The most comprehensive and particularly reliable method for non-destructively measuring the residual stress of the surface layer of metals is the sin method. When X-rays were used the relationship of sin measured on the surface layer of the processing metal did not show linearity when the sin method was used. In this case, since the effective penetration depth changes according to the changing direction of the incident X-ray,  becomes a sin function. Since  cannot be used as a constant, the relationship in sin cannot be linear. Therefore, in this paper, the orthogonal function method according to Warren’s diffraction theory and the basic profile of normal distribution were synthesized, and the X-ray diffraction profile was calculated and reviewed when there was a linear strain (stress) gradient on the surface. When there is a strain gradient, the X-ray diffraction profile becomes asymmetric, and as a result, the peak position, the position of half-maximum, and the centroid position show different values. The difference between the peak position and the centroid position appeared more clearly as the strain (stress) gradient became larger, and the basic profile width was smaller. The weighted average strain enables stress analysis when there is a strain (stress) gradient, based on the strain value corresponding to the centroid position of the diffracted X-rays. At the 1/5 max height of X-ray diffraction, the position where the diffracted X-ray is divided into two by drawing a straight line parallel to the background, corresponds approximately to the centroid position.

A strain-gradient crystal plasticity finite element method(SGCP-FEM) was utilized to simulate the compressive deformation behaviors of single-slip, (111)[101], oriented FCC single-crystal micro-pillars with two different slip-plane inclination angles, 36.3o and 48.7o, and the simulation results were compared with those from conventional crystal plasticity finite element method(CP-FEM) simulations. For the low slip-plane inclination angle, a macroscopic diagonal shear band formed along the primary slip direction in both the CP- and SGCP-FEM simulations. However, this shear deformation was limited in the SGCP-FEM, mainly due to the increased slip resistance caused by local strain gradients, which also resulted in strain hardening in the simulated flow curves. The development of a secondly active slip system was altered in the SGCP-FEM, compared to the CP-FEM, for the low slip-plane inclination angle. The shear deformation controlled by the SGCP-FEM reduced the overall crystal rotation of the micro-pillar and limited the evolution of the primary slip system, even at 10% compression.

A strain-gradient crystal plasticity constitutive model was developed in order to predict the Hall Petch behavior of a Ni-base polycrystalline superalloy. The constitutive model involves statistically stored dislocation and geometrically necessary dislocation densities, which were incorporated into the Bailey-Hirsch type flow stress equation with six strength interaction coefficients. A strain-gradient term (called slip-system lattice incompatibility) developed by Acharya was used to calculate the geometrically necessary dislocation density. The description of Kocks-Argon-Ashby type thermally activated strain rate was also used to represent the shear rate of an individual slip system. The constitutive model was implemented in a user material subroutine for crystal plasticity finite element method simulations. The grain size dependence of the flow stress (viz., the Hall- Petch behavior) was predicted for a Ni-base polycrystalline superalloy NIMONIC PE16. Simulation results showed that the present constitutive model fairly reasonably predicts 0.2%-offset yield stresses in a limited range of the grain size.