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GEOL7333-Seismic Wave and Ray Theory - Day1 08-26-2022-Part2
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00:04 so we just had a little technical there, but I think we've
00:08 And so we'll continue this and Utah will patch these recordings together somehow.
00:18 he's on his computer. So we talking about matrix algebra. Matrix algebra
00:32 so uh adding matrices together is really . You just found uh said that
00:39 element was together like Stephanie. But here's something I want to point out
00:46 that it's pretty clear from this definition the order of these is not
00:52 So what the mathematical term for that uh they say that for addition of
01:01 , the matrix addition is communicative. can commute these things back and
01:08 like a commuting to work. Uh now we're gonna multiply these two
01:16 . I remember there were two different for multiplying vectors, but there's only
01:21 definition for multiplying matrices. So here a matrix C. You find as
01:28 matrix product A. And B. so you you can't just uh do
01:34 simple, do the simple together. you do that's not official matrix um
01:43 and you will surely get into And so the uh the smart thing
01:49 do is whenever you're doing matrix algebra paper in your head, you follow
01:57 conventions and there'll be big payoffs if do that because uh the way it's
02:03 up these things don't really matter. that is if you stick to these
02:15 then you and then later if you ask yourself is what I did depending
02:22 my arbitrary decisions about the corporate The answer will be known. So
02:29 so the definition is for multiplication is a simple multiplying together, but instead
02:36 it is. And so for the one it says this matrix C is
02:41 tube is a second right? Just like these. That wasn't
02:47 Wasn't but now but uh now by rule it is obvious. And what
02:53 do is you take the sum of of these products here. You
02:58 you notice that this eye is leading here is the same as the leading
03:03 hear. And the trailing J. the same as the trailing J
03:07 And in between is that is a case. So we're going to sum
03:11 these caves. So that's where this . So uh if you want to
03:20 this 1111 component of C. You this multiply this and then add these
03:30 . That's sort of a picture of what it works. Um And so
03:37 uh that's the definition of this multiplication the 11 component. Uh So how
03:47 the 12 component here. It says take uh this uh we're gonna deal
03:53 this element here, take this component 11 and multiply it by this
04:00 It's the same one. We need leading one and a trailing two.
04:05 we need a leading one here and trailing two over here. And in
04:10 the two are gonna be repeated one , one here. So multiply those
04:17 together and then add multiplying these two , you get them same first index
04:25 all these, same second index for these and some over that repeated interior
04:33 that's the way to remember. So is Einstein's rule. Yeah, if
04:39 you see matrix repeated index, you some over there and if you have
04:49 some sort of a formula that has example, I repeated three times index
04:55 repeated three times, you know, screwed up somewhere that shouldn't be like
05:02 . So we're not gonna show this summation some explicitly. We're gonna write
05:07 like this. So just know that sending over K. Is repeated.
05:22 , here is the definition, we're talking about consider this product and we're
05:26 call this product D. Instead of . And here it's gonna be B
05:30 A. Instead of eight times And so by the same rule,
05:38 I. J. Component B is look like this and you can see
05:43 this is different from this. So matrix C is not the same as
05:49 D. So uh way mathematicians described you say it's matrix multiplication is not
06:01 . Even though matrix addition is let's go on. Can multiply a
06:10 biomatrix. So uh let's think this X. And multiply it from the
06:18 by a matrix A. And we're get another um uh vector Y.
06:25 the components of Y are given like . Okay. And this leading high
06:32 the same as as I make. this is how to remember uh make
06:43 expect your multiplication. So the first you multiply these two together and added
06:50 these, the second component the second element. It's this combination. Now
07:01 saw an example of this previously when uh talked about the rotation matrix,
07:08 this expression here and here was our matrix. So this is a matrix
07:14 . One easy little line like And it's notation for all this complicated
07:20 . So you see how we really ourselves a favor if we um uh
07:27 we rely on this well defined uh spell out what we mean.
07:34 that's just matrix. Now consider a rotation. Okay, so we're gonna
07:44 the vector X. Prime and rotate again by another rotation matrix. And
07:50 gonna call that X double prime. so what is this rotation matrix?
07:54 , it's the same as the other . Except with five prime in
07:57 Instead of let's consider these rotations in . So here's what we're gonna
08:05 We're gonna rotate by angles by That that vector X prime. And
08:13 gonna write vector X. Prime like inside brackets. And then from the
08:18 this is what we used for the definition. And then we're gonna simply
08:25 the brackets. And that's called uh say that that operation is associative.
08:35 is we can associate these together like . And uh here's an interesting thought
08:43 um uh we got all this it turns out that uh this product
08:50 the same as a single rotation by angles. And so you can prove
08:58 to yourself if you uh remember your from high school. So all of
09:05 is really easy to generalize the three . For example write a three D
09:12 written like this row vector or column and to rotate the three D.
09:18 system about the X three axis. looks just very similar and the formula
09:23 definition is like this see it's it's over jay and we have a subscript
09:30 to remind us that this is a dimensional rotation. And so here's what
09:35 looks like and you will recognize this of it here is the same as
09:39 had for the 22 D rotation. we did was we made it three
09:46 by adding a one here in the components 33 position and zeros elsewhere.
09:59 suppose we want to rotate this ex vector which we just defined right here
10:07 the new x one axis. It's rotation vector for uh the rotation matrix
10:19 rotating about the one axis looks like . See it's got a one up
10:24 instead of down here and this part looks like it did with and so
10:32 we rotate these in sequence, then we can rearrange the brackets just like
10:38 did before and in this case And uh theta and phi are called
10:46 angles so that um that guy was german guy and although you might think
10:53 name should be pronounced Euler, his called the Oiler. So we should
10:58 the same. Uh Tai could you me a favor and get me some
11:05 ? You do, thank you Utah will make sure you come equipped with
11:12 bottles. Will will we pay Stephanie this one? Uh tomorrow and uh
11:20 um that way I won't one Thank you. Stephanie Oiler was a
11:31 famous uh mathematician in the 19th And notice here that um we can't
11:42 in this case since the rotations are different axes, we can't say that
11:46 repeated um, rotation is a simple like this because the angles are brought
11:56 axes. Yeah, we can't change order either. You see here,
12:06 have one rotating on uh we have rotating first and then uh one rotating
12:13 that. Down here, we have rotating first on X and three rotating
12:18 that. And this is different from because Rachel small certification is not
12:24 So uh if you look at the of what we mean um specifically in
12:31 of indices, I readily see that one is not equal to this.
12:38 here's the same places before Matrix some easy. Uh Stephanie did that in
12:43 head and the matrix product is where boggled. And so let's just check
12:50 out. Let's do the 111 11 component is this times this that's
12:55 plus this times this so we need look down below for something with a
13:00 in the +11 direction, here's 1 years to 35. Whoever uh made
13:08 quiz was obviously making it hard So so let's do the second one
13:13 . Um One times five is five um mhm. We want to do
13:25 12 components. So three times five 15 plus one times 10. Uh
13:32 makes 25. And so is that ? Oh yeah, so we're looking
13:54 the 12 components. Three times five 15 plus one times 10 is 10
13:59 25. It is, the answer C. Now uh what's this?
14:09 vector? Uh product? So you can probably do that one.
14:22 . Right, wait just a we have three times 10 is 30
14:28 five is 35. So it's gonna because yeah, so it's gonna be
14:35 okay. Now I don't know if noticed that but when a matrix multiplies
14:41 vector, it's just like multiplying the column of the matrix. So um
14:47 is a matrix multiplying a vector here the same matrix multiplying another bank but
14:53 one here is the same as this here. So this is like a
14:56 vector like we have here. And answer is the first color says uh
15:05 that's uh using that observation is it's very issue to generalize this to any
15:13 of ranks and any number of vector is just like a one column
15:21 and in fact mathematician, it is matrix, it's a matrix of rank
15:30 . And Furthermore, the Scalar is matrix of rank zero. Okay,
15:38 so much for from interest now we to tensors, some major cities are
15:44 and those are called tensors for example a deformed Iraq. So now we're
15:51 talk about a strain in Iraq. point is displaced from its equilibrium position
15:58 its displaced from its equilibrium position and call the that the displacement. So
16:06 imagine a position. Uh an atom Iraq and it gets moved by stress
16:13 and the stress causes strain. And the new position is displaced from the
16:18 position by this factor here. the displacement may vary with position that
16:27 , this atom might be displaced in different way than this happened. So
16:34 now consider the displacement gradient vector. so here is uh a matrix,
16:41 can see a three by three matrix each element is the gradient uh is
16:47 derivative of the corresponding displacement with respect the corresponding position. So we got
16:56 all on this uh road and we excess all on this row. And
17:02 have mixtures everywhere else. And there's notation for that which is called the
17:07 tensor. This upside down delta. in this upside down delta operates on
17:14 vector. It yields a matrix with form. That's a definition, see
17:19 it says. So that's an example attention. And so uh the easy
17:29 to think about sensor is tensors are which are constructed from vacuums. And
17:39 uh uh huh Mathematica, the mathematical is mathematical definition is mathematical definition is
17:53 matrix which transforms with rotation like a is a tensor. And so here
18:00 the transformation. This is what we by like a tensor. So the
18:06 ij component of this tensor is given this product of um of it's uh
18:16 going in and out of my front . Uh Yeah that's the ring.
18:22 uh the workmen are working at my and they're going in and out the
18:27 door. My wife is there. What? So uh so uh here
18:37 um two rotation matrices multiplying by another . And so this is the definition
18:45 tensor uh rotation. And so uh in case you think that this is
18:52 exotic? Well we'll never need to that. Well it's gonna turn out
18:57 be uh geophysical examples of tensors are and strain, things like that.
19:02 we definitely are going to talk about , not all matrices er cancers.
19:08 this cancer? Uh No, it's structured shopping list. So we got
19:13 our fruits on this uh on this and we got our vegetables on the
19:20 row. So it's just a structured . Now, let me show you
19:27 trick here. Here's our definition of . Like a tensor. That's what
19:34 showed you a couple slides previous. what I'm gonna do is I'm gonna
19:39 this rotation matrix and transform it so the order of of indices is not
19:46 and M M and J. So the that's the transposed fencing so that
19:53 know, the 12 component becomes the . And then I'm going to move
19:59 one over here and I can do since this is just ordinary. Uh
20:04 is just ordinary Uh multiplication like you in 5th grade. And so we
20:12 we can change the order when we that. And now look what we
20:23 . This second index is the same this first index. So this rotation
20:28 and I was operating on the same Ak m. So now these cases
20:33 together and these MSR together and on outside we got I and J,
20:40 is the same as we have. in matrix notation we can write that
20:46 like this. You need to have pay attention here because when you want
20:52 do this in matrix notation, the is important. And so we got
20:56 have uh the transposed rotation matrix on end like this, operating from the
21:05 a from the left and our is on a from the right now some
21:13 are symmetric. For example, you see here that this 13 component is
21:17 same as this 13 mathematician mathematical description that special case because they call that
21:28 gone and we'll turn, it turns that lots of the geophysical tensor.
21:34 right now, here's a really nifty of orthogonal answer if you take a
21:46 here and I'm using a towel here a reason. We're gonna later use
21:53 for stress and I'm already using it . Uh stress is an orthogonal tensor
21:58 that means that it's uh the same across the axis. And what this
22:08 say is that you can find a quarter system, transform this by rotation
22:14 the original court system, which you up for your own reason.
22:18 there's a special court system which which tensor knows about in this special coordinate
22:28 . Uh we got zeros off diagonal three different numbers on the diagonal.
22:37 this was all invented by the Germans in the 19th century. And so
22:40 have German words for this. Uh three quantities are called Eigen values and
22:46 new directions of the new court system convicted again is a german word,
22:52 means problem properly. So uh you , I wouldn't have chosen in English
23:01 I had been inventing this in the century, I wouldn't have called these
23:05 , values. I would have called special values or something like that,
23:09 I wasn't around to guide them and so they they chose the german
23:18 Yeah, I told you we want set up the physics as independent of
23:31 arbitrary choice of foreigner system. And a tensor a second rank It has
23:41 different invariants. Second rank has two . Uh and we're talking about three
23:51 . Let's see here, I think is true for if it has 17
23:58 anyway, there so let's uh Let's on three dimensions. And so what
24:08 says is that if you rotate into principal coordinate system and you find the
24:14 different Eigen values, some of those very and then if you rotate back
24:24 the other corner system into this gutter , this sum is still the
24:28 This some of these three is the the same as the sum of this
24:33 . They're all different. This one not equal to this one, but
24:36 sum of these three is the same the sum of these three. That's
24:40 of remarkable is it? And then this uh here is a second
24:47 you take uh uh principal values, Eigen values and combine them in this
24:56 and that's also in variant and then third one is called the determine.
25:01 you probably haven't heard of this but you probably have heard of this
25:05 . This determinant has a complicated I think I'm gonna show it to
25:09 in a minute. Um but it's of a good measure of the size
25:15 that tension. So here's how to to find the determinant for uh to
25:30 matrix. It's simply like this and furthermore for uh greedy. It's uh
25:38 like this uh let's not worry about these details Later, it's gonna get
25:49 . We're gonna have the elastic stiffness with four industries and the electric compliance
25:56 also with foreigners. Okay, so that generalization, let's um I have
26:06 little quiz here. What is the , A. B C. Or
26:17 , definitely. Okay, so I'm just remind you. Uh Right,
26:51 , here it is. Right Yeah, remember the transposed version goes
26:56 the end uh Here it is. so these others are just garbage.
27:16 , um now this will uh will getting into uh things like vector
27:24 And so uh I think uh you have had this uh as an
27:33 Did you did you talk about this symbol? Del Okay, so
27:40 so let's remember that, that's different uh than uh delta. It's sort
27:49 an upside down hills and it's a operator, but we don't uh normally
27:55 an error on it. And the on it are uh the elements are
28:02 these partial differentiation operations. And so uh were gonna encounter Dell operating on
28:12 and tensors but it's only going to from on the left side from the
28:16 side because it has this partial derivative it. So when it's applied to
28:23 scalar it makes a vector. And you can see how we get this
28:29 out of the scalar fight, which a function which agrees with position
28:36 But it's a scalar everywhere. And you can see this operator disappearing everywhere
28:48 and three different uh when we operate this del operator uh with the dot
28:59 on a vector youth, we get . And you see uh the eyes
29:05 repeated here. So we're gonna sum I and when we take McDowell with
29:15 cross product on another victim, we this more complicated uh expression here,
29:23 following the rules of scalar multiplication of . Jimmy uh vector multiplication of
29:34 So here is this a cross product this vector. And this vector.
29:39 this definition here is exactly as we before, where the X cross,
29:45 . You can go back and see now without the without the dot or
30:05 cross. So here's the dot and a cross but without the dot of
30:08 cross. It's simply making um uh makes it tensor by applying this operator
30:21 this vector. And the components of tensor are just like that and it
30:29 be applied to attention to make a . And uh why is it a
30:35 ? Well, because this thing is tensor, it has two indices.
30:38 . They are our NJ. And we're gonna um take multiplied by Dell
30:46 has uh partial with respect to I. And then we're gonna some
30:53 these eyes gonna leave only one index J. That's a vector. And
31:02 , it can be uh applied twice make another scalar and this is called
31:09 operation um named after the french Mathematician the 19th century of applause. And
31:20 the definition of that uh for a quarter system. And I should remind
31:28 that when we say Cartesian, we're to dig cart. There was another
31:32 mathematician in the 19th century now. There are some useful vector identities.
31:42 which uh I just stayed here without . Um and I'll just describe here
31:49 this one, take any scalar, the gradient of that scalar and make
31:56 cross product with the del operator. always his evil. So we say
32:02 curl the gradient is even and then similarly, if you take a vector
32:11 make the curl of it and then the dot product with doubt.
32:14 you get zero. So all of things are independent of the um of
32:20 arbitrary choices we make when we choose corporate system. Yeah. We are
32:35 always gonna use a Cartesian order Here's a perfect example. Suppose we
32:42 making a seismic wave for the charge dynamite and the dynamite is gonna send
32:48 out in cheers. Of course the is gonna just start that. But
32:55 the earth were homogeneous, it would out a uh a wave as a
33:01 . And you obviously want to describe wave in terms of performance systems,
33:06 um Kardashian Fortis's. So because of uh the beauty of the way we
33:14 up this mathematics, we can still the expression del squared and now it
33:21 this in terms of their food. , that's what it says.
33:33 Look at this. I suppose uh probably know that as this spherical wave
33:40 out from the dynamite, it uh in applicator. And in a simple
33:47 the the decrease in amplitude is um is exactly given by the the factor
33:57 over our. And so what this is when you take the laplace operator
34:03 the scalar function. One over This is the definition of it from
34:08 previous pace. Let me just back . Here's the previous base for the
34:13 . Component. So one over Doesn't have any derivative. Respected data
34:20 have any derivative. So when we're this operator to one over R.
34:27 get this here. And you can that at the at the origin where
34:34 set off the dynamite at that position we're gonna call that R. Equals
34:40 . And you can see that there's gonna arise here because we're dividing by
34:46 . And so uh there are special so this issue came up, you
34:52 , in the 19th century and they it out that uh it's not it's
35:00 infinity, it's not zero. It's we call the uh minus delta of
35:07 where delta of R. Is the D. Durant delta notation. And
35:14 what what is delta? R. a quantity Which is like a
35:24 And and it's zero everywhere except at equals zero. And exactly R.
35:32 zero. It's infinite. But it's just any infinity. It's a special
35:40 of infinity, which I will uh a special, it has special properties
35:48 I will describe to you shortly. gonna skip over this. Oh,
36:07 I didn't have what I thought I have. But remember that this delta
36:13 , delta is a special quantity which zero for uh or are not equal
36:24 zero, but exactly at R equals . It's infinity. And it has
36:29 special property which I'll tell you about . And we're gonna skip this
36:40 So uh this is a good I think for a break. So
36:47 take uh let's take a break here and come back at three o'clock,
37:01 you see this screen on your Okay, so this is basically where
37:08 left off. So now, now we refresh your mind about the
37:14 uh let's talk about stress and then gonna talk about strain and putting them
37:19 . So what is stress? This the distributed force per unit area.
37:28 it we're gonna give it the symbol and it needs two indices. One
37:34 to uh describe the orientation of the area. So here is a unit
37:40 , like a postage stamp. And it's um uh orientation is described by
37:46 perpendicular vector here. So the length this vector is the size of the
37:54 . So we're gonna have a unit , meaning it's gonna be describing the
37:59 of unit area of one unit of . And then on this unit area
38:07 going to have a certain force per area. And so all across this
38:13 unit area, there's gonna be a and the force might be lying in
38:17 plane of the of the area, it might be perpendicular to that
38:24 And so uh it's gonna have a vector. And that is going to
38:32 be accounted for by the second So the magnitude of each of the
38:39 , indicates the magnitude of that component the forest area. Yes,
38:48 So here's the stress tensor. It's special matrix whose components are made from
38:53 vectors. So that earlier we call a tensor and the industry's referred to
39:00 court in directions. And so if change the coordinate system, then all
39:08 elements will be different. So you according to set up a recording system
39:13 that and uh measure stress and and it in terms of those cornices.
39:20 maybe you later change into this or . Uh isn't for all of
39:27 So um let me interrupt myself. is my wife forgot saying this 100
39:44 . Oh uh is he finished? no, I will definitely not
39:59 But if you're happy with the you can give him a check uh
40:10 . Yeah. Lost her somewhere. , let's continue as your public call
40:24 . And this is how we uh how to transform a cancer.
40:29 so if we have a stress this dancer just uh change the a to
40:37 towel and then we can have the towel prime in another quarter system,
40:43 by this operation here. So let's at some examples. So here's a
40:48 system and uh so here's our unit in the one direction. Okay.
40:54 the force also in the one So that is gonna give us the
41:01 um component of Uh stress. So the same thing in the 33
41:09 And there is the same forces in same direction as a unit area.
41:13 the 33 component. And here is 13 component where the was where the
41:21 are entered in the one direction and forces are entered in the three
41:26 And so obviously, uh for a which is neither lying in the plane
41:33 perpendicular plant that can be broken down components in the plane and perpendicular.
41:40 we have to do is consider it these. So here is now where
41:47 um Uh stress component 3, 1 the unit area is in the three
41:55 and the forces in one direction. , who said, which annex a
42:04 . More important? Who cares? is why we don't care. Because
42:14 these two were different, then a stress would cause infinite spinning. It's
42:23 to show that that that if these are not exactly the same, then
42:28 body of the rock would would So I'm not talking about simply
42:35 We're talking about spinning. So those have to be exactly equal. So
42:40 uh an orthogonal cancer. So like said before, these uh faculty are
42:49 across uh across the diagonal. So , I've written it without recognizing
42:55 So we've got 31 here and 13 . But since it's symmetrical, we
42:59 that 13 is equal to 31. now you can see here that there's
43:07 of the stress sensor, only six are independent. Uh these these other
43:18 , I repeated history. Now suppose had a good reason to be describing
43:27 . And uh uh this other quarter . Well, we could uh describe
43:34 same stress, call it tau prime in this way and we know how
43:40 compute the towel primes from the towels the rotation matrix between this uh quarter
43:49 and the original one. And like said before one court system is special
43:56 this for lots of different part of . We would eventually find one system
44:01 looks like this and we call that principal court system and the three
44:11 I can values along here are called stresses. Now we don't have to
44:17 this corner system by trial and Their standard algebraic ways for things.
44:28 . In the kind of rocks where are exploring principal coordination, the principal
44:37 system has one access for it. that's pretty uh pretty obvious because the
44:46 sedimentary strata are laid down horizontally and they remain horizontal, horizontal, then
44:53 , that support system is going to one access vertical. But who knows
45:00 the other axes are um are You might think that if you have
45:11 uh center sentimentally patient of course, alternating sands and shales, all the
45:19 , all the as michael directions are . And that is the way that
45:24 thought about that this situation when I joined the L. A.
45:27 Now we realize that because of geologic , most of these um sedimentary
45:37 players are not contain a natural hydraulic . And so what that means is
45:45 uh all these asthma and directions are equivalent. And we found that out
45:51 trial and error, A lot of , sweat and tears found that out
45:57 my career uh corn bP. And everybody in the industry um understands
46:07 And basically because of that geologic these three stresses are um not equal
46:15 each other, but the horizontal stresses minimum maximum are significantly smaller than the
46:22 stress. Um uh about um uh 60 or 70% of the vertical
46:32 And uh and that's of course because is point is always pointing downwards.
46:38 so um uh the downward stresses the and the two horizontal stresses are significantly
46:46 , but they are normally close to other within a couple of percent of
46:51 other. So if the maximum horizontal say 70% of the Vertical stress,
46:59 minimum is maybe 68 or 69% very . Oh you do. Oh
47:12 Okay. Mhm. Right. Okay. How about that?
47:24 that's a lucky break. So you tell us next week whether uh what
47:29 said here is borne out by your . Okay, So um uh this
47:37 what I said here that these two uh normally similar to each other and
47:44 less than the maximum test, which vertical because of course gravity is always
47:49 things down, but there's no obvious way uh to know in many cases
47:58 not obvious how these two horizontal principal are oriented. Um uh It might
48:07 this way or might be this or might be this way and uh
48:12 do not know without doing some I don't know what sort of experiment
48:18 are doing, but um you can us about it uh next week or
48:23 week after. When you have some . Now, if there are fractures
48:34 and the subsurface as you see then it's probably gonna be pretty obvious
48:39 uh these fractures when they when they opened up, they were opened up
48:49 their flat faces perpendicular to the least stress. So when these rocks
48:57 they fail in the easiest way is . So, so this direction here
49:04 was the least compressive stress. And the plane of the fracture, that's
49:09 intermediate uh compressive stress. So that true. Obviously when the fractures were
49:19 , it might still be true or might not be. So uh I
49:25 okay. It's very common for your to say that the fractures give the
49:35 of the stresses well, uh but may or may not be true.
49:39 might be true that the fractures give orientation of the stresses some time ago
49:46 those fractures were formed and the stresses have changed since then. I mean
49:52 of years have gone by, who , snow were changed. So uh
49:57 you should keep an open mind on and I'll say it again. The
50:05 lie in the same plane as uh the vertical fence. So these are
50:10 by necessity because the vertical stress is and the greatest horizontal stresses also in
50:17 planes and stresses perpendicular because they open with their flat faces perpendicular to the
50:27 compressive stress. Now it looks to like in this outcrop, there are
50:32 fractures here. So that's an indication uh See here, you see it
50:39 . So that's an indication of a complicated um geologic history or it might
50:47 that these are um um shear These fractures here, we can tell
50:55 the orientation that their tensile fractures. they opened up, they opened up
51:00 move like this where the displacement is to the plane. In a shear
51:05 in particular is applied in the And that's what earthquakes. You're mostly
51:11 this kind of movie. And so one might be a shear fracture.
51:17 sort of suspect that from the way oriented. But um uh might have
51:26 because of uh geologic history translate your , stress is changing over geologic
51:35 within a fluid like the ocean and three principal stresses are equal. And
51:41 uh the convention is that a competitive uh means uh It's a pressure corresponding
51:55 matrix has a minus sign in so is positive minus sign. It's just
52:02 convention. Now, this situation here uh the stress tensor looks like
52:14 Uh We can describe it uh index using the chronicle delta, which is
52:21 than the direct delta notation talked about . And uh so the could the
52:32 of a delta is zero off the and equals to one on the
52:37 And so we can call that the to identity me. We I wish
52:43 Conacher had used the letter I instead the letter delta, but he didn't
52:49 so we're stuck with it and we and I think again he was a
52:54 working in the 19th century. So , so here's here's a quiz for
53:00 , Stephanie. Uh Got unit area picture here pointed in the Appointed in
53:10 two directions. This unit areas pointed of the plane. So you can
53:18 the square here, the flat square see that and then you can also
53:24 that the the force is in the in the one direction. So what
53:30 you call that? Which component of that B But it could also be
53:38 to one right, tell to one exactly the same as to how 12
53:44 to one is not one is not choice. So you you you chose
53:50 . Okay, so now we understand stress. Now we do strength strained
53:56 dimension dimension list defamation. So I to know to know five points this
54:04 and this uh and where they're located the original quarter system. We make
54:10 up. We decide where the origin . We decide what the directions
54:14 And uh after uh because I was this slide I made those decisions
54:20 These two points lying in the 13 flying in the 13 plane. And
54:26 are separated by the vector delta That's their position. Dis Inspector between
54:37 two points as a magnitude. As square of the magnitude given by elsewhere
54:44 delta X. Not delta X. these are not these are not Dell's
54:49 are adults. So uh forget don't what I said about the operator.
54:56 these are deltas not whereas Del is down delta. So these are
55:02 That's the name of this of this has a magnitude where adult X divided
55:10 itself is the square of the Now consider the same 2 2 points
55:18 by displacement field. So this one over to here and this one goes
55:23 here. You can see that the is now different. There's the difference
55:29 distance now delta X. Prime. so delta X. Crime is defined
55:34 this way. Remember what you said about addition of. So this is
55:41 applying that we have the original delta plus the difference between uh your ex
55:56 you at X plus del fax that's you. This Inspector has magnitude squared
56:03 by this and see I wrote this than I did before. There's no
56:07 product here. Instead there is uh in the indices I which is another
56:17 of writing the same. And since two points are close together, the
56:23 at the second point can be approximated this way. So this is a
56:29 expansion. And if you go to 101, you see a lot of
56:34 about the taylor expansion. And so taylor expansion is saying, if you're
56:39 at uh um quantity defined it at at this position, X plus delta
56:47 or delta X is a small number that is approximately equal to the quantity
56:53 at the original position plus the derivative that separation uh factor. Uh And
57:05 it's first order because we have only term here. If we had a
57:09 order, there would be another term with delta X square. So you're
57:17 are you Stephanie with taylor approximation? , so yeah, so so let
57:28 show you keep the recording running. And what I wanna do is I
57:35 to find where that show you where can find. So I'm gonna open
57:47 file and uh think they can't see on zoom but you can see it
58:00 and say it again. Oh So I'm gonna browse instead you will
58:33 on the blackboard, I had to Blosser and you can see that it's
58:42 a bunch of of uh of uh terms in alphabetical order and some technical
58:53 here. Uh So here's compressibility for here's a bunch of stuff about the
58:58 operator. Uh But what I want do is I want to go to
59:06 huh. I forgot what I was for here. Yeah, the taylor
59:14 . So so let's just come down looking for keys. Okay. See
59:23 there are several here. Okay. the taylor expansion and the next here
59:29 that there's another portrait. So here's expert. So uh that's all on
59:35 blackboard so you can refresh your Um that okay, so let us
59:50 to the lecture. Mhm. You're this position now I want to
60:19 Um Sure. Okay, this is we were, I think the recording
60:39 been running so uh what I just us how to find the taylor expansion
60:48 the glossary file which is on the . So we're gonna make this
60:56 And uh so then the new distance is this expression which reserved for the
61:03 distance vector with this um uh This field headed in and we're gonna replace
61:13 one with this approximation just like we up here and then notice here this
61:21 cancels this one exactly sort of left . Okay so uh now the two
61:28 are separated by this distance, original plus this derivative uh times. Uh
61:37 measure of how separate they were Now uh look what I did right
61:45 very cleverly. You see in this here we have two J's or
61:50 It means we're gonna sum over That's implied up here already. And
61:56 so what I did over here was simply changed the name of that.
62:00 call it a dummy verbal. If going to some, it doesn't really
62:04 which is what we call it. gonna some over 123 anyway, but
62:09 can't change the name of this one because that's the same as we have
62:14 . Okay, so then the new vector has this magnitude. And you
62:21 here we have a uh the ice and the ice component here. Uh
62:32 we're summing over eyes and in each these were summing over J. So
62:37 that out, uh expanding this product pretty complicated but we are going to
62:46 uh fits like this um notice Yeah, we're gonna rename certain of
62:57 repeated dynasty. So uh right two, we're sending over I so
63:12 means we can change the name of . So let's let's change uh those
63:18 to J. We're not gonna do here. We're gonna do it only
63:21 this term here, change I to . Right here and right here next
63:28 gonna change K. Two I in like this and finally we're gonna change
63:35 I to em in places like this we're gonna do all that, Then
63:41 can uh collect terms like this and we can see the change in in
63:52 distance, in the change in the of the distance is given by a
63:58 off L. Prime square from And we're left only with this term
64:09 along with this term. And so define the stress chancer to be
64:13 that's our definition of the stress. it involves derivatives of the displacement respective
64:24 . And if we and we have one half here for reasons which will
64:28 out to be clever later. Um so then the difference in the square
64:36 the distance is now given by this expression. So after all that
64:41 we came down to this where the are hidden inside this definition of the
64:52 . Now, normally in psychic application are small. So we can't neglect
65:01 . So, uh if this is , it means this one's smaller and
65:05 one is small and this is time small. So we're gonna neglect
65:11 . And so that's what we're gonna for our definition of strain when we
65:16 about seismic ways that to be honest you, uh, that's not always
65:23 . For example, suppose you have seismic source and a bunch of seismic
65:29 , but near the source. The are not small, right? Got
65:34 or a gun or something like It might be the strains near the
65:41 source are not small. So that's I showed you the exact expression
65:46 but then as soon as the wave a little bit away from the source
65:51 it doesn't have to be far less a wavelength. That's your wife,
65:57 know? And uh so that it's as it expands, its getting a
66:05 smaller according to geometric spreading basically when got all this energy in the wave
66:12 here and the same energy in the front here but spread out in a
66:17 a bigger wave front. So uh density at any point is given by
66:24 amplitude is uh gotta be decreasing and according to one over R. And
66:31 the time it gets less a wave away from the source or less
66:36 Uh this approximation is a better. here's some example launch single strain
66:44 We have uh Strain in the three . Changing this box into this rectangle
66:53 squeezing it down. And here is Who? Okay, sure. Um
67:08 I'm gonna be cheer strain in a . But next I'm gonna, this
67:13 a long longitudinal strain and the biometric is called the dilatation. So it's
67:22 sum of these three here. So talked before about how the some if
67:31 have um um you have an orthogonal like strain which is symmetric across the
67:44 . Then in that case the sum the elements along that, I can
67:51 the same no matter what um no matter what orientation the coordinate system
67:59 . Some is important, some is variant. So now let's look at
68:06 strain uh epsilon 13 is defined in way with ones and three scattered around
68:13 . And uh here it shows that displacement in the one direction, Which
68:23 in the three directions. So at point has moved from here, from
68:29 to here, it's moved in the direction, but it's moved by an
68:36 which is different depending on its three . So this one here is moved
68:42 in the one direction, but less it's less further away. So uh
68:48 gradient is measured by this. And now the question for you, Stephanie
68:54 what, which strand is shown Yes, yes, because it's got
69:06 uh displacement in the two directions, in the three directions. So it's
69:12 one. I'm gonna back up here a minute and here you can see
69:17 here that uh that uh the string is symmetric just by the definition.
69:28 , now we get the hook. is an interesting issue. Hook does
69:35 look like a very pleasant person I would say, but he was
69:38 extremely famous physicist in his day He was the first President of the
69:48 Society, which is the most prestigious of intellectuals in the UK. And
69:55 those days it was the most prestigious the world. And he was the
69:58 president. Um but why is it that we don't have um statues of
70:06 and paintings of it? Well, not quite clear. But here here's
70:11 interesting speculation. He lived in the time as Isaac and they didn't like
70:17 other. And uh he died before . Newton died. And so the
70:24 is that Newton uh caused all the the images of him to be
70:31 paintings and and uh statues and so . I don't know if that's true
70:36 not, but that's an interesting Okay, So this was in Hook's
70:43 was formulated a long time ago. not talking about the 19th century
70:48 We're talking about the 17th century and uh um wanted to describe the properties
70:56 springs or maybe any solid homogeneous spring or copper or whatever. And
71:05 says, Hook's Law says that stress strain are proportionate. That's all it
71:13 . Modern notation counting uh complexity of of all kinds of stresses and all
71:28 of strains, uh write it this , this strain is proportional to
71:35 And here are the set of proportional constants. You can see here that
71:41 since strain tensor, it's got two and stresses attention. It's got two
71:48 if they're gonna be proportionally to each . We have to have all these
71:58 elements to describe the proportionality between all different elements of tau and epsilon.
72:08 99 elements here. Nine elements And this quantity X. Is obviously
72:15 to have three times three times three 3 81 different elements spread across four
72:22 of tensor to describe this. And we're gonna some over ems and
72:29 So leaving the jays in the case unsung. So that asked. This
72:35 of the ask is called the elastic tensor. So intensive notation, we
72:45 this way this right here with a uh shoulder Now. Alternatively we can
72:55 the same idea as saying that uh uh stress is proportional to strain in
73:01 case we have a different set of councils which we call the elastic stiffness
73:08 in index notation. That looks like . Now we know. What we
73:15 say is that when we write books this way and this way, alternatively
73:22 way it's got to be the same . So if we take this component
73:28 strain and put in this product right , that's what we did here and
73:35 the parentheses. So this component of better be the same as this
73:43 So this one better be the identity . And so it's the fourth rank
73:49 indian matrix defined in this way. terms of the chronicle deltas, which
73:55 told you before, second rank any . No, in the general
74:08 This is very complicated. For Hook's law says that the 11 stress
74:14 upon nine components to strain. See these nine components are strained here.
74:21 icy tropic cases simpler than this. we will examine the ice tropic case
74:27 some detail before we got to get this. Let me pose you the
74:34 , the stress caused strain or the cause stress definitely. What do you
74:46 you think strain causes stress? Utah what do you think? That's
75:05 Hook said. Okay, so uh what hook said. And uh so
75:14 this is hooks lock right here. says that the two are linearly um
75:23 Oh then nearly related. And if write it in the other way,
75:32 versus stress, then it would look this. But it's still be the
75:37 thing does not know or care about causes which. Right? That couldn't
75:51 right. Uh The effect always follows cause, right? So Stephanie thinks
76:01 comes of stress and uh you tie didn't answer either way. So what
76:10 what we're gonna say, we're gonna it at this point at this point
76:13 gonna go with but we know that know it's not right. We know
76:20 uh things don't happen instantaneously. Uh gotta be a delay between the cause
76:29 the effect and it might be a delay, but it's gonna we can't
76:35 instantaneous effects like this even so I tell you that all of seismology is
76:44 on this false assumption. Not all , but a lot of size models
76:52 on this false assumption demonstrably false. so this is the first time that
76:57 encountered concept Rich is ignored in most courses like this. And we're gonna
77:13 on with this Hook's Law assumption, causality for the next seven elections.
77:19 then we're going to find out that all mobile. Okay, so,
77:25 not in the Senate election, but the eighth lecture, we find out
77:28 it's all wrong, what you don't to throw out what we learned in
77:33 first seven lectures, we'll be able use that with the caveat that in
77:39 in your mind that everything we're going talk about uh Next is wrong.
77:45 by the way, we already mentioned thing here, Hook's Law is assumes
77:51 uh it's applied to homogeneous materials like , because he was interested in
77:57 So, in the in the we don't have homogeneous materials. We
78:01 rocks which are composed of grains at minimum, many grains, and probably
78:09 minerals. So solid ist heterogeneous and , we got the pore space.
78:15 , rocks are so we shouldn't be applying Hook's Law to Iraq's at
78:22 Because back in 1660 said this is homogeneous material. So, again,
78:28 I'm going to tell you is that we're gonna ignore this for the next
78:33 lectures ignore this inconsistency. And we're apply anyway to just to think about
78:42 and race. And then in the election, we're gonna find out that
78:46 a more elaborate theory beyond elasticity. it's called moral elasticity. It's the
78:55 of uh shows how rocks deform under even and where they have minerals and
79:06 . And so uh real walks are obey the laws for elasticity. And
79:12 gonna find out then uh we don't to throw away everything. We're learning
79:18 seven lectures, we'll be able to an easy transition to uh apply to
79:25 in the first six lectures to Uh That's it. Another example of
79:33 uh when you when you're listening to for the next six lectures, I
79:39 you're gonna be thinking this is really , and it's true. We're gonna
79:43 it up to you towards the end the course. So, now to
79:49 answer to this, uh this stress train or vice versa. We gotta
79:54 analyze that with experiments, not with not with theory. Okay,
80:08 let's see what his relationship between Hook's and the science of thermodynamics. The
80:16 law of thermodynamics says that energy is . And so in a case like
80:21 uh says that uh the change in is given by uh the work
80:29 which is coming from outside some minus heat injected. Never mind the minus
80:36 . That's just a convention. And 3rd 2nd law of thermodynamics says that
80:43 amount of of heat is related to entropy, whether of course with the
80:51 constant of temperature. So yeah, apply these thermodynamic ideas to rock
81:02 Work done is infinite Testable deformation. given by the stress times. The
81:11 both of these are this is a , so this better be a scalar
81:16 the right side. So we're gonna over Jason case. Okay, so
81:23 gonna express the the stress in this , using hooks law, leaving strain
81:28 there and work uh rearrange these parentheses so on. And so then what
81:35 expressing, changing internal energy density is by this work term minus the heat
81:45 . Uh Remember here we're summing over . S. And ends. So
81:50 a lot of terms here. Um we're doing it separately for all the
81:55 and all the case, we're summing at Jason case also because we've got
81:59 end up with a scalar energy. , now, during wave propagation,
82:09 will assume that the uh energy, of the wave is happening so uh
82:19 . The frequency is so high that there's no opportunity for heat to flow
82:30 or out of the system. There's enough time now. Lower. Lower
82:37 , you know. Later on, going to be making low frequency
82:41 flying into seismology and the low frequency uh wave, we're gonna again assume
82:50 no heat flows out because in a frequency wave, the wave length is
82:55 . And so the heat has further flow. So, again, I
82:59 , to get out. And so assuming there's no flow of heat in
83:05 . That may or may not be . But let's go with that assumption
83:09 the next six lectures. Okay. , because of the symmetric form of
83:20 internal energy function or something over all things, uh it's got to be
83:25 that uh that the stiffness tensor has um symmetry that is. You can
83:36 these pair wise exchange the JK for mm and the mm for the
83:41 And you can convince yourself uh by at this for a while this is
83:47 . Then we gotta have this symmetry the assistance element. And that's a
83:52 good thing because these stiffness elements have different albums. It's a lot of
83:58 to get our minds around. And we're gonna have, we want to
84:02 to uh we want to take advantage arguments like this to decrease the number
84:08 independent elements in the stuff. And same is true for the compliance.
84:17 that's a good, very good thing these rank four tensors, we're gonna
84:22 it down to uh we can look it and think about it even though
84:29 a 43, there are some more , but it's got to have,
84:35 want you to look at that uh yourself that because of this, of
84:42 uh form of this function here, got to have all these other symmetries
84:48 the same for their compliance is now uh here's the payoff for all of
85:00 . Oh, yeah. Okay. . And then you get pain in
85:05 neck and your knees. Okay. . So this is an important slide
85:12 of these symmetries, the fourth rank may be mapped onto a six by
85:17 matrix with over only to uh And I'm sorry. This should be
85:24 and beta, not A and I'll fix that up later. So
85:30 is a matrix of rank two. that's really good because we can show
85:35 on the screen and it has all information that the uh has contained in
85:41 fourth rank tensor, which we can't look at. And here's the mapping
85:46 gets map para wise, for JK when J K is equal to
85:51 and one, then that maps onto . Well, that's pretty obvious,
85:55 for two and three. But here gets a little bit less obvious to
85:59 or alternative 32 maps onto a four 13 maps onto the five and 12
86:05 under six. And that's why this is six by six. Whereas this
86:09 is three by three by three by . So we with that by the
86:16 , this is this was invented by guy named Voight, it's exactly like
86:21 said here, boy. And so in the glossary as well. So
86:28 this is very good. Now we write this stiffness tensor on one screen
86:33 it's not a tensor anymore. It's matrix 6.6 Majors. Furthermore, it's
86:41 because of the previous symmetries. So really good. That means that it
86:48 uh only um Um 21 different uh here. Um 15 in the upper
87:00 , six on the long diagonal. that makes 21. And these are
87:05 same mistake. So we can just the lower triangle. And now it's
87:10 to look maybe semi understandable. We have a long ways to go.
87:16 corresponding compliance matrix looks just like this same kind of skip this. So
87:27 a little quest uh is this statement or false Stephanie? That's true.
87:37 . Is this statement true or That's true. Also, I like
87:47 way you're thinking about this because some these are trick questions. So uh
87:53 a good idea to think about Okay, so um um in the
88:00 general and I should remember that this is a property of the of the
88:05 stress and strain is what you do it. But this thing is a
88:09 of materials. So it's gonna be for every rock and it's going to
88:12 different whether it's tropical and subtropical. the question is in the most general
88:20 psychotropic case. How many independent Now this is the first time.
88:25 think I mentioned the word and I you but I gave you the clue
88:30 Because it's six x 6 symmetric. gonna have 21 different elements.
88:39 So now let's now turn our attention um compliance. So it's now
89:03 Okay. So so this is the um so you know, it's a
89:10 easier to think that we're gonna use the stiffness is to do wave
89:16 But it turns out that it's easier think about your guy waiting these things
89:21 terms of compliance is and then at last minute will change to stiffness.
89:26 you'll see how we go along It's easier to think in terms of
89:29 is. So this is what the matrix looks like. It's not a
89:34 anymore because it's six by six does transform like a tensor. Okay.
89:43 right here it uh it tells you most general anti psychotropic cases is
89:49 I don't I don't know if anybody ever measured all these compliance is on
89:56 crystal or anything. Uh Maybe but not very many. And all
90:05 are different from each other in Uh huh. Crystal. I think
90:13 think calcite Might have 21 different. two. I'm not sure but we're
90:22 figure out the actual topic case one at a time. So for example
90:27 the 11 strain. So here's the of the 11 strain in terms of
90:32 . Is and this some Or stresses that out. There's nine terms because
90:41 all the different amazon at ends using two index sport notation. This uh
90:51 like this now. It says what to the 2? Well, there
91:05 any twos, but you know that this one is uh you know,
91:11 to one is equal to tau one , you know that. And
91:16 uh you know that uh these uh are equal. So these two some
91:22 to make it to what uh uh no twos down here. So that's
91:34 and uh slide that I just skipped . So maybe I shouldn't skip
91:44 So let's go back, let's go here and not skip over this thing
91:52 here. Yes. So I gave the recipe for converting the uh the
92:08 tensor to this matrix. And then said that for the compliance is it
92:15 uh the same. But there's uh a little thing, a little thing
92:22 when you uh convert from the four notation to the to index notation for
92:30 is you need to have these factors one half and 1/4. Let's make
92:36 all work out. And so uh the uh in the glossary, under
92:43 I think in Math 101. Uh an explanation of why these factors to
92:50 . I should not have skipped over one. And then, so when
92:57 sum this together with this that uh up for the missing tooth sam.
93:06 is this comes out of the conventions we have adopted. And uh you'll
93:18 immediately why that's not important for the the argument that follows next. So
93:25 gonna consider a special case uh actual on a horizontal ice. A tropic
93:33 . So here's a cartoon of the . And we got the stress uh
93:38 on one end. And of course got to push back on the other
93:42 . Otherwise the cylinder would move. we have that we can also measure
93:49 strain. And so I think you that if uh if the stress is
93:57 in this way, the strain is be uh it's gonna the cylinder is
94:03 get shorter and the strain is going be a negative number. So um
94:11 go back here. These are all current uh in this, in this
94:17 case, in this case of horizontal . Okay, um horizontal compression of
94:28 an isotopic cylinder. That's a That's a zero. That's a
94:32 That's a zero. That's a The only term which is non
94:35 The only stress which is non zero this one? That's what we
94:40 That's why it's simplified chair. So this makes the definition of young's
94:51 . This ratio of 11 stress divided 1111 strain divided by one month
95:00 That gives you the Youngs Marshals and market has has to be defined in
95:07 way um uh as an inverse of compliance, Oliver because um I wanted
95:26 have the same physical dimensions as the markets in the sugar bowl. And
95:32 this was done by Mr Young who a english Physicist back in the
95:40 That's the ratio of these two numbers property. So in this way we
95:49 Um identified, we have analyzed the component, the 11 component of the
95:56 compliance matrix for anisotropy writing. since the rock as I stop
96:03 it's the same. No matter how rate, it's very easy for us
96:07 populate these next two positions. Now 11 stress, there will also be
96:15 tutu strength. So here is the strength written out in terms of all
96:20 components. And so all these stresses are uh the this stress is zero
96:29 this stress is not zero. And we're analyzing uh SG1 all the other
96:39 . So this is the one that analyzing this ratio here of two
96:48 uh radio strain as a result of stress. So I think you know
96:56 as the uh huh If you squeeze cylinder it gets fatter. Okay,
97:04 that's gonna involve Dawson's ratio. So gonna see that in just a second
97:10 is personals ratio. And that's the of these two strains, same
97:16 These two strains? It's non And we have a minus sign in
97:21 because uh epsilon stresses 11 strain is and um uh to to strain as
97:37 . So after these manipulations you reduce S12 0 - Parsons ratio over uh
97:50 models. So that's what this And because the rock is I should
97:55 , we have the same thing Now we ask question here, uh
98:07 there any share strength in response to normal stress? And the answer is
98:13 know to get a shear strain. gotta twist it or twist it,
98:19 it. You can't squeeze it. this sure strain is zero but it's
98:26 uh Right in terms of folks law these terms now these these terms are
98:38 , but that term is not That stress is not zero. So
98:42 must be that the compliance uh element you. So that means that uh
98:50 got a zero here. And since psychotropic all those are zero.
99:07 let's think about a different kind of , sheer stress. Taking that same
99:15 oriented in the same one direction and sharing it in the two directions.
99:21 we define a sure Marcus, I know who did this the first
99:25 but it wasn't young and in his was it wasn't hooked, I don't
99:31 . And it's the ratio of these properties. See it's it's the ratio
99:36 stress strain and that way this marvelous the same dimensions as yellow because of
99:48 of stress and strain er in verses each other. We know that s
99:57 over C 61 over new by that . So we reduce this and because
100:05 rock has ice a topic, it's same for these others and the sheer
100:11 in the other direction. So we've the complete compliance matrix in this
100:19 Very simple. Thinking about a cylinder all and we introduced uh three different
100:30 . Young's models share models and processes . Yeah, this is not quite
100:39 obvious. Uh So I'm just stating result that if we rotate the sample
100:45 any direction, the compliance tensor must the same. And so uh with
100:52 requirement we deduce that Parsons ratio is to young smarts and care models in
101:00 way. Um We need to look , you know, uh Mhm.
101:08 for to verify that. Or you be able to do it yourself.
101:15 see here. I think, I you really have to, we're gonna
101:26 to put up better. Yeah. let's consider the case of ice.
101:33 tropic pressure which we thought about This is the case of uh public
101:41 for example, in the ocean. it's got a positive pressure everywhere.
101:46 the same pressure in all directions. change in volume. Is this some
101:53 stress? Um some of strange components see Hook's law. We reduce uh
102:03 this sunk here is a single um single string component. And we're summing
102:11 all J's and also all M. . And ends. That comes from
102:16 law. But the stress is in special case, it's simplified like we
102:23 before. And so what that means that we need only to some over
102:28 and Adam's multiplying by a minus And so uh these are the terms
102:36 seminal work. Just these uh with in the Matrix uh it's easy to
102:49 these in the matrix. Uh and easier to write them. And uh
102:56 the index notation like so so when sum these up, listen take a
103:03 and there's the song. And furthermore know that this uh ratio of the
103:13 change as a fraction divided by the which caused that divide by the pressure
103:20 the minus side. That is the of the buck Marcus. And we
103:25 call that the in compressibility and that's in terms of uh other quantities.
103:36 found a few pages earlier this relationship these three. So combining that uh
103:44 we just learned we find in terms K. And mute the apostles ratio
103:49 be written in this way. Let me ask you Stephanie. Do
103:58 know what is the maximum value of ratio? In an ice and tropic
104:13 zero. Uh That sounds to me you're thinking of the minimum value.
104:18 the maximum value? Yeah, it low. Okay, so uh so
104:26 is uh, here's the answer in next one. There are two special
104:31 of interest. So if the share zero, then you put zero in
104:35 , you gotta have that's the case fluids. The other special case of
104:42 is when K is equal to zero equals zero. In here, you
104:46 a minus one. So what this is that theoretically you could have um
104:54 With approximates racial less than zero. , most students, when I asked
104:59 the question that I asked you, say, well the maximum is one
105:04 . That's for sure. Normally say minimum zero. But this argument has
105:10 you the minimum -1. And I to have a piece of uh sponge
105:17 which was especially made to have a process ratio. So when you squeeze
105:24 cylinder of it, it actually got fat. Uh that was a specially
105:32 sponge. Well, that dispirit I don't know uh in real life
105:38 always find positive association and furthermore dependent the saturation state. Um and
105:51 you can show these $2.01 half and one are politically the absolute maximum.
106:02 because they come from these two simple 008. Now for action tropic
106:11 it depends on all these factors and usually lies between 0.1 and 0.4 point
106:18 is pretty close to 40.5. And you can imagine that's a pretty mushy
106:22 , uh muddy rock or maybe a itself. Uh And uh the opposite
106:29 would be a stiff rock all Uh but uh you know, so
106:40 it's really not legitimate for us to talking about this, because at this
106:46 we're dealing with Hook is only dealing copper and glass and iron. He's
106:50 dealing with rocks, we're gonna get to rocks. Uh at the end
106:55 this uh series of lectures and all things are going to come into it
107:02 uh it's going to be an important for scientific exploration. Okay,
107:11 Stephanie, this is the same question we had before, which I answered
107:15 you. But now this is for and tropic loans and has how many
107:20 components. Well, that's for, the general case for tropic rocks.
107:29 many independent? Well, let's let's back and just look at see that's
107:40 value of this. Okay, there's picture of it. How many independent
107:44 are there? Well, there are different components. But are those all
107:58 , Remember the sigma is related to immune. So there's only so there's
108:04 only two independent components here and sort the obvious way to pick up is
108:10 would be one and this would be other one and these are then uh
108:16 in terms of those uh through the the uh the formula that I showed
108:23 the next page which is right Or you could do it another
108:26 You could say that this one is and this one is independent and this
108:30 gets derived from those two, but it's more natural to say that act
108:38 these two are independent and these are functions of that through this formula
108:51 Okay, so that was the question we just looked at. Next question
108:58 is this statement true or false? , that's a trick question. It's
109:08 to one over young that that was issue. Tropic rocks parcels ratio always
109:18 between 0.5 uh Stephanie put you on spot. Read this carefully.
109:31 What the .4 I said usually. this says always. Okay, so
109:39 gonna call that true. Uh and about the negative process? Inspiration.
109:44 , that's not for. What's Okay, so that one's false.
109:50 , now we get to elastic Yeah, stiffness is related to stress
110:06 to all the strings. So we cooks law. Not in this
110:11 but rather in this form. They to all the strength and this is
110:21 this is the form. We're gonna for wave propagation and uh for a
110:26 , which will become clear tomorrow. , So remembering that compliance and stiffness
110:37 in versus for each other. For inverse. For our fourth rank matrix
110:43 defined in this complicated way. Then sixth component. The 6 6 component
110:54 um Oh yes, let's let's apply to the 66 components. So up
111:13 we're gonna put jake april's 12, uh making it six. And
111:19 Q. Is uh is uh uh to 12. That's uh also a
111:27 . Now we're gonna sum over all M. S. And ends for
111:34 situation and we end up with a half. So why is that?
111:44 , delta 11 is one, delta is one. Delta 12 is
111:49 So uh this uh when we do this, some summing up all these
111:56 , uh We get for this case 121, 2 components of the identity
112:04 is a 1/2. So now let's that one half on the left side
112:09 this thing up here. And uh then uh implement all the uh what
112:21 learned about the eye stopping materials with those zeroes and everything. Uh This
112:26 borrows down to this Um uh two only. And so we know that
112:33 uh Yeah. Uh No, the 1212 converts to S 6,
112:47 Remember that page? Where I said for the compliance is you can't just
112:53 that s one uh Rebecca for the we have a recipe that says that
113:01 C1212 is given by C 66. for the correspondent compliance we needed a
113:08 of 1/4 has explained in that previous . And uh same thing for the
113:15 term. These two terms add make a half and uh then this
113:22 one half cancels this one half. so we deduce then that C 66
113:28 given by mute. C66 is the . So that's why what we did
113:37 we went through the compliance is element element for uh the topic material worked
113:46 all out and now we're converting we have three things to convert um
113:53 models to the young models and Parsons . We're gonna convert those three and
113:59 how they show up in the own it's different. So what we've
114:04 just now is proven the stiffs and . We already know that it has
114:09 form and we just proved that Sure that goes into these these three
114:20 And furthermore, because of all the in the compliance matrix, you know
114:25 all these things are zero to and component is given in this way uh
114:32 this expression here, uh work out that means. So this fourth rank
114:40 matrix with 11111 an actual topic we can work out the uh this
114:52 and you find young models is given this air by this form. Similarly
115:02 12 models worked with the same kind uh argument and you come up with
115:09 answer And some of those simultaneous mint for C11 and C12. And you
115:21 then that the stiffness elements, stiffness looks like this. You will not
115:29 familiar I think with this marginal em but you're as soon as I won't
115:43 it out, you're immediately uh ceo recognize that K plus four thirds
115:50 That's the modules which uh governs the wave velocity. But you never find
116:00 any wave propagation formula, you never K by itself. You always find
116:08 in the combination K plus 4 So let's give it a name.
116:14 called the launching all stiffness. And that that is gonna govern p
116:23 propagation and it's gonna be the same all three directions, 1122 and
116:29 Because this rock is a property. then off diagonal here we have some
116:35 uh non dementia, some non zero . And if you remember uh your
116:47 I'm not sure if you ever would covered this in a previous court,
116:51 gonna see that these terms are calculated terms of these two um independent
117:02 And and so if this if this independent and not calculated from these
117:18 then you would have a third kind weight this one. Government share,
117:24 ? This one governs pre write this would be a third kind of
117:28 but it's calculated from from these So it's the same. Uh it's
117:37 it doesn't imply there's a third kind and so this is what you already
117:44 that uh eight plus four thirds over governs p wave velocity, you overall
117:53 , velocity, yeah, quantities in propagation, young drones and prosperous ratio
118:02 appear anywhere anywhere. This element uh we defined as m minus to mu
118:12 actually has a name. LeMay. was a french actually, he was
118:18 priest, but he had time on hands and he did some mathematics,
118:23 century mathematics. And uh this parameter named after and it's defined in this
118:32 . So you put in K plus 30 me right here and you can
118:36 the lambda, K and new right . What? Just like in the
118:43 of Ian and sigma and wave propagation wave propagation equations, lambda never appears
118:53 . Uh A lot of people are about lambda and this statement is
119:07 It never appears in a way for prepublication will encounter it. Um
119:14 in this course, from time to , mainly to make equations look simpler
119:22 you can replace it. Some like with a single parameter. Maybe it's
119:28 it super. Yeah, there it some parts of but you have to
119:34 yourself out to the side, the is related. Now there's a
119:49 classroom exercises, you know, I that's not true. I think I
119:54 to put the spreadsheet there. But will give it to be an Excel
119:58 . And you can just put in numbers you want. Actually not any
120:03 . It's gonna be asking you to inside me crime. Because we're gonna
120:10 that you have some intuitive guess in mind about what suitable parameters for segment
120:19 is. So I will apologize in . I will tonight I will put
120:26 spreadsheet um uh Like okay, so let me ask you Stephanie, you
120:39 a similar question or compliance is and stiffness is what's the answer to
120:47 Great two Yeah. Yeah. Or could do it and then land it
120:54 . Uh and then derive you from . But there is no way of
120:58 to lambda. Uh I think you it right. Uh share waves are
121:03 by mu and P waves are given M. And lambda doesn't cover
121:12 Okay. Um Here's a this question similar to the one that you uh
121:17 you before This is false, It's equal to M. Not two
121:25 . Okay. So uh I made reputation in your physics by anisotropy.
121:33 naturally we're gonna consider anti sophistry Not because uh of what I
121:39 not because I'm an expert in But those rupture anisotropy. So we
121:45 return to that in our last lecture say a few words about it
121:51 So uh here it says that if really want to deal with complicated an
121:56 crystals, you need this complicated stiffness this, The simplest geophysical case,
122:06 simpler than that, but still pretty . Um simplest case is unstructured
122:16 answer in that case uh Stephanie count the number of independent uh justice elements
122:25 are here, let's count here. got one, 2, 34 and
122:40 . And then this one is calculated five. So that's a bummer.
122:44 in the simplest case We go from , it doesn't mean that there's five
122:52 waves propagating uh shales. What it is that there are We will learn
123:00 the in the 10th lecture that there three different ways not to means a
123:06 wave and to share with you and uh philosophy which vary with direction and
123:15 polarization and um uh the equations are complicated And they involve all five of
123:23 programs. And then if you I that in the icy tropic case,
123:32 these two are the same. So think those and I he was an
123:39 isotopic equations. And put in equivalence like these two here reducing them to
123:45 top inflation. And sure enough they exactly. So um here's our summary
123:55 is the study of the deformation of materials under stress. What is
124:01 It's the 4th per unit area Which applied to material, since the forces
124:07 vector in the area is specified by normal vector. The stress is a
124:13 three x 3 tension strain on the hand, is the non dimensional measure
124:20 deformation. Also symmetric, three x tents. Books. Law is the
124:28 that stress and strain are linearly proportional each other. Either one is specified
124:34 hook is either calls or effect apply stress and the strain appears instantaneously or
124:43 apply a strain. And the stress is back instantaneously. That's according to
124:53 . Plastic compliance is the ratio of of stress. It is 1/4 rank
124:59 , although we can write it for purposes as a six by six maiden
125:06 your topic materials. Various components are in terms of only two independent
125:12 It can be taken as young's models shares function to me. The other
125:21 , elastic stiffness is the ratio of to strain. Also a right for
125:27 press your topic materials. You have two independent parameters which can be given
125:32 the launch vehicle modules and the I always say about elastic. And
125:42 that brings us to the end of lecture miraculously we're pretty well on
125:47 Even given all of the interruption delays have. So what we will
125:54 Um um For tomorrow, Yes, one, we will be meeting at
126:05 30. We'll work for four have one hour break at work four
126:10 hours before that. Here is your assignment Stephanie you're gonna bring into class
126:20 a written question concerning um question that to you probably during this what.
126:34 um so we'll spend the first and if you have more than one since
126:40 you're the only student here uh you special dispensation and you can bring in
126:47 questions but at least one written. like to do a written questions because
126:55 students are shy. They try to up their hands and you know exposing
127:00 . But that's the best way to response to. Actually. The best
127:05 to learn is to talk with your students but you can't do that.
127:10 uh maybe uh Utah can answer some your questions out of battle plaza.
127:20 Next next best question is ask your questions and get get your direct answer
127:26 . So uh at the end of be a long day tomorrow end of
127:36 I will hand out to your And the quiz will count for um
127:44 of your grade. Now this quiz be a quiz. Unlike anything you
127:50 ever taken before. I expect. gonna be unlimited time. You can
127:57 as much time as you want. book. The only question is the
128:05 requirement is you do it all at time. So uh you're not gonna
128:10 uh hours and hours you can spend much but find find a place where
128:19 won't be disturbed by your baby for for a couple of hours at least
128:27 then do the question will open it and that's you will be given to
128:31 in paper form, Open it up that time and starting to take as
128:37 time as you want. But at end I'm gonna ask you how much
128:41 you took uh and that's not gonna for or against you. But I
128:46 want to know because I'm not very at composing test questions. Uh I
128:55 to make them too easy and I to make them too hard. And
129:00 this uh this question will be only a few questions on the quiz and
129:12 might be able to do it in hour but you might want to take
129:15 than my parents. Give yourself that . You can have with you whatever
129:23 you want, you can have with these course notes, you can download
129:27 from uh from blackboard, you can the textbook by the way which is
129:33 available on amazon out of print. it's a good text that it was
129:42 by our own bob sharp professor in department a few years ago cost a
129:50 bucks but it has a lot of about this thing. Uh and uh
129:57 can buy that you want, you open in front of you, anything
130:01 want. And the only thing is can't consult with others. So so
130:06 I will not be sitting by your . Okay? And uh so then
130:12 you're done, u um um post paper and you can write out your
130:18 even on the sheet of paper that give you, there'll be enough room
130:21 that or you might need extra sheets you might attach those or uh then
130:28 want you to return it to your . Uh maybe in digital form.
130:35 you want to scan it and send him. You have his email address
130:40 you can find him if you wanted senate to give them papers uh
130:46 You can do that. You're working between uh and your deadline is gonna
130:51 on Wednesday at midnight. So I'll you plenty of time to uh find
130:58 couple of hours maybe you only need uh On at least one hour for
131:06 gonna be undisturbed. Got all your with, you got some water with
131:11 . Uh and you do the close the exam, pass it back
131:17 Utah. He will pass it to and I will grade it by um
131:24 friday afternoon to meet. And so you can, you can write to
131:31 and decide after you drive home today tomorrow when we meet, you will
131:36 how the friday afternoon commute looks like you can tell me tomorrow whether we
131:42 to meet the next friday at five 5 30. Okay so um um
131:54 think that's all. Don't forget the . And me my homework is to
131:58 the exam, which I have not yet. I will do that and
132:02 bring it to class tomorrow, give to you in paper form even
132:08 Yeah. And so any questions. so um Um so we won't convene
132:22 uh today tomorrow at 8:30 AM and with lecture
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