Homework+5.0 - Aero 315 Homework 5 February 1st 2008 Full...

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(1) A prismatic, isotropic, homogenous, linear elastic, circular cylinder:b r a, -c x 3 c , is shown below. This structure has a displacement vector field u where the components of displacement are given below in the given Cartesian coordinate system. Here α , β , , and are given, small, positive constants. Here is Poisson ʼ s ratio. Assume that b=0 (the cylinder is solid)and that the displacement field is provided below u 1 ( x 1 , x 2 , x 3 ) = α 2 ( x 3 2 υ x 2 2 + υ x 1 2 ) υ β x 1 γ x 3 x 2 δ υ 2 ( c x 3 )( x 1 2 x 2 2 ) x 3 3 6 + c x 3 2 2 u 2 ( x 1 , x 2 , x 3 ) = υα x 1 x 2 υ β x 2 + γ x 3 x 1 δ ( c x 3 ) x 1 x 2 u 3 ( x 1 , x 2 , x 3 ) = α x 1 x 3 + β x 3 + δ ( 3 4 + ν 2 ) a 2 x 1 + 1 4 ( x 1 3 3 x 1 x 2 2 ) + x 1 x 2 2 + x 3 x 1 ( c x 3 2 ) For each of the following scenarios: (i) What are the stresses inside the body? Provide an equation and some type of plot that describes an interesting component of stress as a function of position.
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