Fig. 7 (a) Typical three-element strain-gage rosette. (b) In-plane strain components caused by release of residual stress through introduction of a hole. Source: Ref 21

The partial strain relief measured by one of the three strain gages in the rosette in Fig. 7(b) is related to the principal inplane residual stresses by:

where Omax and Omin are maximum and minimum principal residual stresses, and a is the angle from the gage axis to the maximum principal stress direction. A and B are calibration constants, the values of which depend on the specimen material properties, the rosette geometry, the hole diameter, and the hole depth. ASTM E 837-92 tabulates the calibration constants for the standard rosette pattern shown in Fig. 7.

Equation 17 can be inverted to determine the principal residual stresses from the measured strain reliefs. The result is:

tfmax, S min = [(«3 " A] ± {[(£3 - ^ + (£3 + - 2g2)Y2/4B}

where P is the angle measured clockwise from the location of gage 1 to the direction of omax. The above equations are valid for a homogeneous isotropic material that is wide, when compared with the hole diameter, and thick, when compared with the hole depth. The equations also apply to a through-the-thickness hole in a material in the form of a thin sheet, provided that the sheet thickness is uniform near the hole. The associated calibration constants have slightly different values than those of the thick-material case. Table 2, reproduced from ASTM E 837-92, lists the dimensionless calibration coefficients for both the thin-sheet and thick-material cases. A and B can be determined from the dimensionless coefficients using:

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