**Unformatted text preview: **β ( x 3 ) = αx 3 which varies linearly along the shaft. The displacement field is then u 1 = x 1 [cos β ( x 3 )-1]-x 2 sin β ( x 3 ) u 2 = x 1 sin β ( x 3 ) + x 2 [cos β ( x 3 )-1] u 3 = 0 . (i) Compute the components of the Green strain tensor E = 1 2 ( ∇ u + ∇ u T + ∇ u T · ∇ u ) . (ii) Show (in components) that when β ¿ 1 the Green strain is approximately equal to the small strain tensor ² = 1 2 ( ∇ u + ∇ u T ) . Problem 4 (Problem 2.1 from Shames & Dym) Given the functional: I [ y ( x )] = Z x 2 x 1 [3 x 2 + 2 x ( y ) 2 + 10 xy ] dx, (i) Compute the first variation δI . (ii) Find the Euler-Lagrange equation. Problem 5 (Problem 2.10 from Shames & Dym) By setting the first variation to zero δI = 0 , find the Euler-Lagrange equation for the Brachistochrone problem, with functional I [ y ( x )] = 1 √ 2 g Z x 2 x 1 s 1 + ( y ) 2 y dx. 1...

View
Full Document

- Fall '04
- Klug
- Stress, Strain Energy density, Shames, strain tensor, constitutive law