 --> 00:28:30,470 That sounds like some description of a corn patch 390 00:28:30,470 --> 00:28:31,970 that requires a tensor. 391 00:28:31,970 --> 00:28:33,670 But no. 392 00:28:33,670 --> 00:28:36,530 We probably don't really think of what we mean 393 00:28:36,530 --> 00:28:37,530 when we say a field. 394 00:28:37,530 --> 00:28:40,380 When we say an electric field, you think of an e-vector. 395 00:28:40,380 --> 00:28:41,560 You think of a vector. 396 00:28:41,560 --> 00:28:47,520 But what a field is is just a set of coordinates, XYZ, and 397 00:28:47,520 --> 00:28:49,820 that describes an area or a volume. 398 00:28:49,820 --> 00:28:54,340 And just like the things that corn grow in are referred to 399 00:28:54,340 --> 00:28:57,860 as corn fields, it's an area that has corn on a lattice 400 00:28:57,860 --> 00:28:59,980 very often also. 401 00:28:59,980 --> 00:29:04,460 So a field is just a set of coordinates in space to which 402 00:29:04,460 --> 00:29:07,310 a value of something is a sign. 403 00:29:07,310 --> 00:29:10,430 So if we have an electric field, and we express that 404 00:29:10,430 --> 00:29:13,480 electric field as a function of X, Y, and Z, that is a 405 00:29:13,480 --> 00:29:15,090 field of vectors. 406 00:29:15,090 --> 00:29:19,540 If we take something that has intrinsically, that 407 00:29:19,540 --> 00:29:22,650 intrinsically needs to be described as a tensor, then we 408 00:29:22,650 --> 00:29:28,000 have to every point in space XYZ a tensor sigma IJ, whose 409 00:29:28,000 --> 00:29:31,610 value and whose components value vary with position 410 00:29:31,610 --> 00:29:32,700 within the space. 411 00:29:32,700 --> 00:29:38,810 So talking about the field as assigning values of something 412 00:29:38,810 --> 00:29:44,760 to every coordinate in the space that could be a tensor. 413 00:29:44,760 --> 00:29:47,370 What we refer to as an electric field really is a 414 00:29:47,370 --> 00:29:48,430 field vector. 415 00:29:48,430 --> 00:29:51,790 So that's Nye's term, but the fact that it took me three 416 00:29:51,790 --> 00:29:55,220 minutes to explain why he uses it explains as well why I 417 00:29:55,220 --> 00:29:56,090 don't care for it. 418 00:29:56,090 --> 00:29:58,880 So we'll talk about these as property tensors. 419 00:29:58,880 --> 00:30:04,040 And property tensors are something that's 420 00:30:04,040 --> 00:30:05,000 innate to the material. 421 00:30:05,000 --> 00:30:09,550 Something like stress, strain, and other second rank tensors 422 00:30:09,550 --> 00:30:18,110 are external stimuli, just like an electric field. 423 00:30:18,110 --> 00:30:23,510 And they have value that is imposed on the crystal, and 424 00:30:23,510 --> 00:30:28,000 the stress field is defined as a function of position. 425 00:30:28,000 --> 00:30:31,000 So that's the distinction between a property tensor, 426 00:30:31,000 --> 00:30:34,690 which is subject to symmetry constraints, and a tensor that 427 00:30:34,690 --> 00:30:38,390 represents a stimulus, an external stimulus applied to 428 00:30:38,390 --> 00:30:42,320 the crystal, and it's defined as a function of position. 429 00:30:42,320 --> 00:30:45,360 And that is a field tensor. 430 00:30:45,360 --> 00:30:46,940 OK, so that's the difference. 431 00:30:46,940 --> 00:30:49,640 Since this is something that's imposed externally, it can 432 00:30:49,640 --> 00:30:52,760 have any form of the tensor that you wish to oppose. 433 00:30:52,760 --> 00:30:56,300 And there are no symmetry restrictions on it. 434 00:30:56,300 --> 00:30:58,760 The only restriction is, as we'll show next time, that it 435 00:30:58,760 --> 00:31:02,401 must be symmetric if the body is to be in equilibrium. 436 00:31:05,287 --> 00:31:05,770 All right. 437 00:31:05,770 --> 00:31:08,540 It is time to quit, and that I think is a good 438 00:31:08,540 --> 00:31:11,490 place to leave things. 439 00:31:11,490 --> 00:31:14,100 And we'll say more about stress next time.