Figure

2 below schematically shows a circular hole of radius a drilled into a specimen subject to in-plane residual stresses components

x, andxy. As result of hole drilling, the specimen

surface around the hole deforms into three dimensions. For each surface point,

there are radial, axial

and circumferential displacement components.

For in-plane loading

around an axisymmetric feature such as circular hole, it is convenient to

transform the axial residual stress into equivalent isotropic and shear stresses.14

P (x + y)/2, Q = (x

This

transformation is useful because; for linear elastic material properties, the

associated deformations have

simple

trigonometric forms. The trigonometric relationship in the following equations

are “exact”, 15

and not approximate as they are sometimes reported.

For

isotropic loading, P acting alone, the deformations are

Uz(r,) Uz(r), Ur(r,) ur(r), U(r, ) = O, where Uz (r, ) is the axial

(out-of-plane) displacement at surface point with cylindrical co-ordinates (r, ). Uz(r) is radial profile of

the axial displacement. This profile can be evaluated using finite element

analysis. 15 The

first two equations of equations (2) indicate that the axial and radial

displacements Uz (r,) and Ur(r,) are independent of the angle

, i.e., it is axisymmetric.

This is as expected because of isotropic loading associated with P which is

non-directional. For this axisymmetric case, all circumferential displacements’

(r,) are zero.

For shear loading

at 45 to the x—y axes, Q acting alone, the

deformations are Where;

Vz is the axial displacement corresponding to Q, and vz(r)

the profile of axial displacement along the radius at = 0. Analogous

trigonometric relationship applies to radial and also circumferential

displacements.

Similar

equations apply for shear loading in axial directions, T acting alone, where the

superscript * is added instead of using a separate symbol indicating that the

deformations for T loading are essentially the same as for Q, but with rotation

of 45

In addition

to above elastic deformations, ESPI measurements may also include arbitrary

rigid-body motions caused by the small relative movements of the components in

Fig.1. Local temperature change and the bulk movement of part caused by

drilling are common causes of these movements. These rigid-body motions include

translation and rotation about x and y axes. The corresponding axial

displacements are; where , , and , respectively, are the normalized amplitudes of

rigid-body translation and rotations around x and y axes at r a.

The

Cartesian components of surface displacements for the combination of cases

described by equations (2)—(5) are; where

i, j, and k are unit vectors in x—y—z directions. The normalization with

respect of Young’s modulus in equation (6) allows the displacements (Ur (r,), etc., to be expressed in the dimensionless

form. However, these quantities are not completely material independent because

there remain complex dependences on Poisson’s ratio. Although some

approximations of this dependence are possible, 16 to be more conservative, a numerical scheme was

adopted here to interpolate between the displacements calculated at discrete

values of Poisson ‘s ratio.