Diffraction stress analysis of grain interaction in polycrystalline materials

U. Welzel and E. J. Mittemeijer

Max Planck Institute for Metals Research – Heisenbergstraße 3 – 70569 Stuttgart, Germany

The elastic behaviour of polycrystals depends upon the single-crystal elastic constants of its constituents (grains), the crystallographic texture and the microstructure. However, a straightforward calculation of the mechanical elastic constants (relating mechanical (macroscopic) strains to mechanical (macroscopic) stresses) and the diffraction elastic constants (diffraction stress factors for elastically anisotropic samples), relating (diffraction) lattice strains to mechanical stresses from single-crystal elastic compliances (or stiffnesses) of the crystallites composing the specimen is generally not possible without more ado. A so-called grain-interaction model can be adopted, describing the distribution of stresses and strains over the crystallographically differently oriented grains in the specimen.

Extreme cases for grain interaction are given by the Voigt [1] and Reuss [2] proposals (either the strain or stress tensors of all crystallites are taken equal to the mechanical strain and stress tensors, respectively), which are generally incompatible with physical reality but set bounds for the mechanical elastic constants [3]. It is common to all traditional grain-interaction models (like the extreme Voigt and Reuss models and the intermediate models as the Neerfeld-Hill [3,4] and the Eshelby-Kröner [5-7] models) that they involve that a polycrystal is mechanically elastically isotropic in the absence of crystallographic texture, as the same grain-interaction assumptions are adopted along all directions in the specimen. They can therefore be called isotropic grain-interaction models [8]. However, even in the absence of crystallographic texture, polycrystals cannot generally be considered as being mechanically elastically isotropic. It can be anticipated that deviations from an isotropic ‘microstructure’ may have an impact on the elastic properties of polycrystals.

Two microstructural features involving the occurrence of macroscopic elastic anisotropy (even in the absence of crystallographic texture) have been recently considered by the development of appropriate grain-interaction models: the presence of a free surface in thin films, 'surface anisotropy', and morphological (grain-shape) texture, which is also frequently encountered in thin films. The mechanical and diffraction elastic constants can be calculated employing recently developed grain-interaction models. As in such models different grain-interaction assumptions are adopted along different directions in the specimen, these models can therefore be called direction-dependent grain-interaction models.

The present paper presents an overview of recent work on the development of direction-dependent elastic grain-interaction models and the diffraction analysis of elastic grain interaction in polycrystalline material [8-12]. The extension of the applicability of the newly developed grain-interaction models to the plastic deformation regime will also be discussed on the basis of selected examples.

[1] Voigt, W., 1910, Lehrbuch der Kristallphysik (Leipzig-Berlin: Teubner).

[2] Reuss, A., 1929, Zeitschrift für angewandte Mathematik und Mechanik 9, 49.

[3] Hill, R., 1952, Proc. Phys. Soc. London 65, 349.

[4] Neerfeld, H., 1942, Mitt. K.-Wilh.-Inst. Eisenforschg. 24, 61.

[5] Eshelby, J. D., 1957, Proc. Roy. Soc. A 241, 376.

[6] Kröner, E., 1958, Z. Physik 151, 504.

[7] Kneer, G., 1965, Phys. Stat. Sol. 9, 825.

[8] Welzel, U. & Mittemeijer, E. J., 2003, J. Appl. Phys. 93, 9001.

[9] van Leeuwen, M., Kamminga, J.-D. & Mittemeijer, E. J., 1999, J. Appl. Phys. 86, 1904.

[10] Leoni, M., Welzel, U., Lamparter, P., Mittemeijer, E. J. & Kamminga, J.-D., 2001, Philos. Mag. A 81, 597.

[11] Welzel, U., Leoni, M. & Mittemeijer, E. J., 2003, Philos. Mag. 83, 603.

[12] Koch, N., Welzel, U., Wern, H. & Mittemeijer, E.J., 2004, submitted for publication.