On the Separation of Complete Triaxial Strain/Stress Profiles from Diffraction Experiments

Abstract

The fundamental equation in X-ray diffraction relates the measured strain quantities to a superposition of six independent components, three normal and three shear components which can exist as gradients with depth. However, the linear system of equations to solve for the six components leads to a singular matrix. In such a case of nonunique solution is expected. In the literature then so called regularization methods are recommended. All these methods work only if the determinant of the matrix is close to zero, in diffraction experiments it is definitely zero because of the existence of the normal component ε33. Therefore in the past, assumptions were made such as biaxial stress states and so on. It is shown that by a numerical differentiation the shear components can simply resolved. Once the shear components have been subtracted from the fundamental equation, the three normal components remain. By a Taylor series development of the fundamental equation, it is shown that ε33 and its first derivative at ψ = 0 are independent of the rotation angle φ. This requires a special structure of the matrix to analyze the data at different φ rotations. Once these two values are obtained, they serve as the initial conditions of a differential equation of second kind which is solved numerically. The unknown functions in the differential equation are approximated by a Taylor series expansion whose coefficients are determined by a nonlinear optimization procedure. Together with simulated data, first results are presented.