hw3 - e 3 e 1 water/air surface x 3 =0 θ n d n l x 3< 0...

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Unformatted text preview: e 3 e 1 water/air surface; x 3 =0 θ n d n l x 3 < 0 dam face MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed : Wednesday, March 3, 2004 Due : Wednesday, March 10, 2004 Problem 1 20 Points (a) The state of stress at position x in a fluid at rest (say, a liquid) can be characterized by σ ij ( x ) = − p ( x ) δ ij , where p is the [fluid] pressure at x . Assume that the liquid has a constant mass density, ρ , and assume further that the fluid is subject to a gravitationally-induced body force loading (per unit mass) of magnitude b = − g e 3 , where the cartesian basis vector e 3 points “up.” Let atmospheric [air] pressure at the surface of the liquid be p 0 (elevation of air/liquid surface: x 3 = 0) . Using the appropriate equilibrium equations, show that the pressure at generic elevation x 3 < 0 is given by p ( x 1 , x 2 , x 3 ) = p 0 − ρg x 3 . 1 (b) A long, straight dam holds the water in place. The dam extends along the e 2 direction, and the planar surface of the dam makes an angle of θ with respect to the vertical, as shown. At a point on the liquid/dam interface that is at elevation x 3 < 0: 1. evaluate the traction vector exerted by the dam on the fluid surface. The outward normal vector on the liquid is n l , as shown. 2. explain why the traction vector acting on the surface of the dam (that is, the traction exerted by the fluid on the surface element of the dam) is equal and opposite to the previously-determined traction vector in part (1). The outward normal to the surface of the dam is n d , as shown. 3. use the traction vector from part (2) to express 3 linear equations involving the carte- sian stress components σ ij in the dam at elevation x 3 . Problem 2 (20 Points) The isotropic linear thermal/elastic constitutive relations can be expressed in the compact notation 1 + ν ν 3 σ kk + α Δ T δ ij ....
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hw3 - e 3 e 1 water/air surface x 3 =0 θ n d n l x 3< 0...

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