Turbine blades in aircraft engines offer a good story of advanced technologies. Here we focus on...

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Turbine blades in aircraft engines offer a good story of advanced technologies. Here we focus on a key mechanics-materials issue. The blades rotate as a high speed at a high temperature. The centrifugal force causes tensile stress in the blades. All metals, including the nickel-based superalloys used today, creep under stress at high temperature. The creep strain rate of metals is a nonlinear function of stress, often fitted by a power-law: de Bo", dt where B and n are constant best fit the experimental data. A main concern is that the tensile stress in a blade will cause it to elongate to an unacceptable length. (a) Model a blade as a rod, fixed at one end, and rotating at a uniform angular speed w. Derive the differential equation that governs the stress field in the rod. (b) Derive the expression of the stress in the rod. (c) Derive an expression of the elongation of the rod as a function of time. (Hint: (a): use the momentum balance and consider the fact that the radial acceleration of a particle moves in a circle of radius x is w?r. Pay attention to the direction of acceleration. (b): integrate the differential equation that you found in (a) considering the fact that o = 0 at free end (c): first find an expression for € by integrating equation (*). Because €r = U, the elongation of the rod is AL = SEzzdx, which is a function of time. Because n is not given, we cannot evaluate this integral. Just leave the expression for AL in an integral form. ) Fixed end Free end Turbine blades in aircraft engines offer a good story of advanced technologies. Here we focus on a key mechanics-materials issue. The blades rotate as a high speed at a high temperature. The centrifugal force causes tensile stress in the blades. All metals, including the nickel-based superalloys used today, creep under stress at high temperature. The creep strain rate of metals is a nonlinear function of stress, often fitted by a power-law: de Bo", dt where B and n are constant best fit the experimental data. A main concern is that the tensile stress in a blade will cause it to elongate to an unacceptable length. (a) Model a blade as a rod, fixed at one end, and rotating at a uniform angular speed w. Derive the differential equation that governs the stress field in the rod. (b) Derive the expression of the stress in the rod. (c) Derive an expression of the elongation of the rod as a function of time. (Hint: (a): use the momentum balance and consider the fact that the radial acceleration of a particle moves in a circle of radius x is w?r. Pay attention to the direction of acceleration. (b): integrate the differential equation that you found in (a) considering the fact that o = 0 at free end (c): first find an expression for € by integrating equation (*). Because €r = U, the elongation of the rod is AL = SEzzdx, which is a function of time. Because n is not given, we cannot evaluate this integral. Just leave the expression for AL in an integral form. ) Fixed end Free end

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