5 Werkstofftechnik
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The elastic constants are the most basic mechanical properties of a material and are needed for any structural analysis of a component. For example, they have a major influence on the eigenfrequencies of vibrating parts. Single crystals of Ni-base superalloys are strongly anisotropic, which means that the observed properties are orientation dependent. Tensor algebra is then required to mathematically formulate the elastic properties and their relations to the crystal orientation. Hence, this chapter first summarizes some basic definitions and calculation rules for Rotation matrices, including the definition of the Euler angles, which are most commonly used to define the relative orientations of the crystal and the component. Parts of this chapter closely follow the lines of the excellent exposition of the topic by Olschewski.
The elastic energy of mixing for multi-component solid solutions is derived by generalizing Eshelby's sphere-in-hole model. By surveying the dependence of the elastic energy on the chemical composition and lattice misfit, we derive a lattice strain coefficient λ*. Studying several high-entropy alloys and superalloys, we propose that most solid solution multi-component alloys are stable when λ*<0.16, generalizing the Hume-Rothery atomic-size rule for binary alloys. We also reveal that the polydispersity index δ, frequently used for describing strain in multi-component alloys, directly represents the elastic energy e with e=qδ², q being an elastic constant. Furthermore, the effects of (i) the number and (ii) the atomic-size distribution of constituting elements on the phase stability of high-entropy alloys were quantified. The present derivations and discussions open for richer considerations of elastic effects in high-entropy alloys, offering immediate support for quantitative assessments of their thermodynamic properties and studying related strengthening mechanisms.