θ θ φ EN2210: Continuum Mechanics Homework 2: Polar and Curvilinear Coordinates, Kinematics Solutions 1. The for the vector i , calculate:

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1 EN0: Continm Mchanics Homwok : Pola and Cvilina Coodinats, Kinmatics Soltions School of Engining Bown Univsity x δ. Th fo th vcto i ij xx i j vi = and tnso S ij = + 5 = xk xk, calclat: a. Thi componnts in sphical-pola coodinats b. Th gadint of v in sphical-pola coodinats c. Th divgnc of S in sphical-pola coodinats (a) Not that xi / = so w Fo S ij th fist tm is isotopic so v= δ ij xx i j + = ( ) 5 + θ θ + φ φ + [ POINTS] (b) v= + θ + φ ( ) sin = + θ θ + φ φ θ θ φ [ POINTS] (c) It is asist to s th fomla S S S S S θ cot θ φ θ + ( Sθθ + Sφφ ) θ sinθ φ S S S S θ θ S S φθ S cot φφ S + + θθ + θ θθ + + θ cotθ θ sinθ φ S φ Sφ sinθ Sθφ Sθφ S φφ cosθ + + ( Sφ + Sφθ ) θ sinθ φ S S + ( Sθθ + Sφφ ) S S φφ = cotθ θθ + cotθ 0 0 = = [ POINTS]

2 . Paabolic Coodinats a sd to simplify th soltion of PDEs fo solids with paabolic bondais. Thy spcify th position of a point sing th paamtic coodinats ( vθ,, ) as = v(cosθi+ sin θj) + ( v ) k (a) Find th componnts of nomalizd basis vctos fo this coodinat systm = v = θ = θ v θ Show that thy a othogonal; and calclat thi divativs with spct to ( vθ,, ) (xpss yo answ in {, v, θ } coodinats. = vcosθi+ vsinθj+ k v = cosθi+ sinθj vk θ = ( sinθi+ cos θj) Ths a claly othogonal ( ) ( ) Not that k = ( vv) cosθi+ sin θj= ( v + v) + v + v = ( vcosθi+ vsinθj+ k) + k / ( ) v = ( ) + vv = v ( ) ( ) v = ( vcosθi+ vsin θj+ k) + (cosθi+ sin θj) v / ( ) v = ( ) + v + v = v ( ) v = ( vsinθi+ vcosθj) = θ θ By symmty And finally v v v v = = = θ v θ θ θ = = 0 θ = (cosθi+ sin θj) = ( v + v) v θ + v [5 POINTS]

3 (b) Find an xpssion fo th gadint opato in th ( vθ,, ) systm (xpss yo answ in {, v, θ } coodinats. Consid a scala fnction f( vθ,, ). Its divativ is Similaly, not that by dfinition f f f d f = d + dv + dθ v θ d = + v d + + v vdv θ dθ Sinc {, v, θ } a othogonal w find that f f f df = + v + θ d v v v v θ + + Th gadint opato thfo follows as f = + v + θ f v v v v θ + + (c) Find th gadint and divgnc of th vcto fild w = w = + v + θ v v v v θ + + v = v + v + v v + v v + θ θ v v + v + Th divgnc is th tac of this: w = + + v ( ) + v + ( ) v = + = / v ( v ) ( v ) ( ) [ POINTS] [ POINTS]

4 . Hlical coodinats a sd to dc th govning qations fo poblms with hlical symmty (sch as flow down a hlical channl) to two dimnsions. A nmb of diffnt hlical coodinat systms hav bn poposd. On xampl is dfind by xpssing position vcto in tms of (, θ, z) ( ξ, ξ, ξ) as follows L = cosθi+ sin θj+ ( z+ θ) k π Wh L is th pitch of th hlix. (a) Calclat th covaiant basis vctos m i L m = cosθi+ sinθj m = sinθi+ cosθj+ k m = k π (b) Find xpssions fo th cipocal basis vctos It is asist to do this by inspction L m = cosθi+ sin θj m = ( sinθi+ cos θj) / m = (sinθi cos θj) + k π i m [ POINTS] [ POINTS] (c) Calclat th covaiant and contavaiant componnts of th mtic tnso g. Chck yo answ kj by calclating gik g ij i j gij = mi mj = 0 + ( L/ π) L/π g = m m = 0 / L/π 0 L /π 0 L/π + ( L/ π) It is asy to s that kj j gik g = δi as qid. [ POINTS]

5 k (d) Calclat th lmnts of Chistoffl symbol (which satisfis dmi = Γijm kdξ j ) k l k k Not that dmi m =Γijml m dξj =Γijdξj Also dm = ( sinθi+ cos θj) d θ dm = (cosθi+ sin θj) d θ + ( sinθi+ cos θj) d dm = 0 L dm m = 0 dm m = dθ dm m = dθ π L dm m = dθ dm m = d dm m = d π dm m = dm m = dm m = 0 k Th nonzo lmnts of Γ ij a ths L Γ = Γ = π L Γ = Γ = Γ = π [ POINTS] () Calclat xpssions fo th covaiant and contavaiant componnts of th gadint of a scala fnction φ φ φ φ φ = m + m + m θ z φ i ij φ m = g mj ξi ξj φ φ L φ L φ L φ φ = m + m ( ) θ π z m π θ 4π z [ POINTS] (f) Find xpssions fo th contavaiant componnts of th vlocity and acclation of a paticl, in tms of tim divativs of, θ, z d d d d dθ d dz d dθ dz v= = + + = m+ m + m dt d dt dθ dt dz dt dt dt dt dv d ξi dξ dξ j k a= = m i i + Γijmk dt dt dt dt d d θ d z dθ d dθ L d dθ = m + m + m m+ m m dt dt dt dt dt dt π dt dt d dθ d θ d dθ d z L d dθ = m+ + m + dt dt dt m dt dt dt π dt dt [ POINTS]

6 (g) A stady flow down a hlical channl mst hav th fom v= v () m (not that v dos not hav nits of vlocity bcas m is not a nit vcto). Calclat th vlocity gadint tnso i v j i k v= mi + v Γikmj m ξ k dv v= m m + vγm m + vγm m + vγm m d dv v vl = m m + v( ) m m + m m m m d π [ POINTS] 4. Constct (i.. find a displacmnt fild) a homognos dfomation that has th following poptis: Th volm of th solid is dobld A matial fib paalll to th diction in th ndfomd solid incass its lngth by a facto of and is ointd paalll to th + diction in th dfomd solid A matial fib paalll to th diction in th ndfomd is ointd paalll to th + diction in th dfomd solid. A matial fib paalll to th diction in th ndfomd solid psvs its lngth and ointation in th dfomd solid Th a sval ways to do this poblm. H is on. W can xpss th dfomation gadint as F = U. Withot loss of gnality w can assm th pincipal dictions of U a paalll to,,, in which cas U= λii i Th fist, scond and thid conditions giv λλ λ= λ= λ= Finally th fist and scond conditions show that is a 45 dg contclockwis otation abot th axis. This givs 0 = Hnc F is F = = Th displacmnt fild follows as = Fx + c [5 POINTS]

7 5. To mas th in-plan dfomation of a sht of mtal ding a foming pocss, yo manags plac th small hadnss indntations on th sht. Using a tavlling micoscop, thy dtmin that th initial lngths of th sids of th tiangl fomd by th th indnts a cm, cm,.44cm, as shown in th pict blow. Aft dfomation, th sids hav lngths.5cm,.0cm and.8cm. (c) cm (a) cm.44cm (b) (c) cm (a).8cm.5cm (b) 5. Calclat th componnts of th Lagang stain tnso E, E, E in th basis shown. By dfinition th Lagang stain is latd to th lngth changs by l l mem = 0 l This givs E E.5 [ 0] E 0.65 E E 0 = = = E E 0 [ 0 ] E.5 E E = = = E E.8 [ ] ( E E E) /.46 E 0.98 E E = + = = = 4 [ POINTS] 5. Calclat th componnts of th Elian stain tnso * * *,, E E E in th basis shown. Th a vaios ways to do this. On way is to fist calclat F, as follows T Not that U = F F= E+ I This givs U = W know F = U and also that cosθ sinθ = sinθ cosθ W s fom th fig that th lin [0,] is mappd to [0,] by dfomation. W can solv th qation cosθ sinθ = θ = 0.5 sinθ cosθ W can now calclat F F = U = = Finally

8 * T E = ( I F F )/ = [5 POINTS] 6. Th fig shows th fnc and dfomd configations fo a solid. Th ot-of-plan dimnsions a nchangd. Points a and b a th positions of points A and B aft dfomation. Dtmin A L B L m 4L m 45 0 a b L/4 6. Th ight sttch tnso U, xpssd as componnts in,,. (A x matix is sfficint). Th is no nd fo lngthy calclations yo may wit down th slt by inspction. Th dfomd configation can b achd by a sttch paalll to th two basis vctos, followd by a otation. Ths can b takn to b th two dfomations in th dcomposition F=U 0 0 /4 [ POINTS] 6. Th otation tnso in th pola dcomposition of th dfomation gadint F=U=V [ POINTS] 6.Th dfomation gadint, xpssd as componnts in m, m, m. Ty to do this withot sing th basis-chang fomlas. Th dfomation gadint can b dcomposd as F=V, and V has componnts 0 0 /4 in m, m, m, whil has th sam componnts in both,, and m, m, m. Thfo 0 0 /4 = /4 /4 [ POINTS]

9 7. Show that th Lagang stain E, th ight Cachy-Gn dfomation tnso C and th ight sttch * tnso U hav th sam pincipal dictions (ignvctos). Similaly, show that E, BV, hav th sam pincipal dictions. W can wit Similaly U= λii i C= U = λi i i E= ( C I)/ = ( λi ) i i / V = λivi vi B= V = λi vi vi * E = ( I B )/ = ( / λi ) vi vi / [ POINTS]