UNIVERSITY OF CALIFORNIA BERKELEY Stuctual Engineeing, Depatment of Civil Engineeing Mechanics and Mateials Fall 6 Pofesso: S. Govindjee A Quick Oveview of Cuvilinea Coodinates Intoduction Cuvilinea coodinate systems ae geneal ways of locating points in Euclidean space using coodinate functions that ae invetible functions of the usual x i Catesian coodinates. Thei utility aises in poblems with obvious geometic symmeties such as cylindical o spheical symmety. Thus ou main inteest in these notes is to detail the impotant elations fo stain and stess in these two coodinate systems. Shown in Fig. ae the definitions of the coodinate functions. Note that while the definition of the cylindical coodinate system is athe standad, the definition of the spheical coodinate system vaies fom book to book. Both systems to be studied ae othogonal. The pecise definitions used hee ae: Cylindical x = cos(θ) x = sin(θ) x 3 = z () Spheical x = sin(ϕ) cos(θ) x = sin(ϕ) sin(θ) x 3 = cos(ϕ) (3) = x + x θ = tan (x /x ) z = x 3 () Basis Vectos = x + x + x 3 ϕ = cos x ( 3 x +x +x 3 θ = tan (x /x ) ) (4) Fo convenience in some of the equations to be given late we will denote ou cuvilinea coodinates as z k whee (z, z, z 3 ) = (, θ, z) in the cylindical case and (z, z, z 3 ) = (, ϕ, θ) in the spheical case. Fo the basis vectos we will intoduce fo two types of basis vectos. The natual basis vectos and
x3 x3 ϕ z x x x θ x θ Figue : Definition of the cylindical and spheical coodinate systems. the physical basis vectos. Both bases ae othogonal but the physical basis has the additional popety of othonomality. The basic definitions ae g k = xi z k e i (5) e k = g k g k. (6) To diffeentiate between the physical basis vectos and the usual Catesian ones we typically wite e, e θ, etc. Fo the cylindical coodinate system one has: and g cos(θ) sin(θ) g sin(θ) cos(θ) g 3 (7) e = g e θ = g e z = g 3. (8) Note that the components have been expessed in the standad othonomal Catesian basis. Fo the spheical system one has that sin(ϕ) cos(θ) cos(ϕ) cos(θ) g sin(ϕ) sin(θ) g cos(ϕ) sin(θ) cos(ϕ) sin(ϕ) g 3 sin(ϕ) sin(θ) sin(ϕ) cos(θ) (9)
and e = g e ϕ = g e θ = sin(ϕ) g 3. (). Physical and Natual Components As with all bases we can expess the components of vectos and tensos with espect to ou new cuvilinea bases. In this egad, it is vey impotant with the cuvilinea coodinates to know whethe o not the components ae with espect to the natual basis vectos o with espect to the physical basis vectos. To help maintain the distinction we use supescipt numeals with the components in the natual basis and subscipt lette (Latin and Geek) fo components in the physical basis. Conside fo example a vecto v = v θ e θ = v g, then we have the elation v θ = v. () If fo instance we have a vecto v = v ϕ e ϕ + v θ e θ = v g + v 3 g 3, then we have that v ϕ = v () v θ = sin(ϕ)v 3. (3) Simila elations can be deived fo tenso components. 3 Dual Basis Vectos When dealing with non-catesian coodinate systems one often intoduces the so called dual (o contavaiant) basis vectos; they ae denoted by the symbol g k note the aised index. The defining popety of these basis vectos is that they ae othogonal to the fist basis intoduced; i.e. g i g j = δ i j, (4) whee δ i j is simply the Konecke delta symbol. The i index is aised so that it matches the othe side of the equation. The meaning is still the same ( if i = j and othewise). Anothe way of witing this is g k = ( z k / x i )e i. 3
Note that fo the usual Catesian coodinates thee is no diffeence between the dual basis and the egula basis. Fo the cylindical system we have: cos(θ) sin(θ)/ g sin(θ) g cos(θ)/ g 3. (5) Note again that the components have been expessed in the standad othonomal Catesian basis. Fo the spheical system one has that sin(ϕ) cos(θ) cos(ϕ) cos(θ)/ g sin(ϕ) sin(θ) g cos(ϕ) sin(θ)/ cos(ϕ) sin(ϕ)/ g 3 4 Gadient of a Scala Function sin(θ)/ sin(ϕ) cos(θ)/ sin(ϕ). (6) Conside a scala function f. Its gadient is given as f. This can be conveted though the use of the chain ule into cuvilinea coodinates as: f = f x = f x i ei = f z k z k x i ei = f z k gk. (7) Typically, howeve, esults ae expessed using the physical basis vectos and not the natual basis vectos. Fo ou two coodinates systems we have upon expansion: f = f e + f f = f e + 5 Gadient of a Vecto θ e θ + f z e z (8) f ϕ e f ϕ + sin(ϕ) θ e θ (9) To compute the gadient of a vecto expessed in cuvilinea coodinates we need to be able to compute the gadient of the basis vectos as they ae 4
functions of position (unlike in the Catesian case). Impotantly we will need to know the deivatives g i z j = z j x k z i e k = x k z j z i e k. () The components of these vectos ae usually expessed in the dual basis as Γ k ij = g k g i z j, () whee Γ k ij is called the Chistoffel symbol. Fo the cylindical coodinate system all of the Chistoffel symbols ae zeo except Γ =, Γ = Γ =. () Fo the spheical coodinate system we have all of the Chistoffel symbols ae zeo except Γ =, Γ 33 = sin (ϕ) Γ = Γ =, Γ 33 = sin(ϕ) cos(ϕ) (3) Γ 3 3 = Γ 3 3 =, Γ3 3 = Γ 3 3 = cot(ϕ) We can now conside taking the gadient of a vecto. This gives v = ( ) v i v i g x i = x g i + v i g i x (4) = vi z g k i g k + v i g i z k gk (5) ( ) v i = z + k +vj Γ i jk g i g k. (6) Fo the cylindical coodinate system we can expand this esult to detemine the needed components of the gadient. When expessed in tems of the physical basis we find that u is given by u, (u,θ u θ ) u,z u θ, (u θ,θ + u ) u θ,z. (7) u z, u z,θ u z,z 5
The components of the symmetic pat of this tenso give the stain, ε, (in the physical basis as) ( u, u ),θ + (u θ /), (u,z + u z, ) (u θ,θ + u ) (u θ,z + u z,θ). (8) sym. Fo the spheical coodinate system we can also expand this esult to detemine the needed components of the gadient. When expessed in tems of the physical basis we find that u is given by u z,z u, (u,ϕ u ϕ ) (u sin(ϕ),θ u θ sin(ϕ)) u ϕ, (u ϕ,ϕ + u ) u θ cot(ϕ) + u sin(ϕ) ϕ,θ u θ, u θ,ϕ u sin(ϕ) θ,θ + u / + u ϕ cot(ϕ)/ (9) The components of the symmetic pat of this tenso give the stain, ε, (in the physical basis as) ( u ) (,ϕ + (u ϕ /), u, ( (u ϕ,ϕ + u ) ) u sin(ϕ),θ + (u θ /), u θ cot(ϕ) + u θ,ϕ + sin(ϕ) u ϕ,θ sym. sin(ϕ) u θ,θ + u / + u ϕ cot(ϕ)/ 6 Gadient and Divegence of a Tenso ) (3) The basic pocedue fo finding the gadient and divegence of a tenso follows exactly as we did above. Fo simplicity conside the stess tenso σ. Its gadient is given by σ = ( ) σ ij σ ij g x i g j = x g i g j + σ ij g i x g j + σ ij g i g j x (3) = σij = z g k i g j g k + σ ij g i z g k j g k + σ ij g i g j z k gk (3) ( ) σ ij z + k +σlj Γ i lk + σ il Γ j lk g i g j g k. (33) The divegence is obtained by contacting upon the j and k indicies to give ( ) σ ij σ = z + j +σlj Γ i lj + σ il Γ j lj g i. (34) 6
Fo ou two coodinate systems we can expand this last expession and convet to physical components. The esult in the physical basis fo the cylindical coodinate systems is: e ( σ) = σ, + σ θ,θ + σ z,z + σ σ θθ e θ ( σ) = σ θ, + σ θθ,θ + σ θz,z + σ θ (35) e z ( σ) = σ z, + σ zθ,θ + σ zz,z + σ z The esult in the physical basis fo the spheical coodinate systems is: e ( σ) = σ, + σ ϕ,ϕ + e ϕ ( σ) = σ ϕ, + σ ϕϕ,ϕ + e θ ( σ) = σ θ, + σ θϕ,ϕ + sin(ϕ) σ θ,θ + σ σ ϕϕ σ θθ + σ ϕ cot(ϕ) sin(ϕ) σ ϕθ,θ + 3σ ϕ + (σ ϕϕ σ θθ ) cot(ϕ) sin(ϕ) σ θθ,θ + 3σ θ + σ ϕθ cot(ϕ) (36) 7