Selfconsistent Evaluation of Residual Shear Stress Profiles near Ground Surfaces

The presence of large residual shear stresses near ground surfaces are often revealed by x-ray diffraction analysis’. With the X-Ray-Integral-AJethod(RIM)Z it is now possible to determine the depth profiles of strains and stresses as a function of the true depth below the surface. However, any shear stress component parallel to a free surface must be zero at the surface itself Therefore, the detection of shear stress implies the existence of steep gradients, but often they are at variance with the laws of static equilibrium at a boundary when the conventional Cauchy definition of stress is assumed. The observations can be reconciled with the laws of static equilibrium if a torque-stress theory is adopted3, but this requires the existence of a particular deformation structure in a thin region below the ground surface. Some measurements of residual strain patterns near ground surfaces are analyzed and interpreted according to the phase compensation hypothesis and according to the torque stress formulation as suggested by Gola and Coppa3. In this couple-stress hypothesis, the residual shear stress 013 (z) must have a sign inversion in depth. The results obtained with RIM seem to favor this hypothesis but also other explanations will be given. With the aid of a selfconsistency criterion it is possible to find a proper normalization of the measured strains which allows for the first time an independent calculation of the stress free lattice spacing which is often not known. NORMALIZATION OF STRAINS AND STRESSES Residual strain and stress profiles measured by X-rays in the near surface region of polycrystalline materials are always averaged quantities because the counted intensities are averages over the diffracted volume. Since the true z-profiles of residual stresses are generally of more interest in evaluation the effects of surface treatments, several analytical and numerical approaches have been employed to retrieve these z-profiles from the measured z-profiles 2P4-12. The author introduced a Fourier method in which it is assumed that the z-profile E(Z) can be represented by trigonometric basis functions. The mathematical details are described elsewhere2. One serious advantage of this procedure is the capability to find a proper and selfconsistent normalization of the measured strains which allows an in-situ evaluation of the stress free lattice spacing. The principles are summarized in the flow chart diagram of Fig. 1. From the measured d-values in a first step a do start value is evaluated. Because we deal with an iterative technique this step is called iter=O. With the use of orthonormal trigonometric basis functions as a representation of strain gradients, the true strain z-profiles are determined. Now, Hooke’s law is used to convert strains to stresses with the known x-ray elastic constants. The general form of Hooke’s law in terms of stress (oij), strain (aij), and the elastic stiffness (C&l) or compliance (Sijkl) tensors read as follows: Copyright 0 JCPDS-International Centre for Diffraction Data 1997 Copyright (C) JCPDS-International Centre for Diffraction Data 1997 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website – www.dxcicdd.com ICDD Website www.icdd.com Selfconsistent normalization of strains