The ESPI

observed surface displacement D(r,) is the component of the surface displacement vector

d(r,) in direction of the sensitivity vector The direction of this vector bisects to directions of

the illumination and object beams in Fig. 1. Combining equations (2)—(6), and

using trigonometric identities gives the ESPI observed in surface displacement:

In equation

(7), the displacement field D (r, ) is experimentally determined and the quantities (r), etc., are

known through finite element calculations. The objective is to calculate

residual stresses P, Q, and T (and possibly rigid-body motions . The mathematical challenge (and opportunity) is to

evaluating these few quantities from many thousands of measured data D (r, ).

Despite

its substantial length, equation (7) is algebraically simple. It retains in the

linear form trigonometric characteristics of its origin in equation (6). This

feature makes the equation (7) amenable to Fourier analysis, by which means

that it can be divided into manageable parts. In addition, and also very

importantly, the Fourier method reduces the thousands of the measured D (r, ) data to a much smaller number of the representative

quantities.

The

analysis proceeds from the orthogonality properties of trigonometric functions:

The

solution procedure involves weighting of the measured D (r, ) data using a trigonometric function, say cos 2, and integrating over an annular region around the

hole bounded by the inner and outer radii and. These radii are chosen so that they enclose to the

region of significant surface displacement. All the integrals corresponding to

individual terms in equation (7), except the one containing cos 2, equal zero because of the orthogonality properties

in equation (8). Equation (7) reduces to

from which

Q can be evaluated explicitly. A similar calculation using the sin 2 as a weighting

function yields T. However, there is no similar direct way of evaluating P

because all the associated trigonometric terms also appear elsewhere in the equation

(7). Even without this limitation, it turns out that equation (9) is not an

ideal way of proceeding because of the calculation uses only the axial

displacements (r). This

displacement is much smaller than the in-plane displacements (r) and (r), and so are more sensitive to measurement noise. A

different approach is therefore required.

The idea

of using orthogonality to divide the terms in equation (7) can be generalized

beyond the direct application in equation (9). When carrying out practical the calculations,

it is useful to defining the following dimensionless “calibration

constants”•

A radial

weighting function f (r) is included in the equation (10) to provide in future

mathematical needs. The displacement profiles (r), etc., can

be determined from the finite element calculations, and f (r) and other

quantities that are explicitly known. Thus, , , etc.., reduce

to dimensionless numbers. These “calibration constants” are functions

of the hole depth and Poisson’s ratio, and appropriate values of these latter

two quantities must be used for the associated finite element calculations.

However, these calculation needs only to be done once. The results can be

organized in a tabular form, from which future needed values can be extracted through

interpolation.

The

following weighted integrals of measured data is useful

(11)

and similarly,

for the analogous integrals using cos 2, sin 2, cos 3, and sin 3 as

circumferential weighting functions. In equation (11), the radial weighting

functions f (r) are the same as chosen for use in equation (10). The additional

factor 2 in definition of CO corresponds the same factor appearing in the first

of orthogonality conditions (8).

In

practice, equationss (l l) are evaluated from CCD data, which is in discrete

pixel format. In terms of the pixels, the second equation of equations (l l)

becomes

(12)

where i is

a pixel index and N is the number of the pixels within the integration area.

The other integrals can be evaluated also in the same way. Equations (10) and

(l l) were defined including a division by integration area to normalize equation

(12) and to make it independent of pixel density.

The main

obstacle to direct extraction of P, Q, and T from equation (7) is the sharing

of the trigonometric terms with the rigid-body motions . This obstacle can be removed by generalizing the

orthogonality of the idea contained in equation (8) to the radial direction, by

requiring

Use of

appropriate weighting function eliminates the integral associated with the term. Any

function f (r) obeying equation (13) is acceptable. Here, a simple polynomial

chosen

where

for CO, C2 and S2 =

and

for C l, Sl, C 3 and S3 ß =

The two

different values for ß shown in equation (14) are chosen to accommodate the

additional factor r in WI and un terms of equation (7). The first value of ß in

equation (14) is used for the out-of-plane constants in equation (10), and for

the even numbered integrals in equation (l l) and (12). The second value of ß

is used for in-plane constants in equation (10), and for the odd numbered

integrals in equation (l l) and (12).

There are

seven different trigonometric terms in the equation (7), and so seven weighted

integrals can be evaluated. Using the quantities defined in equation (10) and

(12), the results of seven integrations can be expressed compactly in matrix

form:

The three

rows above the dashed dividing line refer to the out-of-plane displacements,

and four rows below refer to the in-plane displacements. Three unknowns, P, Q,

and T are required and a total of seven equation are available. A solution can

be found using any of three rows. The upper three rows form diagonal matrix,

and therefore provide a particularly simple solution. However, as already

observed, the associated out-of-plane displacements are relatively small and

consequently are more sensitive to the measurement noise. Choosing from among

lower (in-plane) rows gives better solution. Even better is to choose all rows

because this more fully uses the available data. The least-squares method i7

provides a convenient way in achieving that objective. The procedure involves

pre-multiplying both sides of equation (15) by the transpose of the left-side

matrix to form the “normal equations” These comprise 3 x 3 matrix equation

whose solutions are P, Q, and T. The Cartesian stresses can then be determined

using;