D032 X-ray stress analysis employing the lattice parameter ellipsoid

Abstract

If we assume a spherical element with a radius “r” within a specimen and the specimen is then subjected to a state of stress, the sphere will be deformed into an ellipsoid. The semi-axes of the ellipsoid will have the values of (r + ex), (r + ey), and (r + ez), which are replaced by dx, dy, and dz, or for the cubic case, ax, ay, and az. In this technique, at a particular Phi angle (see Figure 1), the two-theta position of a high angle (hkl) peak is determined at Psi angles of 0, 15, 30, and 45°. These measurements are repeated for 3 to 6 Phi angles in steps of 30°. The “d” or “a” values are then determined from the peak positions. This data is then fitted to the general quadratic equation for an ellipsoid by the method of least squares. From the coefficients of the quadratic equation, the angle between the laboratory and the specimen coordinates (direction of the principle stress) can be determined. Applying the general rotation of axes equations to the quadratic, the equation of the ellipse in the x-y plane is determined. The ax, ay, and az values for the principal axes of the lattice parameter ellipsoid are then evaluated. It is then possible to determine the unstressed ao value from Hooke’s Law using ax, ay, and az, and the magnitude of the principal stresses/strains. . INTRODUCTION In 1925 H. H. Lester and R. H. Aborn [1] published their paper entitled “The Behavior Under Stress of the Iron Crystals in Steel”. They could not have foreseen that during the next 81 years there would be such a dramatic change in the way the x-ray stresses measurements would be carried out. The use of computer controlled x-ray diffractometers, with sophisticated software to reduce the data and determine the residual stress/strain tensor has made stress analysis a routine technique. Examples of these developments include the work of B. B. He and K. L. Smith [2], where they introduce the use of the two-dimensional detector for x-ray stress analysis. The twodimensional x-ray detector is able to record a large fraction of the diffraction cone, which will be distorted by the residual stress in the sample. It is this relationship of the stress tensor and the distortion of the diffraction conic that make this a very powerful method for stress analysis. The work of D. Balzar, R. Von Dreele and H. Ledbetter[3] have employed the Rietveld refinement of the diffraction patterns collected at different specimen orientations, the refined lattice parameters yield the strain components in the specimen. At the present time the sin(ψ) method is probably the most popular technique used, it is very simple to use and the tabulation of the elastic constants for most materials of interest have been tabulated. This paper presents a different approach to x-ray stress analysis, which employs the lattice parameter ellipsoid. This method has proven to give results that are in good agreement with the conventional sin(ψ) technique. However, it has one advantage over the sin(ψ) technique, that is it integrates the data over many φ angles, where the sin(ψ) technique only evaluates the data in one plane at a time. 145 Copyright ©JCPDS-International Centre for Diffraction Data 2007 ISSN 1097-0002