Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection. Quantities that only need thei magnitude to be fully epesented ae called scalas. We use bold lettes to denote vectos, e.g. v. The magnitude of a vecto is denoted by v. A vecto of unit length is called a unit vecto. The unit vecto in the diection of vecto v, is obtained by scaling the vecto by the invese of its magnitude: e v = v v, We can also expess v as v = v e v. A.2 Components of a vecto A basis in R 3 is a set of linealy independent vectos 1 such that any vecto in the space can be epesented as a linea combination of basis vectos. We will epesent vectos in a catesian basis whee the basis vectos e i ae othonomal, i.e. they have unit length and they ae othogonal with espect to each othe. This can be expessed using dot poducts e 1 e 1 = 1 e 2 e 2 = 1 e 2 e 3 = 1 e 1 e 2 = 0 e 1 e 3 = 0 e 2 e 3 = 0. 1 A set of vectos v 1, v 2, v 3,..., v n ae linealy dependent if β 1 v 1 + β 2 v 2 + β 3 v 3 +... + β n v n = 0 whee β 1, β 2,...β n ae not all eo. In R 3, the maximum numbe of linealy independent vectos is 3. 107
108 APPENDIX A. VECTOR ALGEBRA We can wite these expessions in a vey succinct fom as follows e i e j = δ ij whee the symbol δ ij is the so-called Konecke delta: δ ij = { 1 : if i = j 0 : if i j Then we can epesent any vecto v in thee dimensional space as follows v = v 1 e 1 + v 2 e 2 + v 3 e 3 = 3 v i e i, i=1 (A.1 whee v 1, v 2 and v 3 ae the components of the vecto in the basis e i, i = 1, 3 A.3 Indicial notation Fee index: A subscipt index ( i will be denoted a fee index if it is not epeated in the same additive tem whee the index appeas. Fee means that the index epesents all the values in its ange. Latin indices will ange fom 1 to 3, (i, j, k,... = 1, 2, 3, Geek indices will ange fom 1 to 2, (α, β, γ,... = 1, 2.
A.4. SUMMATION CONVENTION 109 Examples: 1. e i, i = 1, 2, 3 can now be simply witten as e i, no need to make the explicit mention of i = 1, 2, 3, as i is a fee index. 2. a i1 implies a 11, a 21, a 31. (one fee index 3. x α y β implies x 1 y 1, x 1 y 2, x 2 y 1, x 2 y 2 (two fee indices. 4. a ij implies a 11, a 12, a 13, a 21, a 22, a 23, a 31, a 32, a 33 (two fee indices implies 9 values. 5. σ ij x j + b i = 0 has a fee index (i, theefoe it epesents thee equations: σ 1j x j + b 1 = 0 σ 2j x j + b 2 = 0 σ 3j x j + b 3 = 0 A.4 Summation Convention In expessions such as: 3 v i e i, i=1 we obseve that the summation sign with its limits can be eliminated altogethe if we adopt the convention that the summation is implied by the epeated index i. Then, a vecto epesentation in a catesian basis, Equation (A.1, can be shotened to v = 3 v i e i = v i e i, i=1 Moe fomally:
110 APPENDIX A. VECTOR ALGEBRA Summation convention: When a epeated index is found in an expession (inside an additive tem the summation of the tems anging all the possible values of the indices is implied Examples: 1. a i b i = 3 i=1 a ib i = a 1 b 1 + a 2 b 2 + a 3 b 3 2. a kk = a 11 + a 22 + a 33. 3. t i = σ ij n j implies the thee equations (why?: t 1 = σ 11 n 1 + σ 12 n 2 + σ 13 n 3 t 2 = σ 21 n 1 + σ 22 n 2 + σ 23 n 3 Othe impotant ules about indicial notation: t 3 = σ 31 n 1 + σ 32 n 2 + σ 33 n 3 1. An index cannot appea moe than twice in a single additive tem, it s eithe fee o epeated only once. a i = b ij c j d j is INCORRECT 2. In an equation the lhs and hs, as well as all the tems on both sides must have the same fee indices a i b k = c ij d kj fee indices i, k, CORRECT a i b k = c ij d kj + e i f jj + g k p i q INCORRECT, second tem is missing fee index k and thid tem has exta fee index When the summation convention applies, the index is dummy (ielevant: a i b i = a k b k. A.5 Opeations Scala poduct between vectos is defined as a b = (a i e i (b j e j = a i b j (e i e j = a i b j δ ij = a i b i. Coss poduct between two basis vectos e i and e j is defined as e i e j = ε ijk e k, whee ε ijk is called the altenating symbol (o pemutation symbol and defined as follows 1, if i, j, k ae in cyclic ode and not epeated (123, 231, 312, ε ijk = 1, if i, j, k ae not in cyclic ode and not epeated(132, 213, 321, 0, if any of i, j, k ae epeated.
A.6. TRANSFORMATION OF BASIS 111 In geneal, the coss poduct of two vectos can be expessed as a b = (a i e i (b j e j = a i b j (e i e j = a i b j ε ijk e k. ε δ identity elates the Konecke delta and the pemutation symbol as follows ε ijk ε imn = δ jm δ kn δ jn δ km. (A.2 Poblems: 1. Veify the ε δ identity by the definition of Konecke delta and the pemutation symbol. 2. Use the ε δ identity to veify a (b c = (a c b (a b c. Dyadic poduct (o tenso poduct between two basis vectos e i and e j defines a basis second ode tenso e i e j o simply e i e j. In geneal, the dyadic poduct a b = (a i e i (b j e j = a i b j e i e j, esults in a second ode tenso which has component a i b j on the basis e i e j. The following identities ae popeties of the dyadic poduct (αa b = a (αb = α(a b, fo scala α, (a b c = (b ca, a (b c = (a bc. A.6 Tansfomation of basis Given two othonomal bases e i, ẽ k and a vecto v whose components in each of these bases ae v i and ṽ k, espectively, we seek to expess the components in basis in tems of the components in the othe basis. Since the vecto is unique: v = ṽ m ẽ m = v n e n Taking the scala poduct with ẽ i : v ẽ i = ṽ m (ẽ m ẽ i = v n (e n ẽ i But ṽ m (ẽ m ẽ i = ṽ m δ mi = ṽ i fom which we obtain: ṽ i = v ẽ i = v j (e j ẽ i Note that e j ẽ i ae the diection cosines of the basis vectos of one basis on the othe basis: e j ẽ i = e j ẽ i cos êjẽ i = cos êjẽ i
112 APPENDIX A. VECTOR ALGEBRA A.7 Tensos Tensos ae defined as the quantities that ae independent of the selection of basis while the components tansfom following a cetain ule as the basis changes. When the basis changes fom {e i } to {ẽ i }, a scala does not change (e.g. mass, the vecto component tansfoms as ṽ i = v j (e j ẽ i (see Section A.6, and the stess component tansfoms as σ ij = σ kl ( ek ẽ i ( el ẽ j (see Equation 1.11, which defines a second ode tenso. Similaly, the vecto is a fist ode tenso, and a scala is a eoth ode tenso. The ode of the tenso equals the numbe of fee indices (see Section A.3 that the tenso has, and it can also be intepeted as the dimensionality of the aay needed to epesent the tenso, as detailed in the next table: # indices Tenso ode Aay type Denoted as Rule of Tans. 0 Zeoth Scala α 1 Fist Vecto v = v i e i = ṽ j ẽ j 2 Second Matix σ = σ ij e i e j = σ ij ẽ i ẽ j ṽ i = ( v j (e j ( ẽ i σ ij = σ kl ek ẽ i el ẽ j Highe ode tenso can be defined following the tansfomation ule. Fo instance, the fouth ode tenso can be defined as C = C ijkl e i e j e k e l with C ijkl = C pqs (e p ẽ i (e q ẽ j (e ẽ k (e s ẽ l upon the change of basis. A.8 Tenso opeations Tensos ae able to opeate on tensos to poduce othe tensos. The scala poduct, coss poduct and dyadic poduct of fist ode tenso (vecto have aleady been intoduced in Sec A.5. In this section, focus is given to the opeations elated with the second ode tenso. Dot poduct with vecto: σ a = (σ ij e i e j (a k e k = σ ij e i (e j a k e k = σ ij e i a k δ jk = σ ij a j e i a σ = (a k e k (σ ij e i e j = a k (e k (σ ij e i e j = a k (σ ij δ ki e j = a i σ ij e j Double-dot poduct: σ : ɛ = (σ ij e i e j : (ɛ kl e k e l = σ ij ɛ kl (e i e k (e j e l = σ ij ɛ kl δ ik δ jl = σ ij ɛ ij Double-dot poduct with fouth ode tenso: C : ɛ = C ijkl e i e j e k e l : ɛ pq e p e q = C ijkl ɛ pq e i e j (e k e p (e l e q = C ijkl ɛ kl e i e j.
Appendix B Vecto Calculus B.1 nabla opeato( In a Catesian system with othonomal basis {e i }, the nabla opeato is denoted by The gadient of a scala field φ is defined as e i + e 2 + e 3. x 1 x 2 x 3 gad φ = φ = e i φ x i. Poblems: 1. Veify that φ is pependicula to the suface {x φ(x i = const}, and it is associated with the maximum spatial ate of change of φ. (Hint: the spatial ate of change of φ along unit vecto ˆl is given by φ x i (e i ˆl. In geneal, the gadient of a tenso geneates a new tenso with a highe ode. Fo instance, the gadient of a fist ode tenso v (vecto poduces a second ode tenso: v = e i (v j e j = v j e i e j = v j e i e j. x i x i x i It is also common to decompose the gadient of vecto as v = 1 ( vj + v i e i e j + 1 2 x i x j 2 The divegence of a vecto is defined as ( vj x i v i x j e i e j. which esults in a scala. div v = v = (e i (v j e j = v i x i x i 113
114 APPENDIX B. VECTOR CALCULUS In geneal, the divegence of a tenso geneates a tenso with a lowe ode. The divegence of a tenso Φ = φ ij e i e j poduces a vecto, as shown in next equations: The cul of a vecto is defined as Φ = (e i x i (φ mn e m e n = φ in x i e n. cul v = v = ε ijk v j x i e k. The Laplacian opeato 2 (sometimes witten as on a scala field φ is defined as the divegence of the gadient vecto φ: B.2 Integal Relations 2 φ = ( φ = ( φ = 2 φ x i x i. The integal elations ae equations that elate the volume integal to the suface integal, o the suface integal to the line integal. Conside an abitay egion in space of volume V which is suounded by suface S with a unit oute nomal ˆn. Gadient Theoem: fo any scala field φ, gad φ dv = ˆnφ ds. (B.1 V S Witten in component, V φ dv = ˆn i φ ds. x i S Divegence Theoem (o Gauss Theoem: fo a fist o second ode tenso A, V div A dv = S ˆn A ds (B.2 Fo vecto A = A i e i, the divegence theoem gives ise to one identity A i dv = ˆn i A i ds. V x i S While fo second ode tenso A = A ij e i e j, the divegence theoem gives ises to thee identities: A ij dv = ˆn i A ij ds. x i S V
B.3. CYLINDRICAL AND SPHERICAL COORDINATE SYSTEMS 115 B.3 nabla opeato in Cylindical and Spheical Coodinate Systems The nabla opeato has aleady been intoduced in Catesian coodinate system in the pevious section. Let ê x be a unit vecto along the x-axis, and define ê y and ê by analogy. Then {ê x, ê y, ê } foms a basis of the space. On the basis, the nabla opeato is denoted as The Laplacian opeato is given by = ê x x + ê y y + ê. 2 = 2 x 2 + 2 y 2 + 2 2 In cylindical and spheical coodinate systems, the basis is defined locally, and will change as the position changes. As as a esult, the nabla opeato has diffeent expessions. B.3.1 Cylindical coodinates In a cylindical coodinate system (R, φ,, the othonomal basis vectos associated with the coodinates ae defined by ê R = cos(φê x + sin(φê y ê φ = sin(φê x + cos(φê y ê = ê whee {ê x, ê y, ê } ae the thee basis vectos in the Catesian coodinate system (x, y,. e e φ Z e R φ R y x The nabla opeato in cylindical coodinate system is denoted as: = ê R R + ê 1 φ R φ + ê
116 APPENDIX B. VECTOR CALCULUS Facto 1 in the second tem is a little supise at the fist sight, while it makes the hs dimensionally consistent. The new expession can be deived fom the coodinate tansfomation R as below. (R, φ, coodinates can be witten as functions of (x, y, : R = x 2 + y 2 φ = actan y x = which means given any scala field ψ(r, φ,, ψ(r(x, y,, φ(x, y,, is a scala field in (x, y, coodinate system. Recalling the expession of in Catesian coodinate system, ψ ψ = ê x x + ê ψ y y + ê ψ = ê x ( ψ φ R R x + ψ φ x + ê y( ψ R R y + ψ φ φ y + ê = ψ ( R ê x R x + ê R y + ψ ( φ ê x y φ x + ê φ y y = ψ ( x ê x R R + ê y y + ψ ( ê x ( y R φ R + ê x 2 y R 2 = ψ R (cos(φê x + sin(φê y + ψ φ ψ 1 = ê R R + ê φ R ψ φ + ê ψ, ψ + ê ψ + ê ψ 1 R ( sin(φê ψ x + cos(φê y + ê which denotes the opeato in (R, φ, coodinates on the basis {ê R, ê φ, ê }. Consequently, Laplacian opeato on a scala field ψ can be witten as 2 ψ = ( ψ = (ê R R + ê 1 φ R φ + ê (ê ψ R R + ê 1 ψ φ R φ + ê ψ = 2 ψ R + ê 2 φ 1 R φ (ê R ψ R + 1 2 ψ R 2 φ + 2 ψ 2 2 = 2 ψ R + ê 2 φ 1 ψ Rêφ R + 1 2 ψ R 2 φ + 2 ψ 2 2 = 2 ψ R + 1 ψ 2 R R + 1 2 ψ R 2 φ + 2 ψ 2 2 = 1 ψ (R R R R + 1 2 ψ R 2 φ + 2 ψ 2. 2
B.3. CYLINDRICAL AND SPHERICAL COORDINATE SYSTEMS 117 B.3.2 Stain and Stess components in cylindical coodinates The cylindical coodinates have vey impotant applications in plane stess and plane stain poblems. Conside only the in-plane coodinates (pola coodinates (, θ. The displacement fields (u, u θ can be expessed as u = cos θ u 1 + sin θ u 2, u θ = sin θ u 1 + cos θ u 2 whee (u 1, u 2 ae the displacements in the Catesian coodinates system. At the same time, it is easy to see that (u 1, u 2 can be expessed in tems of (u, u θ as u 1 (, θ = cos θ u sin θ u θ, u 2 (, θ = sin θ u + cos θ u θ. The stain tenso (ɛ, ɛ θ ; ɛ θ, ɛ θθ can be obtained though the otation of the Catesian coodinates by θ, i.e. ɛ = ɛ 11 cos 2 θ + ɛ 22 sin 2 θ + ɛ 12 sin 2θ ɛ θθ = ɛ 11 sin 2 θ + ɛ 22 cos 2 θ ɛ 12 sin 2θ ɛ θ = ɛ θ = ɛ 11 ɛ 22 sin 2θ + ɛ 2 12 cos 2θ. Using the definition and the chain ule deivative, the stains ɛ ij can be expessed in tems of (, θ, u, u θ, and finally the stains in pola coodinates can be obtained by substitution. Fom chain ule, it is staightfowad to show that x 1 = cos θ, x 2 = sin θ, θ x 1 = sin θ, θ x 2 =. And consequently, cos θ ɛ 11 = u 1 x 1 = u 1 cos θ + u 1 θ ( sin θ, ɛ 22 = u 2 x 2 = u 2 sin θ + u 2 θ (cos θ, ɛ 12 = 1 2 (u 1 + u 2 x 2 x 1 = 1 ( u1 2 (sin θ + u 1 θ (cos θ + u 2 (cos θ + u 2 θ ( sin θ. Since sin 2θ = 2 sin θ cos θ, ɛ = u 1 cos3 θ + u 1 θ ( sin θ cos2 θ + u 1 (sin2 θ cos θ + u 1 = u 1 cos θ + u 2 sin θ = (u 1 cos θ + u 2 sin θ = u. θ sin θ θ (cos2 + u 2 sin3 θ + u 2 θ (cos θ sin2 θ + u 2 (cos2 θ sin θ + u 2 θ cos θ θ ( sin2
118 APPENDIX B. VECTOR CALCULUS Similaly, ɛ θθ = u 1 cos θ sin2 θ + u 1 θ θ ( sin3 + u 2 sin θ cos2 θ + u 2 θ θ (cos3 u 1 (sin2 θ cos θ u 1 θ sin θ θ (cos2 u 2 (cos2 θ sin θ + u 2 θ cos θ θ (sin2 = u 1 θ ( sin θ + u 2 cos θ θ = 1 ( ( u1 sin θ + u 2 cos θ + u 1 cos θ + u 2 sin θ θ = 1 ( uθ θ + u, and ( u1 ɛ θ = cos θ u 1 sin θ u 2 θ sin θ u 2 cos θ sin θ cos θ θ ( u1 + (sin θ + u 1 θ (cos θ + u 2 (cos θ + u 2 θ ( sin θ (cos 2 θ 1 2 = 1 ( u1 2 ( sin θ + u 1 cos θ + u 2 θ cos θ + u 2 sin θ θ = 1 ( ( u1 sin θ + u 2 cos θ + 1 (u 1 cos θ + u 2 sin θ 1 2 θ ( u 1 sin θ + u 2 cos θ = 1 ( uθ 2 + 1 u θ u θ. In sum, the stain components in pola coodinates can be witten as follows: ɛ = u ɛ θθ = 1 u θ θ ɛ θ = ɛ θ = 1( u θ 2 + u + 1 u θ u θ The isotopic elastic constitutive elation between stain and stess emains the same in the pola coodinates. In constitutive elation, the only diffeence is the change of index fom {i, j} to {, θ}. The stess tenso (σ, σ θ ; σ θ, σ θθ can be obtained though the otation of the Catesian coodinates by θ, i.e. σ = σ 11 cos 2 θ + σ 22 sin 2 θ + σ 12 sin 2θ σ θθ = σ 11 sin 2 θ + σ 22 cos 2 θ σ 12 sin 2θ σ θ = σ 11 σ 22 sin 2θ + σ 2 12 cos 2θ. The stess tenso can also be expessed as deivatives of the Aiy stess function. See Unit 4 fo moe details. The equilibium equations can be also deived though the tansfomation of coodinates and chain ule as we have done fo the stain components. Hee, we adopt an altenative tensoial appoach and use the esults fom pevious section. In the pola
B.3. CYLINDRICAL AND SPHERICAL COORDINATE SYSTEMS 119 coodinates, ê = cos θê 1 + sin θê 2, and ê θ = sin θê 1 + cos θê 2, whee ê i, i = 1, 2 ae the basis vectos in the Catesian system. It is staightfowad to see that ê = ê θ = 0, ê = ê θ θ and ê θ = ê θ. The equilibium equation in tensoial fom eads 0 = σ =(ê + êθ θ (σ ê ê + σ θ ê ê θ + σ θ ê θ ê + σ θθ ê θ ê θ = σ ê + σ θ êθ + σ ê + σ θ êθ + 1 ( σ = + 1 σ θ θ + σ σ θθ ê + σ θ θ ê + σ θ êθ + 1 + 1 σ θθ θ ( σθ σ θθ + σ θ + σ θ θ êθ + σ θθ ( ê ê θ. Consideing the symmety of the stess tenso, the equilibium equations in pola coodinates can be witten as follows B.3.3 Spheical system σ + 1 1 σ θθ θ σ θ θ + σ θ + σ σ θθ + 2σ θ = 0, = 0. In a spheical coodinate system (, θ, φ, the othonomal basis vectos associated with the coodinates ae defined by ê = sin(θ cos(φê x + sin(θ sin(φê y + cos(θê ê θ = cos(θ cos(φê x + cos(θ sin(φê y sin(θê ê φ = sin(φê x + cos(φê y whee {ê x, ê y, ê } ae the thee basis vectos in the Catesian coodinate system (x, y,. e e φ φ θ e θ y x The nabla opeato is witten as: = ê + 1 êθ θ + 1 sin(θêφ φ.
120 APPENDIX B. VECTOR CALCULUS The deivation follows the same way as in the cylindical coodinates. Main tools ae the chain ule of deivative and the fact that (, θ, φ coodinates can be witten as functions of (x, y, : = x 2 + y 2 + 2 θ = accos x2 + y 2 + 2 φ = actan y x. The Laplacian opeato on a scala field ψ can be witten as 2 ψ = 1 ( 2 ψ ( 1 + sin(θ ψ 1 2 ψ + 2 2 sin(θ θ θ 2 sin 2 (θ φ. 2