Figure = (x This transformation is useful because; for

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

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P (x + y)/2,    Q = (x             

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

trigonometric forms. The trigonometric relationship in the following equations
are “exact”, 15
and not approximate as they are sometimes reported.

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

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.

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.