## Xray Diffraction Method

Basic Principles. Stress measurement, using the x-ray diffraction method, is based on the change in the interplanar spacing (strain) close to the surface of the specimen material. The details of the theory and interpretation of residual stress measurements are well described in the article "X-Ray Diffraction Residual Stress Techniques" in Volume 10 of the ASM Handbook, as well as in Ref 9, 10, and 11. Consider an isotropic material with a lattice parameter d0hkl for unstressed material in the sample plane normal. The strain in a direction inclined by an angle y to the surface normal of the coating and the stress acting in the surface plane of the coating at an angle 9with the principal axis of the specimen are related by:

+ [(1 + vmi) / oz - (Va / Em) (ox + oy + oz) + [(1 + V ) / (t xzCoscp + T yzsincp)sin2\|/

where h,k, and l are the indices of the Bragg reflection; ehl and djy dtyy are the strain and interplanar spacing of (hkl) in the direction of (9, y) respectively; vhkl and Ehkl are Poisson's ratio and Young's modulus in (hkl) in the coating; ox, oy and oz are normal stresses; t t yz, and t xz are shear stresses; ox is the normal stress acting in the x direction on a plane perpendicular to the x axis; and t yz is the shear stress on a plane normal to the y axis (the first subscript) in the z direction (the second subscript). Residual stresses of the coating in various stress states can be determined using Eq 8.

Biaxial Stress. At a free plane, the out-of-plane stress components oz, Txz, and Tyz are all zero, at a free surface. Because the penetration depth of x-rays is very small, the resulting measurements refer specifically to near-surface material. Plane stress conditions therefore often apply to x-ray measurements, and Eq 8 is simplified to:

where 09= oxcos29 + oysin29 is the macrostress in the coating parallel to its surface at an angle 9with the principal axis of the sample.

For a biaxial stress state, ox = oy = 09, and at y= 0,

From Eq 9 and 10, dhy = C = 0 + 09 dhhkl [(1 + vhkl) / Ehkl] sinV (Eq 11)

In practice, high-angle diffraction peaks of an (hkl) reflection are obtained from -y to at a given angle 9. Lorentz polarization, absorption, and background corrections are applied to the diffraction peak profile. The peak positions are determined by profile fitting or other methods and are subsequently converted to interplanar spacing djy for stress analysis.

In the linear plot of d™ vs. sin2y, the intercept is I = d™ = 0 and the slope is M = 09 dh0kl [(1 + vhkl)/E hkl] .The stress of the coating can then be determined by:

Triaxial Stress State Without Shear Stress. For a material in a three-dimensional (triaxial) stress state without shear stress, but with the stress component oz having a finite value within the x-ray penetration volume, Eq 8 becomes (Ref 9, 11, 12):

= [(1 + v™) IE™](oxcoszcp + Gysin cp - oz)sin>

J Oz

Two data sets, 9= 0° and 9= 90°, are needed to obtain ox, oy, and oz. The slopes and intercepts of a linear function of dj' vs. sin2y at 9= 0° and 9= 90° are given by:

[(1 + vhkl) / EhklJ (ox - oz); [(1 + vhkl) / EhklJ (oy - oz);

I = dh0kl {[(1 + vkhl) / EhklJ Oz ~(yhkl / Ehkl) (gx + gy + gz)}

Stresses ox, Oy, and Oz can be determined from the sum of the slopes and the intercept in Eq 14.

Triaxial Stress State With Shear Stress. A coating with a three-dimensional (triaxial) stress state, including shear stresses, is fully described by Eq 8. The shear stresses, t xz and t yz, have a sin2y dependence. The djy vs. sin2y distribution is no longer linear and has two branches of an ellipse for 0 and 0. This effect is termed "y splitting," which is an indication of the presence of shear stress.

To obtain these stress-tensor components, three data sets (9 = 0°, 9= 45°, and 9= 90°) are obtained for both -y and The average strain a1 and the deviation a2 from the strains of "y splitting" are determined to be (Ref 9, 11, 12):

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