Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 7

Material creep is time-dependent permanent deformation under sustained stress. Its rate is often appreciable at a large homologous temperature $T/T_m$, because diffusion and obstacle-escape processes become rapid. In an activated regime a useful form iswith additional elastic, geometric and temperature prefactors absorbed into $A$ for the chosen approximation. The activation energy belongs to the rate-controlling process: bulk diffusion, grain-boundary diffusion and dislocation motion need not have the same value.

Boundary sliding can accommodate grain-shape changes during diffusional material creep; it is not a reason to identify either diffusion field as dislocation glide. The supplied map's A/B/C placement, including the boundary between B and C, determines these assignments. Its numerical contours apply to the material state used to construct that map, not to arbitrary grain sizes.

Lower the operating temperature or stress, suppress diffusion, and remove or stabilize the paths that accommodate material creep. Increasing the melting temperature can reduce $T/T_m$ at a specified operating temperature. Chemistry that reduces diffusivity or impedes dislocation climb can reduce the material creep rate; stable precipitates or dispersoids can resist glide and climb, provided they do not dissolve or coarsen rapidly in service.

Large grains reduce both Nabarro-Herring and Coble creep, according to their $d^{-2}$ and $d^{-3}$ dependencies. Directional solidification can reduce boundaries normal to the dominant tensile stress, while a single crystal removes grain-boundary transport and sliding from the ordinary polycrystal mechanism inventory. It does not remove bulk diffusion or dislocation material creep.

This is why room-temperature grain-refinement strengthening is not automatically a high-temperature material creep strategy: fine grains create more boundary transport paths. Any protective microstructure must remain stable at the service temperature, and the design must also avoid stress concentrations and unwanted thermal loading.

At height $z$ above the free lower end, a uniform vertical filament carries the weight of length $z$. With cross-sectional area $A$,The maximum stress is at the top and is independent of diameter. The diameter cancels because both mass and load-bearing area scale with $A$. Neglecting self-heating gradients, evaporation and initial fabrication stress, this top point sets the first creep-strain limit.

For a clearly labelled external reference estimate, take $\rho =1.93\times 10^4\mathrm{kg}{\mathrm{m}}^{-3}$. Then ${\sigma }_{\max }=1.89\times 10^4\mathrm{Pa}=0.0189\mathrm{MPa}$; the area is $3.14\times 10^{-8}{\mathrm{m}}^2$ and the total suspended mass is about $6.06\times 10^{-5}\mathrm{kg}$. The density is independently tabulated, not invented as a supplied 2016 value.^[ref]

Use the original map's top temperature axis at $2500^{\circ }\mathrm{C}$ and its right-hand tensile-stress axis, so no uncertain shear-stress conversion or additional shear modulus is required. The point lies in C, between the $10^{-9}$ and $10^{-8}{\mathrm{s}}^{-1}$ contours, nearer $10^{-8}$. A logarithmic interpolation of the scanned contours gives about $6\times 10^{-9}{\mathrm{s}}^{-1}$, with only graphical precision. Thusor of order three months in this steady-rate, reference-density map estimate. Reading only the bracketing contours gives the appropriately broad interval $5\times 10^6$ to $5\times 10^7\mathrm{s}$; the graph does not support an exact service-life number.

For the exact prescribed input, the answer remainsTODO: verify the density convention in the historical Data Book and the applicability of the supplied map to the filament's grain structure. Thermal expansion lowers density, and an evolved area or tertiary material creep changes the rate; a steady initial-rate estimate is not a guaranteed lifetime. In particular, the source map has its own fixed grain-size assumptions and no separately specified filament grain size. Other failure mechanisms could shorten useful life.