1 Equations of linear elasticity

Transcription

1 C5.2 Elasticity & Plasticity Daft date: 17 Januay Equations of linea elasticity 1.1 Hooke s law Robet Hooke (1678 wote... it is... evident that the ule o law of natue in evey spinging body is that the foce o powe theeof to estoe itself to its natual position is always popotionate to the distance o space it is emoved theefom, whethe it be by aefaction, o sepaation of its pats the one fom the othe, o by condensation, o cowding of those pats neae togethe. Hooke s obsevation is exemplified by a simple physics expeiment in which a tensile foce T is applied to a sping whose natual length is L. Hooke s law states that the esulting extension of the sping is popotional to T : if the new length of the sping is l, then T = k(l L, (1.1 whee the constant of popotionality k is called the sping constant. Hooke devised his law while designing clock spings, but noted that it appeas to apply to all spingy bodies whatsoeve, whethe metal, wood, stones, baked eaths, hai, hons, silk, bones, sinews, glass and the like. In a standad expeiment to veify Hooke s law fo some solid mateial, we would subject a od, of length L and coss-sectional aea A, say, to a tension T, as shown in Figue 1.1. Fo most mateials, we would again discove that the stetching of the od obeys (1.1, fo some constant k. Futhemoe, it is obseved in such expeiments (and intuitively easonable that the stiffness k is popotional to the coss-sectional aea of the od and invesely popotional to the length. Thus we can wite (1.1 as ( T A ( l L = E L, (1.2 whee E is a constant fo any given mateial, known as Young s modulus. The quantity on the left-hand side of (1.2, namely the foce applied to the od pe unit aea, is called the stess, while the dimensionless quantity (l L/L, measuing the extension elative to the initial length, is called the stain. Incidentally, a futhe esult of a uniaxial extension test like that shown in Figue 1.1 is that, while stetching along its axis, the ba shinks in the tansvese plane by a facto popotional to the stain. Fo example, if we stetch a cicula od with initial adius R, then the adius afte the tension is applied is found to be given by ( ( R l L = ν, (1.3 R L whee ν is called Poisson s atio and again is constant fo any given mateial. Both E and ν ae well chaacteised fo typical solid mateials such as metals, ocks, ceamics and so on.

2 1 2 OCIAM Mathematical Institute Univesity of Oxfod y L T x A T z Figue 1.1: Schematic of a unifom ba being stetched unde a tensile foce T. Hooke s law woks well fo most solids, povided the stain does not get too lage. Thee ae vaious ways in which it can fail. The fist is that the elation between stess and stain may cease to be linea, as can be seen fo example by stetching a ubbe band. To descibe lage stains of such mateials, it is necessay to use nonlinea elasticity see the complementay couse C5.1 Solid Mechanics. Most mateials, howeve, cease to behave elastically long befoe the stain is lage enough fo nonlinea effects to be impotant. Fo example, a bittle mateial will factue if it is subjected to an excessive stess. On the othe hand a ductile mateial will instead stat to defom ievesibly when it exceeds its elastic limit; this behaviou is known as plasticity. We will show how both of these phenomena can be descibed mathematically late in the couse. Hooke s law (1.2 is the simplest example of the all-impotant constitutive law elating the applied foce to the displacement of a solid body. To genealise this law to a thee-dimensional continuum, we fist need to genealise the concepts of stain and stess. 1.2 Lagangian and Euleian coodinates Suppose that a thee-dimensional solid stats, at time t = 0, in its efeence state, in which no macoscopic foces exist in the solid o on its bounday. Unde the action of any subsequently applied foces and moments, the solid will be defomed such that, at some late time t, a paticle in the solid whose initial position was the point X is displaced to the point x (X, t. Fo any such paticle, the Lagangian coodinate X maks its initial positition, while the Euleian coodinate x maks its cuent position. In othe wods, the Euleian coodinate x is fixed in space, while the Lagangian coodinate X is fixed in the mateial. The displacement u(x, t is defined in the obvious way to be the diffeence between the cuent and initial positions of a paticle, that is u(x, t = x(x, t X. (1.4 Many basic poblems in elasticity amount to detemining the displacement field u coesponding to a given system of applied foces. We assume that the solid is a continuum, so that thee is a smooth one-to-one elationship between X and x, i.e. between any paticle s initial position and its cuent position. This will be the case povided the Jacobian of the tansfomation fom X to x is bounded away fom zeo: ( xi 0 <J <, whee J = det. (1.5 X j

3 C5.2 Elasticity & Plasticity Daft date: 17 Januay The physical significance of J is that it measues the change in a small volume compaed with its initial volume: dx 1 dx 2 dx 3 = J dx 1 dx 2 dx 3. (1.6 Thus the mateial is in a state of net expansion if J > 1 o compession if J < Stain To genealise the concept of stain intoduced in 1.1, we conside the defomation of a small line segment joining two neighbouing paticles with initial positions X and X + δx. At some late time, the solid defoms such that the paticles ae displaced to x = X + u(x and x + δx = X + δx + u(x + δx espectively. Thus we can use Taylo s theoem to show that the line element δx that joins the two paticles is tansfomed to δx = δx + u(x + δx, t u(x, t = δx + (δx u(x, t + O ( δx 2, (1.7 whee (δx = δx 1 + δx 2 + δx 3 = δx k, (1.8 X 1 X 2 X 3 X k using the summation convention hee and hencefoth. Let L = δx and l = δx denote the initial and cuent lengths espectively of the line segment. Then, to lowest ode in L, l 2 = δx + (δx u(x, t 2. (1.9 The change in length of the line element may thus be witten in the fom l 2 L 2 = 2e ij δx i δx j, whee e ij = 1 ( ui + u j + u k u k. ( X j X i X i X j It is clea fom (1.10 that the stetch of a line element in the solid is measued by the dimensionless quantities e ij ; indeed, the stetch is zeo fo all line elements if and only if e ij 0. It is thus natual to identify e ij with the stain. In this couse, we will only conside mateials undegoing small defomations, so that the displacement gadients u i /X j ae all small and we can neglect the nonlinea tem in (1.10. In addition, we note that the chain ule elating diffeentiation with espect to Lagangian and Euleian vaiables eads X i = x k X i x k = x i + u k X i x k, (1.11 so that, to leading ode, the deivatives with espect to Euleian and Lagangian vaiables ae equal. Hence we can wite the lineaised stain as e ij 1 ( ui + u j 1 ( ui + u j. ( X j X i 2 x j x i Anothe consequence of the small-stain limit is that we can lineaise the Jacobian J to obtain ( ( xi J = det = det δ ij + u i (1.13 X j X j ( ui 1 + T u k +. (1.14 X j x k

4 1 4 OCIAM Mathematical Institute Univesity of Oxfod By consevation of mass, the density ρ of the defomed medium is elated to the initial density ρ 0 in the est state by ρ = ρ 0 /J. Hence, the fact that J 1 means that the density is constant to leading ode. The small change in volume is measued by e kk = u k x k = div u, and the mateial is locally expanding if e kk > 0 o contacting if e kk < 0. Now let us ask what would have happened if we had calculated the stain using a diffeent set of coodinate axes. Suppose we define new coodinates x = P x, whee P is an othogonal matix (satisfying P P T = I so this just epesents a otation of the axes. Then the stain calculated with espect to the new coodinates is given by e ij = 1 2 ( u i x j + u j x i, (1.15 whee u i ae the components of u with espect to the tansfomed axes. Now it is easy to show using the chain ule that E = (e ij and E = (e ij ae elated by E = P EP T. (1.16 Hence the 3 3 symmetic aay E tansfoms exactly like a matix epesenting a linea tansfomation of the vecto space R 3. Aays that obey the tansfomation law (1.16 ae called second-ank Catesian tensos, and E = (e ij is theefoe called the stain tenso. Almost as impotant as the fact that E is a tenso is the fact that it can vanish without u vanishing. Moe pecisely, if we conside a igid-body tanslation and otation u = c + ω x, (1.17 whee the vectos c and othogonal matix ω ae constant, then E is identically zeo. This esult follows diectly fom substituting (1.17 into (1.12, and confims ou intuition that a igid-body motion induces no defomation. 1.4 Stess Conside a small suface element, whose aea and unit nomal ae ds and n espectively, contained within the defomed medium. Suppose the mateial on (say the side into which n points exets a foce df on the element; by Newton s thid law, the mateial on the othe side will also exet a foce equal to df. In the expectation that the foce should be popotional to the aea ds, we wite df = σ ds, (1.18 whee σ is called the taction o stess acting on the element. Fist conside a suface element whose nomal points in the x 1 -diection, and denote the stess acting on such an element by τ 1 = (τ 11, τ 21, τ 31 T. By doing the same fo elements with nomals in the x 2 - and x 3 -diections, we geneate thee vectos τ j (j = 1, 2, 3, each epesenting the stess acting on an element nomal to the x j -diection. In total, theefoe, we obtain nine scalas τ ij (i, j = 1, 2, 3, whee τ ij is the i-component of τ j, that is τ j = τ ij e i, (1.19

5 C5.2 Elasticity & Plasticity Daft date: 17 Januay x 3 S 1 n S 2 x 2 S 3 x 1 Figue 1.2: A tetahedon; S i is the aea of the face othogonal to the x i -axis. whee e i is the unit vecto in the x i -diection. The scalas τ ij may be used to detemine the stess on an abitay suface element by consideing the tetahedon shown in figue 1.2. Hee S i denotes the aea of the face othogonal to the x i -axis. The fouth face has aea S = S1 2 + S2 2 + S2 3 ; in fact if this face has unit nomal n as shown, with components (n i, then it is an elementay execise in tigonomety to show that S i = Sn i. The outwad nomal to the face with aea S 1 is in the negative x 1 -diection and the foce on this face is thus S 1 τ 1. Simila expessions hold fo the faces with aeas S 2 and S 3. Hence, if the stess on the fouth face is denoted by σ, then the total foce on the tetahedon is f = Sσ S j τ j. (1.20 When we substitute fo S j and τ j, we find that the components of f ae given by f i = S (σ i τ ij n j. (1.21 Now we shink the tetahedon to zeo. Since the aea S scales with l 2, whee l is a typical edge length, while the volume is popotional to l 3, if we apply Newton s second law and insist that the acceleation be finite, we see that f/s must tend to zeo as l 0. Hence we deduce an expession fo σ: σ i = τ ij n j, o σ = T n. (1.22 This impotant esult enables us to find the stess on any suface element in tems of the 9 quantities (τ ij = T. Now let us follow 1.3 and ask what happens to τ ij when we otate the axes by an othogonal matix P. If we define τ ij to be the x i-component of stess on a suface element

6 1 6 OCIAM Mathematical Institute Univesity of Oxfod τ 22 τ 12 τ 21 x 2 τ 11 τ 21 δx 2 δx 1 G τ 11 x 1 τ 12 τ 22 Figue 1.3: The foces acting on a small two-dimensional element. whose nomal points in the x j -diection, whee x = P x, then it is staightfowad to show that T = P T P T. (1.23 Thus T, like E, is a second-ank tenso, called the Cauchy stess tenso. We can make one futhe obsevation about τ by consideing the angula momentum of the small two-dimensional solid element shown in Figue 1.3. The net anticlockwise moment acting about the cente of mass G is (pe unit length in the x 3 -diection 2 (τ 21 δx 2 δx (τ 12δx 1 δx 2 2, whee τ 21 and τ 12 ae evaluated at G to lowest ode. By letting the ectangle shink to zeo and insisting that the angula acceleation be finite, we deduce that τ 12 = τ 21. This agument can be genealised to thee dimensions and it shows that fo all i and j, i.e. that T, like E, is a symmetic tenso. τ ij τ ji ( Consevation of momentum Now we deive the basic govening equation of solid mechanics by applying Newton s second law to the mateial an abitay volume : d u i dt t ρ d = g i ρ d + τ ij n j ds. (1.25 The tems in (1.25 epesent successively the ate of change of the momentum in, the foce due to an extenal body foce g, such as gavity, and the taction exeted on the bounday

7 C5.2 Elasticity & Plasticity Daft date: 17 Januay of, whose unit nomal is n. We diffeentiate unde the integal (using the fact that the density ρ is appoximately constant and apply the divegence theoem on the final tem to obtain 2 u i t 2 ρ d = g i ρ d + τ ij x j d. (1.26 Assuming each integand is continuous, and using the fact that is abitay, we aive at Cauchy s momentum equation: ρ 2 u i t 2 = ρg i + τ ij x j. (1.27 This may altenatively be witten in vecto fom by adopting the following notation fo the divegence of a tenso: we define the ith component of T to be Since T is symmetic, we may thus wite Cauchy s equation as 1.6 The constitutive elation ( T i = τ ji x j. (1.28 ρ 2 u = ρg + T. (1.29 t2 We can now genealise Hooke s law by postulating a linea elationship between the stess and stain tensos. This is consistent with ou pevious assumption that the stain components ae small enough that we can lineaise the stain tenso E and also set /X i /x i. Assuming that the stess is initially zeo in the undefomed mateial, we ae appaently led to the poblem of defining 81 mateial paametes C ijkl (i, j, k, l = 1, 2, 3 such that τ ij = C ijkl e kl. (1.30 The symmety of τ ij and e ij only enables us to educe the numbe of unknowns to 36. This can be educed to a moe manageable numbe by assuming that the solid is isotopic, by which we mean that it behaves the same way in all diections. It can be shown that this is sufficient to educe the specification of C ijkl to just two scala quantities λ and µ, such that τ ij = λ (e kk δ ij + 2µe ij. (1.31 The mateial paametes λ and µ ae known as the Lamé constants, and µ is also called the shea modulus. They ae well chaacteised fo eveyday solid mateials like metals, ocks and so on. They both have the dimensions of pessue and they measue a solid s ability to withstand defomation: λ and µ take lage values fo had mateials like steel o diamond, and lowe values fo soft mateials like ubbe. Now we substitute ou linea constitutive elation (1.31 into the momentum equation (1.27 and eplace X with x to obtain the Navie equation: ρ 2 u t 2 = ρg + (λ + µ gad div u + µ 2 u. (1.32 fo the displacement vecto u(x, t. It may altenatively be witten as ρ 2 u = ρg + (λ + 2µ gad div u µ cul cul u, (1.33 t2

8 1 8 OCIAM Mathematical Institute Univesity of Oxfod n solid 2 solid 1 Figue 1.4: Schematic of a small pill-box-shaped egion at the bounday between two elastic solids. by using the vecto identity 1.7 Bounday conditions del squaed equals gad div minus cul cul. (1.34 Suppose that we wish to solve (1.32 fo u(x, t when t is positive and x lies in some pescibed domain D. In elastostatic poblems, in which the left-hand side of (1.32 is zeo, the Navie system is, oughly speaking, a genealisation of a scala elliptic equation. By analogy, it seems appopiate to specify thee scala conditions on u eveywhee on the bounday D. In most physical poblems, we specify eithe the displacement u o the taction T n eveywhee on the bounday. In addition, thee ae some situations in which the taction is specified on some pats of the bounday and the displacement on othes, fo example in contact poblems and in factue. Anothe common genealisation occus when two solids with diffeent elastic moduli ae bonded togethe acoss a common bounday D, as shown in Figue 1.4. Then the displacement vectos ae the same on eithe side of D and, by balancing the stesses on the small pill-box-shaped egion shown in Figue 1.4, we see that T (1 n = T (2 n. (1.35 Thus thee ae six continuity conditions acoss such a bounday. Fo elastodynamic poblems, we may anticipate that (1.32 admits wave-like solutions. It may, theefoe, be viewed as a genealisation of a scala wave equation, such as the familia equation ϱ 2 w t 2 = T 2 w x 2 (1.36 which descibes small tansvese waves on a sting with tension T and line density ϱ. We theefoe expect to pescibe Cauchy data fo the initial displacement u and velocity u/t at t = 0, as well as elliptic bounday conditions such as those descibed above. 1.8 Enegy We can obtain an enegy equation fom (1.27 by taking the dot poduct with u/t and integating ove an abitay volume : ρ 2 u i u i t 2 t d = u i ρg i t d + τ ij u i d. (1.37 x j t

9 C5.2 Elasticity & Plasticity Daft date: 17 Januay The final tem may be eaanged, using the divegence theoem, to τ ij u i x j t d = u i t τ e ij ijn j ds τ ij t d. (1.38 Hence (1.37 may be witten in the fom { d 1 dt 2 ρ u 2 } t d + W d = u i ρg i t d + u i t τ ijn j ds, (1.39 whee W is a scala function of the stain components that is chosen to satisfy W e ij = τ ij. (1.40 With τ ij given by (1.31, we can integate (1.40 to detemine W up to an abitay constant as W = 1 2 τ ije ij = 1 2 λ (e kk 2 + µ (e ij e ij. (1.41 Hee the summation convention is invoked such that (e kk 2 = (T E 2, while (e ij e ij = T ( E 2. The fist tem in baces in (1.39 is the net kinetic enegy in, while the tems on the ight-hand side epesent the ate at which wok is done by the extenal body foce g and the tactions on espectively. In the absence of othe enegy souces esulting fom, say, chemical o themal effects, we can intepet equation (1.39 as a statement of consevation of enegy. The diffeence between the ate of woking and the ate of change of kinetic enegy is the ate at which elastic enegy is stoed in the mateial as it defoms. Theefoe, W is called the stain enegy density, and is analogous to the enegy stoed in a stetched sping. The net consevation of enegy implied by (1.39 eflects the fact that the Navie equation is not dissipative. On physical gounds, we expect W to be a positive-definite function of the stain components, whose unique global minimum is attained when e ij 0. We can easily see fom (1.41 that this is tue if λ and µ ae positive, but in fact it is only necessay to have µ and (λ + 2µ/3 both positive. If these equiements ae met, then it can be shown that the Navie equation is well posed when suitable bounday and initial data ae imposed. The full poof of this is difficult, but we can quite easily pove the simple esult that the solution of the steady Navie equation subject to a given bounday displacement, if it exists, is unique. Suppose, then, that thee exist two solutions u (1 and u (2 of the patial diffeential equation T + ρg = 0 (1.42 in some elastic body B, both of which satisfy the bounday conditions u = u b (x on B. (1.43 Now define u = u (1 u (2, the diffeence, so that u satisfies the homogeneous poblem (1.42 and (1.43 with g = u b = 0. We take the dot poduct of (1.42 with u, integate ove B and use the divegence theoem to obtain u (T n ds = e ij τ ij d = 2 W d, (1.44 B B B

10 1 10 OCIAM Mathematical Institute Univesity of Oxfod whee W is given by (1.41. The left-hand side of (1.44 is zeo by the bounday conditions, while the integand W on the ight-hand side is non-negative and must, theefoe, be zeo. It follows that the stain tenso e ij is identically zeo in D, and the displacement can theefoe only be a igid-body motion (i.e. a unifom tanslation and otation. Since u is zeo on B, we deduce that it must be zeo eveywhee and, hence, that u (1 u (2. We can also use W to pose the steady Navie equation as a vaiational poblem as follows. We can wite the net elastic and gavitational potential enegy in an elastic body B in the fom { U[u] = W(eij ρg u } d. (1.45 B The calculus of vaiations leads to the conclusion that the displacement field u which minimises the functional U[u] must satisfy the steady Navie equation. Hence, instead of tying to solve the patial diffeential equation (1.42, we could instead ty to find the function u that minimises U[u], and this idea foms the basis of the finite element method fo solving (1.42 numeically. 1.9 Coodinate systems It is often useful to employ coodinate systems paticulaly chosen to fit the geomety of the poblem being consideed. Hee we state the main useful esults fo Catesian, cylindical pola and spheical pola coodinates. Catesian coodinates Fist we wite out in full the esults deived thus fa using the usual Catesian coodinates (x, y, z. To avoid the use of suffices, we will denote the displacement components by u = (u, v, w T. It is also conventional to label the stess components by {τ xx, τ xy,...} athe than {τ 11, τ 12,...}, and similaly fo the stain components. The linea constitutive elation (1.31 gives τ xx = (λ + 2µe xx + λe yy + λe zz, τ xy = 2µe xy, τ yy = λe xx + (λ + 2µe yy + λe zz, τ xz = 2µe xz, (1.46 τ zz = λe xx + λe yy + (λ + 2µe zz, τ yz = 2µe yz, whee e xx = u x, e yy = v y, e zz = w z, 2e xy = u y + v x, 2e yz = v z + w x, (1.47 2e xz = u z + w x,

11 C5.2 Elasticity & Plasticity Daft date: 17 Januay and the thee components of Cauchy s momentum equation ae ρ 2 u t 2 = ρg x + τ xx x + τ xy y + τ xz z, ρ 2 v t 2 = ρg y + τ xy x + τ yy y + τ yz z, (1.48 ρ 2 w t 2 = ρg z + τ xz x + τ yz y + τ zz z, whee the body foce is g = (g x, g y, g z T. In tems of the displacements, the Navie equation eads (assuming that λ and µ ae constant Cylindical pola coodinates ρ 2 u t 2 = ρg x + (λ + µ x ( u + µ 2 u, ρ 2 v t 2 = ρg y + (λ + µ y ( u + µ 2 v, (1.49 ρ 2 w t 2 = ρg z + (λ + µ z ( u + µ 2 w. We define the cylindical pola coodinates (, θ, z in the usual way and denote the displacements in the -, θ- and z-diections by u, u θ and u z espectively. The stess components ae denoted by τ ij whee now i and j ae equal to eithe, θ o z and, as in 1.4, τ ij is defined to be the i-component of stess on a suface element whose nomal points in the j-diection. The constitutive elation (1.31 applies diectly to this coodinate system, so that τ = (λ + 2µe + λe θθ + λe zz, τ θ = 2µe θ, τ θθ = λe + (λ + 2µe θθ + λe zz, τ z = 2µe z, (1.50 τ zz = λe + λe θθ + (λ + 2µe zz, τ θz = 2µe θz, whee the stain components ae now given by e = u, e θθ = 1 e zz = u z z, ( uθ θ + u u θ + u θ u θ, 2e θ = 1, 2e z = u z + u z, (1.51 2e θz = u θ z + 1 u z θ. The thee components of Cauchy s momentum equation (1.27 ead ρ 2 u t 2 ρ 2 u θ t 2 ρ 2 u z t 2 = ρg + 1 (τ + 1 = ρg θ + 1 (τ θ + 1 = ρg z + 1 (τ z + 1 τ θ θ + τ z z τ θθ, τ θθ θ + τ θz z τ θz θ + τ zz z, + τ θ, (1.52

12 1 12 OCIAM Mathematical Institute Univesity of Oxfod whee the body foce is g = g e + g θ e θ + g z e z. Witten out in tems of displacements, these become ρ 2 u t 2 = ρg + (λ + µ ( ( u + µ 2 u u 2 2 u θ 2, θ whee ρ 2 u θ t 2 = ρg θ + (λ + µ θ ( ( u + µ 2 u θ u θ u 2 θ ρ 2 u z t 2 = ρg z + (λ + µ z ( u + µ 2 u z, u = 1 (u u i = 1 ( u i u θ θ + u z z,, ( (1.54 u i 2 θ u i z 2 ae the divegence of u and the Laplacian of u i espectively expessed in cylindical polas. Spheical pola coodinates The spheical pola coodinates (, θ, φ ae defined in the usual way, such that the position vecto of any point is given by sin θ cos φ (, θ, φ = sin θ sin φ. (1.55 cos θ Again, we apply the constitutive elation (1.31 to obtain τ = (λ + 2µe + λe θθ + λe φφ, τ θ = 2µe θ, τ θθ = λe + (λ + 2µe θθ + λe φφ, τ φ = 2µe φ, (1.56 τ φφ = λe + λe θθ + (λ + 2µe φφ, τ θφ = 2µe θφ. The lineaised stain components ae now given by e = u, e θθ = 1 e φφ = 1 sin θ ( uθ θ + u u φ φ + u 2e θ = 1, 2e φ = 1 sin θ + u θ cot θ, 2e θφ = 1 sin θ Cauchy s equation of motion leads to the thee equations ρ 2 u t 2 = ρg + 1 ( 2 τ (sin θτ θ sin θ θ (sin θτ θθ ρ 2 u θ t 2 = ρg θ + 1 ( 2 τ θ sin θ ρ 2 u φ t 2 = ρg φ + 1 ( 2 τ φ sin θ θ (sin θτ θφ θ u θ + u θ u θ, u + 1 τ φ sin θ + 1 sin θ + 1 sin θ φ + u φ u φ, (1.57 u θ φ + 1 u φ θ u φ cot θ. φ τ θθ + τ φφ, τ θφ φ + τ θ cot θτ φφ, (1.58 τ φφ φ + τ φ + cot θτ θφ,

13 C5.2 Elasticity & Plasticity Daft date: 17 Januay whee the body foce is g = g e +g θ e θ +g φ e φ. In tems of displacements, the Navie equation eads { 2 u 2u 2 ρ 2 u t 2 = ρg + (λ + µ ( u + µ ρ 2 u θ t 2 = ρg θ + ρ 2 u φ t 2 = ρg φ + (λ + µ (λ + µ sin θ θ φ sin θ { ( u + µ 2 u θ + 2 u 2 { ( u + µ 2 u φ + θ θ (u θ sin θ u θ 2 sin 2 θ 2 cos θ 2 sin 2 θ 2 u 2 sin θ φ + 2 cos θ u θ 2 sin 2 θ φ 2 sin θ u φ φ } u φ, φ }, (1.59 u φ 2 sin 2 θ }, whee u = 1 ( u + 2 u i = 1 2 ( 2 u i sin θ θ (sin θu θ + 1 sin θ ( 1 2 sin θ u i + sin θ θ θ + u φ φ, 1 2 sin 2 θ 2 u i φ 2. (1.60