Continuum mechanics II. Kinematics in curvilinear coordinates. 1. Strain in cartesian coordinates (recapitulation)

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1 Continuum mechanics office Math.7 Decembe 22, 2, Univesité de Fiboug. Stain in catesian coodinates (ecapitulation) Geen stain tenso: Lagange fomulation ε ij = ( ui 2 x j + u j x i + u k u k ) x i x j Cauchy stain tenso: lineaized stain fo small defomations e ij = ( ui 2 x j + u ) j x i Almansi stain tenso: Eule fomulation E ij = ( ui 2 y j + u j y i u k u k ) y i y j

2 . Stain in cuvilinea coodinates Geen stain tenso: Lagange fomulation ε ij = 2 ( j u i + i u j + i u k j u k) Cauchy stain tenso: lineaized stain fo small defomations e ij = 2 ( ju i + i u j ) 2. Example of using cuvilinea coodinates A otational cylinde is being defomed into a otational hypeboloid. Calculate the Cauchy stain tenso. It s advantageous to use the cylindical coodinates: ( ) ξ cos ξ 2 ξ cos ξ 2 ξ3 ( sin ξ2 ) x = ξ sin ξ 2 y(x) = ξ sin ξ 2 + ξ3 cos ξ2

3 2. Example of using cuvilinea coodinates Advantages of using cuvilinea coodinates: Simple analytical fomulae fo paticula defomation modes and paticula geometies Bette intuitive undestanding of defomation modes Paticulaly useful fo shells and membanes o anisotopic mateials Remembe the inflated baloon demonstation? σ t p 3. Cauchy stain in cylindical coodinates Cauchy stain in cuvilinea coodinates: e ij = 2 ( iu j + j u i ) Covaiant deivative: j u i = u i ξ j Γ l ij u l Cylindical coodinates: ξ cos ξ 2 x(ξ, ξ 2, ) = ξ sin ξ 2, [g ij ] = ( ξ ) 2 Chistoffel symbols of 2nd kind: fo cylindical coodinates Γ 22 = ξ, Γ 2 2 = Γ 2 2 = ξ, Γ l ij = othewise.

4 3. Cauchy stain in cylindical coodinates e = u = u ξ e 2 = 2 ( u u ) =» u2 2 ξ Γ2 2u 2 + u ξ 2 Γ2 2u 2 =» u2 2 ξ + u ξ 2 2 ξ u 2 e 3 = 2 ( u u ) =» u3 2 ξ + u e 22 = 2 u 2 = u 2 ξ 2 Γ 22u = u 2 ξ 2 + ξ u e 23 = 2 ( 2u u 2 ) = 2 e 33 = 3 u 3 = u 3» u3 ξ + u 2 2 Note that physical units of e ij ae quite inhomogeneous hee! 3. Cauchy stain in cylindical coodinates Non-homogeneity of physical units fo e ij and u i Units of cylindical coodinates: ξ in [m], ξ 2 in [ad], in [m]. Covaiant basis: g i = x cos ξ 2 g = sin ξ 2 in [] ξ i : ξ sin ξ 2, g 2 = ξ cos ξ 2 in [m], g 3 = {{ in [] Contavaiant basis: sin ξ 2 ξ g = g {{, g 2 = cos ξ 2 ξ in [] in [/m], g 3 = g 3 {{ in []

5 3. Cauchy stain in cylindical coodinates Non-homogeneity of physical units fo e ij and u i Units fo u i and u i : displacement u = u i g i = u i g i should be in [m]: Hence, units fo e ij : coodinate its unit coodinate its unit u [m] u [m] u 2 [] u 2 [m 2 ] u 3 [m] u 3 [m] coodinate its unit coodinate its unit e [] e 22 [m 2 ] e 2 [m] e 23 [m] e 3 [] e 33 [] Coection of unit inhomogeneity: intoduction of physical components e (ij) and u (i) by: e ij = g ii g jj e (ij) and u i = g ii u (i) 3. Cauchy stain in cylindical coodinates Tansfoming covaiant components to physical components Fo cylindical coodinates: e = e () e 2 = ξ e (2) e 3 = e (3) e 2 = ξ e (2) e 22 = (ξ ) 2 e (22) e 32 = ξ e (32) e 3 = e (3) e 23 = ξ e (23) e 33 = e (33) u = u () u = u () ξ j ξ j u 2 = ξ u (2) u 2 ξ = u (2) + ξ u (2) ξ, u 2 ξ = u (2) 2 ξ ξ 2 u 2 ξ = u (2) 3 ξ u 3 = u (3) u 3 = u (3) ξ j ξ j Physical components in cylindical coodinates usually witten u () = u, u (2) = u θ, u (3) = u z e () = e, e (2) = e θ, e (23) = e θz...

6 3. Cauchy stain in cylindical coodinates Tansfoming covaiant components to physical components e = u e θ = uθ + 2 e z = uz 2 + u z u θ u θ ««e θθ = e θz = 2 u θ θ + u u z θ + u «θ z e zz = u z z 4. Back to cylinde hypeboloid A otational cylinde is being defomed into a otational hypeboloid. Calculate the Cauchy stain tenso. Use the cylindical coodinates: ( ) ξ cos ξ 2 ξ3 sin ξ2 u = y x = ( ) ξ cos ξ 2 ξ ξ 3 sin ξ2 ξ sin ξ 2 + ξ3 cos ξ2 ξ sin ξ 2 = ξ cos ξ2

7 4. Back to cylinde hypeboloid ξ ξ 3 sin ξ2 cos ξ 2 u = ξ cos = sin ξ 2 ξ2 {{ g +(ξ ) 2 ξ3 sin ξ 2 ξ cos ξ 2 ξ + g 2 {{ g 3 Hence u = u 3 =, u 2 = (ξ ) 2 ξ3 and u (2) = u θ = ξ ξ3 Resulting Cauchy stain: e = e θ = e z = e θθ = e zz = and e θz = 2 ie. pue shea (ie. distotion of angles) in the (θ, z) tangent-plane. 5. Yet anothe cylinde hypeboloid example But diffeent fom the pevious one! A otational cylinde is being defomed into a otational hypeboloid in the following way (in cylindical coodinates): ( ) ξ cos ξ 2 ξ + 2 cos ξ 2 x = ξ sin ξ 2 y(x) = ( ) ξ + 2 sin ξ 2 The esulting shape is the same, but the defomation tenso is diffeent! Why?