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**Significance and Use**

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 (σ_{33}) 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 σ_{33} influences (12.12). Since the σ_{ij} values will remain constant for a given azimuth, the s_{1}^{{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 sin^{2}ψ data yielding the values σ_{11}, τ_{13}, 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, σ_{11}, will influence the overall slope of the data, while τ_{13} is related to the direction and degree of elliptical opening. Fig. 4 shows a simulated d versus sin^{2}ψ profile for the tensor shown. Here the positive 20-MPa τ_{13} stress results in an elliptical opening in which the positive psi range opens upward and the negative psi range opens downward. A higher τ_{13} value will cause a larger elliptical opening. A negative 20-MPa τ_{13} 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 (σ_{33}) 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 σ_{33} influences (see 12.12). Since the σ_{ij} values and χ_{m} will remain constant for a given azimuth, the s_{1}^{{hkl}} term is renamed C, and the σ_{22} term is renamed D.

The σ_{11} influence on the d versus sin^{2}β plot is similar to omega and chi mode (Fig. 8) with the exception that the slope shall be divided by cos^{2}χ_{m}. This increases the effective ½s_{2}^{{hkl}} by a factor of 1/cos^{2}χ_{m} for σ_{11}.

The τ_{ij} influences on the d versus sin^{2}β 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 sin^{2}β influences of individual shear components for modified chi mode considering two detector positions (χ_{m} = +12° and χ_{m} = -12°). Components τ_{12} and τ_{13} cause a symmetrical opening about the σ_{11} slope influence for either detector position (Figs. 9-11); therefore, σ_{11} can still be determined by simply averaging the positive and negative β data. Fitting the opening to the τ_{12} and τ_{13} terms may be possible, although distinguishing between the two influences through regression is not normally possible.

The τ_{23} value affects the d versus sin^{2}β slope in a similar fashion to σ_{11} for each detector position (Figs. 12 and 13). This is an unwanted effect since the σ_{11} and τ_{23} influence cannot be resolved for one χ_{m} position. In this instance, the τ_{23} shear stress of -100 MPa results in a calculated σ_{11} 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, σ_{11} 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 σ_{11} but shall be approached with caution using one χ_{m} position because of the possible presence of a τ_{23} stress. The combination of multiple shear stresses including τ_{23} 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 σ_{11} 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 σ

FIG. 4 Sample d (2θ) Versus sin^{2}ψ Dataset with σ_{11} = -500 MPa and τ_{13} = +20 MPa

FIG. 5 Sample d (2θ) Versus sin^{2}ψ Dataset with σ_{11} = -500 MPa and τ_{13} = -20 MPa

FIG. 6 Modified Chi Mode Diagram for Measurement in σ_{11} Direction

FIG. 7 ψ and ϕ Angles Versus β Angle for Modified Chi Mode with χ_{m} = 12°

FIG. 8 Sample d (2θ) Versus sin^{2}β Dataset with σ_{11} = -500 MPa

FIG. 9 Sample d (2θ) versus sin^{2}β Dataset with χ_{m} = +12°, σ_{11} = -500 MPa, and τ_{12} = -100 MPa

FIG. 10 Sample d (2θ) Versus sin^{2}β Dataset with χ_{m} = -12°, σ_{11} = -500 MPa, and τ_{12} = -100 MPa

FIG. 11 Sample d (2θ) Versus sin^{2}β Dataset with χ_{m} = +12 or -12°, σ_{11} = -500 MPa, and τ_{13} = -100 MPa

FIG. 12 Sample d (2θ) Versus sin^{2}β Dataset with χ_{m} = +12°, σ_{11} = -500 MPa, τ_{23} = -100 MPa, and Measured σ_{11} = -472.5 MPa

FIG. 13 Sample d (2θ) Versus sin^{2}β Dataset with χ_{m} = -12°, σ_{11} = -500 MPa, τ_{23} = -100 MPa, and Measured σ_{11} = -527.5 MPa

**1. Scope**

1.1 This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of quasi-isotropic bearing steel materials by X-ray diffraction (XRD).

1.2 This test method provides a guide for experimentally determining stress values, which play a significant role in bearing life.

1.3 Examples of how tensor values are used are:

1.3.1 Detection of grinding type and abusive grinding;

1.3.2 Determination of tool wear in turning operations;

1.3.3 Monitoring of carburizing and nitriding residual stress effects;

1.3.4 Monitoring effects of surface treatments such as sand blasting, shot peening, and honing;

1.3.5 Tracking of component life and rolling contact fatigue effects;

1.3.6 Failure analysis;

1.3.7 Relaxation of residual stress; and

1.3.8 Other residual-stress-related issues that potentially affect bearings.

1.4 Units—The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.

1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.

**2. Referenced Documents** *(purchase separately)* The documents listed below are referenced within the subject standard but are not provided as part of the standard.

**ASTM Standards**

E6 Terminology Relating to Methods of Mechanical Testing

E7 Terminology Relating to Metallography

E915 Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement

E1426 Test Method for Determining the Effective Elastic Parameter for X-Ray Diffraction Measurements of Residual Stress

**ICS Code**

ICS Number Code 77.040.20 (Non-destructive testing of metals)

**UNSPSC Code**

UNSPSC Code

**DOI:** 10.1520/E2860-12

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