Characterization of inhomogeneous elastic deformation with x- ray diffraction
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I.
INTRODUCTION
IN most engineering applications, materials such as metals or ceramics are used in their polycrystalline form, with constituent crystallites (grains) of varying shapes, sizes, and orientations. Such materials are usually considered "macroscopically isotropic" if the grain size is much smaller than the overall aggregate dimensions and if there is no texture (preferred orientation) in the distribution of the crystallites. Under these conditions, continuum mechanics formulae, which assume no grains within the material, successfully correlate applied loads to the macroscopic displacements measured within the aggregate body. For example, the average elastic response of macroscopically isotropic bodies is customarily described by the isotropic Hooke's law. t~l l ~Oui Oujl l+v + = eij = - 2 (Oxj OxiJ E
v o'q -
E O'kk~3U
[1]
Where E and v, respectively, are the average Young's modulus and Poisson's ratio of the aggregate, o'ij represents components of the stress tensor, and e 0 represents components of the (elastic) strain tensor calculated from the macroscopic displacements ui. The Kronecker's delta, 6ij, ensures that the normal stresses do not cause shear strains, or vice versa. Similar to most equations in continuum mechanics, Eq. [ 1] contains no terms that limit the size of the specimen or the measurement volume. However, for polycrystalline specimens, it is applicable only if a representative amount of material is contained within the I.C. NOYAN, Research Staff Member, is with IBM, T.J. Watson Research Center, Yorktown Heights, NY 10598. L,S. SCHADLER, formerly Postdoctoral Associate, IBM, T.J. Watson Research Center, Yorktown Heights, NY, is currently Assistant Professor, Materials Engineering Department, Drexel University, Philadelphia, PA 19104. Manuscript submitted January 26, 1993. METALLURGICAL AND MATERIALS TRANSACTIONS A
analysis region.t2,31 This representative volume is defined as the minimum amount of material (number of crystallites or grains) over which the relevant parameters (E, v, and ui) yield averages equal to the macroscopic averages (Figure 1). Continuum formulations describing the plastic flow regime, such as the slip-line field analysis, the Levy-Mises (ideal plastic solid), and the Prandtl-Reuss (elastic-plastic solid) equations, t~l also require measurement volumes equal to or greater than the representative volume for the particular aggregate. Within volumes smaller than the representative volume, the average deformation is inhomogeneous. This inhomogeneity is caused by the elastic and plastic anisotropy of the constituent grains, and by the compatibility requirements at the grain boundaries. The magnitude of the inhomogeneous response also depends on geometric factors, such as grain shape, grain size, and relative grain rnisorientations, t2-71 These effects can cause deviations from the linear deformation theory, such that normal stresses (or,) can cause shear strains (co,,,J) as well. An example of such deformation is shown in Figure 2, which depicts the surface defor
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