Direction Dependent Grain-Interaction Models for the Diffraction Stress Analysis of Thin Films
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Direction Dependent Grain-Interaction Models for the Diffraction Stress Analysis of Thin Films Udo Welzel1, Matteo Leoni2 and Eric J. Mittemeijer1 1 Max Planck Institute for Metals Research, Stuttgart, Germany 2 Università di Trento, Dipartimento di Ingegneria dei Materiali, Trento, Italy ABSTRACT The well-known grain-interaction models for the description of the macroscopic elastic behaviour of polycrystalline specimens, for example the models due to Voigt and Reuss, may be applied to bulk specimens, but they are generally not suitable for thin films because they imply macroscopic elastic isotropy of the body. A thin film is usually at most transversely elastically isotropic, even in the absence of texture. A recently elaborated, alternative model for graininteraction in thin films, adopting grain-interaction assumptions first given by Vook and Witt (J. Appl. Phys. 7, 2169 (1965)), is able to predict the transversely isotropic elastic behaviour. Although this model is more appropriate for thin films than traditional models, it still imposes extreme grain-interaction assumptions, which are in general not compatible with the true elastic behaviour of real specimens. In this paper a more general approach to grain-interaction in thin films is proposed. INTRODUCTION So-called diffraction elastic constants are needed for the evaluation of diffraction stress measurements [1,2]. Generally, diffraction and macroscopic, mechanical elastic constants of polycrystals are calculated from single crystal elastic constants by adopting a grain-interaction model, describing the distribution of stresses and strains over the crystallographically differently oriented crystallites in a polycrystalline aggregate. The most widely used models are the Voigt [3], Reuss [4], Neerfeld-Hill [5,6] and Eshelby-Kröner [7,8] models. Devised for bulk specimens, these models imply macroscopically isotropic elastic behaviour for non-textured polycrystals. However, thin films are usually not macroscopically isotropic but exhibit only transverse isotropy along the plane of the film. Only recently van Leeuwen et al. [9] demonstrated that a grain-interaction model, adopting grain-interaction assumptions first formulated by Vook and Witt [10], can explain experimental findings (curved sin2ψ-plots; see below) for an untextured Ni film that are incompatible with the traditional grain-interaction models (see also [11]). Although the grain-interaction assumptions of Vook and Witt are more appropriate for thin films, still extreme (unphysical) constraints are imposed (see below). A more general approach to grain-interaction in thin films is proposed in this paper and applied to the diffraction analysis of stress in a fibre-textured copper layer. BASIS OF DIFFRACTION STRESS ANALYSIS A plane, rotationally symmetric state of residual stress is often met in thin films. Then, only one independent stress tensor component, the in-plane stress s // , has to be determined. For the diffraction stress analysis of a macroscopically elastically anisotropic specimen (e.g. a s
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