The of the surface displacement vector d(r,) in direction

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, ).

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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;