By Walter Thirring

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The grain boundary velocity as a function of temperature and pressure is shown in (c). ) end structures. Understanding this motion through modeling helps us unravel the complex mechanisms of creep and plastic flow. An example where the mobility of grain boundaries was measured in response to applied pressure is the work of Huang and Humphreys [HH99]. We reproduce some of their results in Fig. 14, where we see that typical grain boundary velocities in this system are on the order of 5–25 µm/s, eight orders of magnitude slower than the speed of sound.

Because summation of products, such as ai xi , appears frequently in tensor operations, a simplified notation is adopted where the Σ symbol is dropped and any index appearing twice in a product of variables is taken to be a dummy index and summed over. For example, S = ai xi = aj xj = am xm = a1 x1 + a2 x2 + · · · + an d xn d . t 24 Essential continuum mechanics and thermodynamics This convention was introduced by Albert Einstein in the famous 1916 paper where he outlined the principles of general relativity [Ein16].

75 µm wide. ) As snowflakes can form directly from water vapor, so too can crystalline metals be made by depositing a vapor, atomic layer by atomic layer – a process referred to as chemical vapor deposition. Once again, surface energetics determines the morphology of the resulting structures and drives microstructural evolution. 8 shows an example of the surface morphology during deposition of copper vapor onto an initially flat copper (111) surface. The steps and islands shown are each 2–4 atomic layers high, and adopt a morphology that is clearly influenced by the underlying surface structure (see the (111) face in Fig.