Numerical simulation of the X-ray stress analysis technique in polycrystalline materials under elastic loading
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INTRODUCTION
STRESS/strain techniques based on X-ray diffraction have found widespread usage in the determination of applied and residual stresses in the surface layers of crystalline materials.[1,2] The theory underlying these techniques is based on continuum elasticity analysis and is rigorously exact for homogeneous materials that possess the same stress/strain tensor at each point within the measurement volume.[3,4,5] On the other hand, polycrystalline aggregates, which comprise the bulk of the samples analyzed by X-ray methods, do not satisfy this assumption. In these materials, homogeneous far-field stresses cause local stress/strain fields that vary over distances comparable to the crystallite size.[6–9] In a specimen with no prior plastic deformation, the local stresses arise to maintain compatibility across boundaries that separate regions of different elastic constants. Consequently, the stress tensor at a point depends on the local elastic constants as well as the orientation and shape of the surrounding crystallites. Any probe with finite volume will measure an average of these stresses, the magnitude of which depends on the population of crystallites contained within the probe volume. In X-ray stress/strain analysis, this probe volume is defined by the diffraction condition and it is restricted to those crystallites whose scattering vectors bisect the incident and diffracting beams. As a consequence, not all possible grain orientations are present in the analysis volume and the volumes analyzed by using different reflections are not identical.[10] In the traditional analysis, these effects are taken into account indirectly by using ‘‘diffraction elastic constants’’ whose values are calculated using various assumptions on the stress/strain states within the material.[3,4,11] In general, these diffraction elastic constants change with reflection. Thus, the tradiD. CHIDAMBARRAO, Advisory Engineer, is with the IBM Semiconductor Research and Development Center, Hopewell Junction, NY 12533. Y.C. SONG, Graduate Student, is with the Henry Krumb School of Mines, Department of Chemical Engineering, Materials Science and Mining Engineering, Columbia University, New York, NY 10027. I.C. NOYAN, Research Staff Member, is with the IBM Research Division, Yorktown Heights, NY 10598. Manuscript submitted March 6, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
tional analysis assumes implicitly that the average strain obtained by X-rays is dependent on the reflection used. That is, this average strain tensor is inhomogeneous in the diffraction space. Furthermore, since in the case of a sample subjected to an applied uniaxial load, the stress from all reflections is assumed to be equal to the applied far-field stress,* the average stress is assumed to be constant in the *This is the case for the experimental determination of diffraction elastic constants.[5]
volumes sampled by all reflections.[12] Experimental validation of these assumptions in a rigorous manner requires the measurement of strains within individu
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