This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of quasi-isotropic bearing steel materials by XRD. Here the stress components are represented by the tensor σ_ij as shown in Eq 1 (1, p. 40). The stress strain relationship in any direction of a component is defined by Eq 2 with respect to the azimuth phi(ϕ) and polar angle psi(ψ) defined in Fig. 1 (1, p. 132).

Alternatively, Eq 2 may also be shown in the following arrangement (2, p. 126):

Using XRD and Bragg's law, interplanar strain measurements are performed for multiple orientations. The orientations are selected based on a modified version of Eq 2, which is dictated by the mode used. Conflicting nomenclature may be found in literature with regard to mode names. For example, what may be referred to as a ψ (psi) diffractometer in Europe may be called a χ (chi) diffractometer in North America. The three modes considered here will be referred to as omega, chi, and modified-chi as described in 9.5.

Omega Mode (Iso Inclination) and Chi Mode (Side Inclination)-Interplanar strain measurements are performed at multiple ψ angles along one ϕ azimuth (let ϕ = 0°) (Figs. 2 and 3), reducing Eq 2 to Eq 3. Stress normal to the surface (σ₃₃) is assumed to be insignificant because of the shallow depth of penetration of X-rays at the free surface, reducing Eq 3 to Eq 4. Post-measurement corrections may be applied to account for possible σ₃₃ influences (12.12). Since the σ_ij values will remain constant for a given azimuth, the s₁^{hkl} term is renamed C.

The measured interplanar spacing values are converted to strain using Eq 24, Eq 25, or Eq 26. Eq 4 is used to fit the strain versus sin²ψ data yielding the values σ₁₁, τ₁₃, and C. The measurement can then be repeated for multiple phi angles (for example 0, 45, and 90°) to determine the full stress/strain tensor. The value, σ₁₁, will influence the overall slope of the data, while τ₁₃ is related to the direction and degree of elliptical opening. Fig. 4 shows a simulated d versus sin²ψ profile for the tensor shown. Here the positive 20-MPa τ₁₃ stress results in an elliptical opening in which the positive psi range opens upward and the negative psi range opens downward. A higher τ₁₃ value will cause a larger elliptical opening. A negative 20-MPa τ₁₃ stress would result in the same elliptical opening only the direction would be reversed with the positive psi range opening downwards and the negative psi range opening upwards as shown in Fig. 5.

Modified Chi Mode-Interplanar strain measurements are performed at multiple β angles with a fixed χ offset, χ_m (Fig. 6). Measurements at various β angles do not provide a constant ϕ angle (Fig. 7), therefore, Eq 2 cannot be simplified in the same manner as for omega and chi mode.

Eq 2 shall be rewritten in terms of β and χ_m. Eq 5 and 6 are obtained from the solution for a right-angled spherical triangle (3).

Substituting ϕ and ψ in Eq 2 with Eq 5 and 6 (see X1.1), we get:

Stress normal to the surface (σ₃₃) is assumed to be insignificant because of the shallow depth of penetration of X-rays at the free surface reducing Eq 7 to Eq 8. Post-measurement corrections may be applied to account for possible σ₃₃ influences (see 12.12). Since the σ_ij values and χ_m will remain constant for a given azimuth, the s₁^{hkl} term is renamed C, and the σ₂₂ term is renamed D.

The σ₁₁ influence on the d versus sin²β plot is similar to omega and chi mode (Fig. 8) with the exception that the slope shall be divided by cos²χ_m. This increases the effective ½s₂^{hkl} by a factor of 1/cos²χ_m for σ₁₁.

The τ_ij influences on the d versus sin²β plot are more complex and are often assumed to be zero (3). However, this may not be true and significant errors in the calculated stress may result. Figs. 9-13 show the d versus sin²β influences of individual shear components for modified chi mode considering two detector positions (χ_m = +12° and χ_m = -12°). Components τ₁₂ and τ₁₃ cause a symmetrical opening about the σ₁₁ slope influence for either detector position (Figs. 9-11); therefore, σ₁₁ can still be determined by simply averaging the positive and negative β data. Fitting the opening to the τ₁₂ and τ₁₃ terms may be possible, although distinguishing between the two influences through regression is not normally possible.

The τ₂₃ value affects the d versus sin²β slope in a similar fashion to σ₁₁ for each detector position (Figs. 12 and 13). This is an unwanted effect since the σ₁₁ and τ₂₃ influence cannot be resolved for one χ_m position. In this instance, the τ₂₃ shear stress of -100 MPa results in a calculated σ₁₁ value of -472.5 MPa for χ_m = +12° or -527.5 MPa for χ_m = -12°, while the actual value is -500 MPa. The value, σ₁₁ can still be determined by averaging the β data for both χ_m positions.

The use of the modified chi mode may be used to determine σ₁₁ but shall be approached with caution using one χ_m position because of the possible presence of a τ₂₃ stress. The combination of multiple shear stresses including τ₂₃ results in increasingly complex shear influences. Chi and omega mode are preferred over modified chi for these reasons.

FIG. 2 Omega Mode Diagram for Measurement in σ₁₁ Direction

Note-Stress matrix is rotated 90° about the surface normal compared to Fig. 2 and Fig. 14.

FIG. 3 Chi Mode Diagram for Measurement in σ₁₁ Direction

FIG. 4 Sample d (2θ) Versus sin²ψ Dataset with σ₁₁ = -500 MPa and τ₁₃ = +20 MPa

FIG. 5 Sample d (2θ) Versus sin²ψ Dataset with σ₁₁ = -500 MPa and τ₁₃ = -20 MPa

FIG. 6 Modified Chi Mode Diagram for Measurement in σ₁₁ Direction

FIG. 7 ψ and ϕ Angles Versus β Angle for Modified Chi Mode with χ_m = 12°

FIG. 8 Sample d (2θ) Versus sin²β Dataset with σ₁₁ = -500 MPa

FIG. 9 Sample d (2θ) versus sin²β Dataset with χ_m = +12°, σ₁₁ = -500 MPa, and τ₁₂ = -100 MPa

FIG. 10 Sample d (2θ) Versus sin²β Dataset with χ_m = -12°, σ₁₁ = -500 MPa, and τ₁₂ = -100 MPa

FIG. 11 Sample d (2θ) Versus sin²β Dataset with χ_m = +12 or -12°, σ₁₁ = -500 MPa, and τ₁₃ = -100 MPa

FIG. 12 Sample d (2θ) Versus sin²β Dataset with χ_m = +12°, σ₁₁ = -500 MPa, τ₂₃ = -100 MPa, and Measured σ₁₁ = -472.5 MPa

FIG. 13 Sample d (2θ) Versus sin²β Dataset with χ_m = -12°, σ₁₁ = -500 MPa, τ₂₃ = -100 MPa, and Measured σ₁₁ = -527.5 MPa