Significance and Use
5.1 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 σij as shown in Eq 1 (1,5 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).
5.1.1 Alternatively, Eq 2 may also be shown in the following arrangement (2, p. 126):
5.2 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.
5.3 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 σij values will remain constant for a given azimuth, the s1{hkl} term is renamed C.
FIG. 2 Omega Mode Diagram for Measurement in σ11 Direction
FIG. 3 Chi Mode Diagram for Measurement in σ11 Direction
Note 1: Stress matrix is rotated 90° about the surface normal compared to Fig. 2 and Fig. 14.
5.3.1 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.
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
5.4 Modified Chi Mode—Interplanar strain measurements are performed at multiple β angles with a fixed χ offset, χm (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.
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 χm = 12°
5.4.1 Eq 2 shall be rewritten in terms of β and χm. Eq 5 and 6 are obtained from the solution for a right-angled spherical triangle (3).
5.4.2 Substituting φ and ψ in Eq 2 with Eq 5 and 6 (see X1.1), we get:
5.4.3 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 7 to Eq 8. Post-measurement corrections may be applied to account for possible σ33 influences (see 12.12). Since the σij values and χm will remain constant for a given azimuth, the s1{hkl} term is renamed C, and the σ22 term is renamed D.
5.4.4 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 cos2χm. This increases the effective 1/2 s2{hkl} by a factor of 1/cos2χm for σ11.
FIG. 8 Sample d (2θ) Versus sin2β Dataset with σ11 = -500 MPa
5.4.5 The τij 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 (χm = +12° and χm = -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.
FIG. 9 Sample d (2θ) versus sin2β Dataset with χm = +12°, σ11 = -500 MPa, and τ12 = -100 MPa
FIG. 10 Sample d (2θ) Versus sin2β Dataset with χm = -12°, σ11 = -500 MPa, and τ12 = -100 MPa
FIG. 11 Sample d (2θ) Versus sin2β Dataset with χm = +12 or -12°, σ11 = -500 MPa, and τ13 = -100 MPa
FIG. 12 Sample d (2θ) Versus sin2β Dataset with χm = +12°, σ11 = -500 MPa, τ23 = -100 MPa, and Measured σ11 = -472.5 MPa
FIG. 13 Sample d (2θ) Versus sin2β Dataset with χm = -12°, σ11 = -500 MPa, τ23 = -100 MPa, and Measured σ11 = -527.5 MPa
5.4.6 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 χm position. In this instance, the τ23 shear stress of -100 MPa results in a calculated σ11 value of -472.5 MPa for χm = +12° or -527.5 MPa for χm = -12°, while the actual value is -500 MPa. The value, σ11 can still be determined by averaging the β data for both χm positions.
5.4.7 The use of the modified chi mode may be used to determine σ11 but shall be approached with caution using one χm 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.
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, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.6 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
