G. S. Schajer Weyerhaeuser Technology Center,

Tacoma, WA 98477 Mem. ASME

Measurement of Non-Uniform Residual Stresses Using the Hole-Drilling Method. Part I—Stress Calculation Procedures The Incremental Strain, A verage Stress, Power Series, and Integral methods are ex­amined as procedures for determining non-uniform residual stress fields using strain relaxation data from the hole drilling method. Some theoretical shortcomings in the Incremental Strain and A verage Stress methods are described. It is shown that these two traditional methods are in fact approximations of the Integral Method. Theoretical estimates of the errors involved are presented for various stress fields. Also, some simple transformations of stress and strain variables are introduced so as to decouple the stress/strain equations and simplify the numerical solution.

Introduction

The hole-drilling method is a popular and widely used technique for measuring residual stresses. Its popularity stems largely from its ease of use in many different applications and materials, its limited damage to the specimen, and its general reliability. A typical application of the hole-drilling method involves drilling a small shallow hole (depth = diameter) in the specimen. This removal of stressed material causes local­ized stress and strain relaxations around the hole location. The strain relaxations are conveniently measured using a specially designed strain gauge rosette, such as the one shown in Fig. 1. During the hole drilling, careful experimental technique is essential so as to avoid introducing additional localized stresses, particularly in materials which strain harden appreciably.

Most often, the hole-drilling method is used when the residual stress field is assumed not to vary with depth below the surface. In such cases, experimental relaxed strain calibra­tion data from test specimens with known uniform stress fields can be used directly [1,2]. For many years, there has also been a great interest in using the hole drilling method to measure non-uniform residual stresses. Two stress calculation pro­cedures, the Incremental Strain Method [3-6] and the Average Strain Method [7] have been widely adopted. The need to use experimental calibration data has limited the theoretical scope of these two stress calculation procedures, and some theoretical shortcomings have recently been identified [8-11].

Finite element calculation of calibration data opens new possibilities for improved ways of calculating non-uniform residual stresses from incremental relaxed strain data. The possibilities and practicalities of experimental calibrations

need no longer be limiting factors. The Power Series Method [9] and the Integral Method [8, 10, 12] both rely on finite ele­ment calculated calibration data, and do not have the theoretical shortcomings of the two traditional methods. This paper describes all four stress calculation procedures and shows that the Incremental Strain and Average Stress methods are in fact just approximations of the Integral Method. Some comparative stress calculations are also presented.

Basic Hole-Drilling Method

Many researchers have contributed, and continue to

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received by the

Materials Division November 4, 1987.

3 Strain Gauge Elements

Fig. 1 Strain gauge rosette for the hole-drilling method. Measurements Group type 062-RE. After Beaney, 1976 [2J.

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contribute, to the extensive literature describing the hole-drilling method. A good historical review is included in a re­cent paper by Hu [13]. Comprehensive practical information and further references are given in the Technical Note [14] supplied by Measurements Group, Inc., a manufacturer of the specialized strain gauge rosette shown in Fig. 1. This section summarizes some of the mathematics used to calculate residual stresses from measured strain relaxations for the uniform stress case. The mathematical method and nomenclature used here differ somewhat from those conven­tionally used. This is done in preparation for later developments for non-uniform stress fields. For uniform stress fields, the calculated results are the same as those from generally used procedures.

For a linear elastic isotropic material, it may be shown theoretically that the following general formula relates the strain relaxation measured at any of the strain gauges in the rosette in Fig. 1 to the principal residual stresses and the angle relative to the maximum principal stress direction

tr =^(ffmax + ffmin)+fi(ffmax-ffmin)COS2a (1)

where er = measured strain relaxation amax = maximum principal stress ffmin = minimum principal stress A,B = calibration constants a = angle measured counterclockwise from the

maximum principal stress direction to the axis of the strain gauge

The two calibration constants A and B depend on the geometry of the strain gauge used, the elastic properties of the material of the specimen, and the radius and depth of the hole.

Since the strain gauge geometry is constant when using the specialized rosette in Fig. 1, only the specimen elastic proper­ties and the hole radius and depth remain as variables. The dependence on elastic properties can be eliminated in practice by working in terms of two closely related dimensionless con­stants a and b [9], where

2EA

\ + v b = 2EB (2)

The factors of two are included so that the constants are associated with the mean biaxial and shear stresses, (<rraax + <7min)/2 and (ffmax - amin)/2, respectively.

Some simplification can also be achieved with the hole radius and depth dependencies. It is desirable to normalize these two dimensions with respect to the mean radius of the strain gauge rosette, rm. When normalized in this way, the constants a and b (and A and B) for a given normalized hole depth, are very nearly proportional to the square of the hole radius. Also, each constant reaches certain limiting values at similar normalized hole depths, irrespective of hole radius. This allows a unified specification of the hole depth limits for stress calculations and for "full" strain relief. The depth variables used here are:

Z = depth from surface, mm z = hole depth, mm H = Z/rm = nondimensional depth from surface h =z/rm = nondimensional hole depth

where, for the strain gauge rosette type shown in Fig. 1

rm = 2.57mm for the MM 062-RE gauge rm =5.15mm for the MM 125-RE gauge

Normalization relative to the mean radius of the strain gauge rosette is in keeping with previous theoretical developments [9] which showed that the sensitivity of the method (i.e., the magnitude of the strain relaxations) depends mainly on the hole radius, while the character of the response (i.e., the shape of the strain relaxation versus depth curve) depends mainly on the rosette geometry. The current ASTM

Standard [15] uses a different approach, and specifies nor­malization of the hole depth relative to the hole diameter. The ASTM approach is not used here because the constants a and b (and A and B) then do not have the properties described above.

For a 45 deg rectangular rosette of the type shown in Fig. 1, the relationship between the Cartesian stress components and the three strain relaxations measured by the strain gauge rosette can be compactly written using matrix notation

A+B

A

A-B

0 A-Bn

IB A

0 A+B

' °\ f\i

. ff3 .

=

" ei '

^2

. £3 _

(3)

where

ffj ,a3 = normal stress components in directions 1 and 3 T13 = shear stress normal to directions 1 and 3

e,,e2,e3 = strain relaxations measured at gauges 1, 2, and 3

Equations (3) may be decoupled using the following transformations of stress variables

P=(<73+(T,)/2 e=( f f 3 - f f , ) / 2 T=T13 (4)

and of strain variables

p=(ei+ei)/2 q=(e3-el)/2 t= (e3 +et - 2 e 2 ) / 2 (5)

Variables P andp conceptually represent the mean "pressure" of the residual stresses, and their corresponding "volumetric" strain relaxations. Similarly, the other four variables concep­tually represent the shear stress and shear strain components. With these transformations, the set of simultaneous equations (3) reduce to three separate equations

a P =Ep/(\ + v) (6)

bQ=Eq (7)

bT =Et (8)

where the dimensionless coefficients a and b have been used instead of A and B so as to give a more general solution. The more familiar Cartesian stress components may be recovered from these equations using

«i=P-Q

a3=P+Q (9)

r13 = r Finally, the principal stresses can be evaluated very com­

pactly in terms of the transformed stresses or strains

--P±-JQ2 + T1 ><V -'max'̂ min

V<72 + 1 2

(10) .a(\ + v) b J

0 = Vi tan ~' (T/Q) = Vi tan - ' (t/q) (11)

where /3 = angle measured clockwise from gauge 3 to the max­imum principal stress direction.

Incremental Strain Method

The Incremental Strain Method for estimating non-uniform residual stresses was first introduced by Soete and Vancrom-brugge [3] and further developed by Kelsey [4]. It involves measuring the strain relaxations after successive small in­crements of hole depth. The stresses originally existing within each hole depth increment are then calculated by assuming that the incremental strain relaxations are wholly due to the stresses that existed within that depth increment. For each

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depth increment, separate values of the calibration constants a and b (or A and B) must be used. These calibration constants for each hole depth increment are determined experimentally by incremental drilling into a specimen with a known, exter­nally applied, uniform stress field.

Although widely used, the incremental strain method has a significant theoretical shortcoming. The assumption that the incremental strain relaxations measured after making an incre­ment in hole depth are wholly due to the stresses within that depth increment is not valid. After the first hole depth incre­ment, subsequent strain relaxations combine the effect of the stresses within the new hold depth increment and the effect of the change in hole geometry. The geometrical change allows further strain relaxations from the stresses within previous hole depth increments. For this reason, strain relaxations can continue to grow, even when the new hole depth increment is totally unstressed [8-10].

Average Stress Method

In order to overcome the theoretical shortcomings of the In­cremental Strain Method, Nickola [7] introduced a new stress calculation method using the concept of equivalent uniform stress. The equivalent uniform stress is the uniform stress within the total hole depth that produces the same total strain relaxations as the actual non-uniform stress distribution. It is calculated by using the values of the calibration constants a and b (or A and B) for uniform stress fields with the measured strain relaxations.

With the Average Stress Method, the equivalent uniform stress is calculated using the strain relaxations measured before and after each hole depth increment. It is then assumed that the equivalent uniform stress after a hole depth increment equals the spatial average of the equivalent uniform stress before the hole depth increment plus the stress within the increment,

cz + Az(z + ^z)=azz + aAzAz (12)

where <x= equivalent uniform stress within subscript region

Z = hole depth before increment Az = hole depth increment

z + Az = hole depth after increment

The stress within the hole depth increment is then determined by solving this equation.

The Average Stress Method also has a significant shortcom­ing. It assumes that the equivalent uniform stress equals the average stress over the hole depth. This would be true only if the stresses at all depths within a given hole depth contribute equally to the strain relaxations measured at the surface. In practice however, the stresses in the material closer to the sur­face contribute much more to the surface strain relaxations than do the stress further from the surface. For this reason, the equivalent uniform stress is actually a weighted average, with a bias towards the stress values closer to the surface. Scaramangas et al. [6] experimentally found that, for a stress field linearly varying through the hole depth, the equivalent uniform stress equals the actual stress at about one quarter of the hole depth. In contrast, the average stress equals the actual stress at one half of the hole depth.

Power Series Method

The Power Series Method was introduced by Schajer [9] as an approximate, but theoretically acceptable method of calculating non-uniform stress fields from incremental strain data. Finite element calculations are used to compute series of coefficients °d(h), xd(h), 2d{h) and °b(h), xb(h), 2b(h), corresponding to the strain responses when hole drilling into

stress fields with power series variations with depth h, i.e., °a(h) = 1, xa(h) = h, 2a(h) = h2, etc. These strain responses are then used as basis functions in a least-squares analysis of the measured strain relaxations. In this way, the measured strains are decomposed into components corre­sponding to the power series stress fields. The actual stress field is then reconstructed by summing the stress fields cor­responding to the individual strain relaxation components.

The least-squares analysis is best done by applying the "nor­mal equations" [16] to each of the transformed strains defined in equation (4). The transformed stresses P(h) are calculated from strains p(h) = (ex{h) + e3 {h))/2 using

E°d(/i)°d(h) E°d(h)'d(h)

Lxd(h)°d(h) Vd(h)xd(.h)

i(h) -

i(h)

E

l + v

• o p -

Sp

' L°d(h)p(h)

Lxc (h)p(h)

stress P(h)=°P+xP h

(13)

(14)

where °P and 'P are the first two power series components of the "P" stress field, and E indicates the summation of the products of the d(h) andp(h) values corresponding to all the hole depths, h, used for the strain measurements. This calcula­tion is repeated for transformed stresses Q(h) and T(h) using strains q{h) and t(h) with coefficients b{h) instead of a (A), and omitting the factor 1 + v. The Cartesian stress field is then recovered using equations (9).

Figures 2 and 3 tabulate the functions °d(h), °b{h), and xd(h), xb(h). These values derive from the finite element calculations described in Part II of this paper [17], and are more accurate than those previously reported [9]. Only the d(h) and b(h) values for uniform and linear stress fields are given because the hole drilling method is not well adapted to giving accurate values for more than the first two power series terms for stresses. For the same reason, the maximum depth below the surface is limited to 0.5 /•,„. In Fig. 2, values of °d(h) and °b(h) are tabulated to greater hole depths because they correspond to the standard d and b values for a uniform stress field.

An advantage of the Power Series Method is that the least-squares procedure forms a best fit curve through the measured strain data. This averaging effect is particularly effective when strain measurements are made at many hole depth increments. A limitation of the method is that it is suitable only for smoothly varying stress fields.

Integral Method

The use of finite element calculations as a calibration pro­cedure has also made application of the Integral method a practical possibility. Initial developments in this area were made by Bijak-Zochowski [8], Niku-Lari et al. [10], and Flaman and Manning [12]. In the Integral Method, the con­tributions to the total measured strain relaxations of the stresses at all depths are considered simultaneously. For exam­ple, let o(H) be the stress at depth H from the surface. Assume for the present that this stress field is equal biaxial, i.e., at any given depth from the surface, the stresses are the same in all directions parallel to the surface. In this case the same strain relaxations will be measured at each of the strain gauges in the three-element rosette. The measured strain relax­ation e(h), due to drilling a hole of depth h, is the integral of the infinitesimal strain relaxation components from the stresses at all depths in the range 0 < H < h

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Hole Depth

ID

.00

.06

. 10

.15

. 20

.26

.30

.35

.40

.45

.50

.60

.70

.80

.90 1 .00

Hole Depth z / r„

.00

.05

. 10

. 15

.20

.25

.30

.35

.40

.45

.50

.60

.70

.80

.90 1.00

0.30

.000 -.011 -.027 -.043 -.058 -.072 -.084 -.093 -. 100 -. 105 -. 108

-.112 -. 112 -.111 -.108 -. 106

0.30

.000 -.021 -.050 -.082 -. 115 -.147 -. 175 -.200 -.220 -.237 -.251

-.271 -.282 -.288 -.290 -.291

Coe

Hole

0.35

.000 -.015 -.035 -.059 -.080 -.099 -. 114 -. 127 -. 135 -.142 -. 146

-. 149 -. 150 -.150 -.145 -. 141

Coe

Hole

0.35

.000 -.028 -.067 -.111 -.155 -.196 -.232 -.264 -.290 -.312 -.329

-.352 -.367 -.376 -.378 -.379

fficients a

Radius,

0.40

.000 -.021 -.049 -.079 -. 107 -.131 -. 150 -. 166 -. 176 -. 184 -. 189

-. 193 -. 193 -. 190 -. 187 -. 163

fficients

Radius,

0.40

.000 -.037 -.088 -.145 -. 201 -.252 -.297 -.335 -.367 -.392 -.413

-.441 -.457 -.466 -.471 -.472

r / i m

0.45

.000 -.028 -.063 -. 102 -. 137 -.167 -. 190 -.208 -.221 -.230 -.236

-.241 -.240 -.236 -.233 -.228

°b

V m 0.45

.000 -.048 -. Ill -.181 -.250 -.312 -.366 -.410 -.447 -.476 -.499

-.530 -.546 -.558 -.563 -.564

0.50

.000 -.033 -.081 -. 1,32 -. 175 -. 210 -. 236 -.256 -.270 -.280 -.286

-.290 -.289 -.285 -.282 -.276

0.50

.000 -.058 -. 140 -.229 -.311 -.382 -.443 -.492 -.531 -.562 -.587

-.619 -.639 -.648 -.653 -.654

Fig. 2 Coefficients °a and °b for a uniform stress field, °o(h)

Hole Depth Z / r m

.00

.05

.10

. 15

.20

.25

.30

.35

.40

.45

.50

Hole Depth z / r „ m

.00

.05

. 10

. 15

. 20

.25

.30

.35

.40

.45

.50

0.30

.0000 -.0003 -.0012 -.0029 -.0050 -.0075 -.0100 -.0123 -.0143 -.0159 -.0169

0.30

.0000 -.0005 -.0024 -.0058 -.0105 -.0163 -.0226 -.0292 -.0356 -.0416 -.0469

Coe

Hole

0.35

.0000 -.0004 -.0017 -.0039 -.0068 -.0101 -.0132 -.0162 -.0186 -.0205 -.0219

fficlent

Radius,

0.40

.0000 -.0005 -.0022 -.0052 -.0089 -.0130 -.0169 -.0203 -.0232 -.0254 -.0269

Coefficient

Hole

0.35

.0000 -.0007 -.0032 -.0077 -.0139 -.0214 -.0295 -.0377 -.0457 -.0532 -.0599

ItctUi Utl ,

0.40

.0000 -.0009 -.0042 -.0100 -.0178 -.0270 -.0368 -.0466 -.0560 -.0647 -.0725

s a

r IT a ro

0.45

.0000 -.0007 -.0029 -.0066 -.0112 -.0161 -.0206 -.0248 -.0281 -.0305 -.0321

9 'b

r IT a m

0.45

.0000 -.0012 -.0052 -.0123 -.0218 -.0328 -.0442 -.0556 -.0662 -.0760 -.0846

0.50

.0000 -.0008 -.0037 -.0085 -.0140 -.0196 -.0247 -.0291 -.0325 -.0350 -.0365

0.50

.0000 -.0014 -.0065 -.0154 -.0265 -.0389 -.0516 -.0639 -.0753 -.0856 -.0947

Fig. 3 Coefficients 1 a and 1 b for a linear stress field, 1 o(h) = h

e(h)=—^-[' A(H,h)o(H)dH Q<H<h (15)

where A(H, h) is the strain relaxation per unit depth caused by a unit stress at depth H, when the hole depth is h. The term (1 + v)/E describes the dependence of the strain relaxations on material properties.

The strain relaxation response, e(h), can be determined ex­perimentally from a sufficiently large number of strain gauge measurements with gradually increasing hole depths. However, it is very difficult, if not almost impossible, to deter­mine the strain relaxation function A(H, h) experimentally. This fact has prevented the practical use of this mathematical approach in the past. However, it is now possible to calculate A(H, h) using finite element calculations. If the functions e{h) and A (H, h) are then known, the unknown stress field o{H) can be determined by solving the integral equation (15). This solution is assumed to exist and to be unique.

In practice, the strain relaxation response e(h) is not con­tinuously determined. Only values at n discrete points, cor­responding to n hole depths after successive increments, h-, = 1, 2, , , n are known. In this case, an approximate solution can be achieved using a discrete form of equation (15)

Y, dijO: = e • 1 <j <j<i<n (16)

where e, = measured strain relaxation after the z'th hole depth increment

Oj = equivalent uniform stress within they'th hole depth increment

ciy = strain relaxation due to a unit stress within incre­ment j of a hole / increments deep.

n = total number of hole depth increments

The relationship between the coefficients dy and the strain relaxation function A (H, h) is

A(H,hi)dH

In matrix notation, equation (16) becomes

aa = Ee/(l + v)

where for four increments

«11

a21 fl22

a3l a32 a33

°41 a42 #43 #44 _

a =

"i

"2

^3

. ff4 .

€ =

£i

«2

^3

_ 6 3

(17)

(18)

(19)

The discrete strain relaxation matrix a is lower triangular. If the matrix coefficients dy are known, a stepwise approximate solution for the stress variation with depth can be found by solving equation (18) using simple forward substitution. The resulting stress values, <jj are the equivalent uniform stresses within each hole depth increment. As previously noted, the equivalent uniform stress within a hole depth increment does not equal the average stress within that increment because of the increased sensitivity to the stresses closer to the surface.

Figure 4 shows a physical interpretation of the coefficients dy of matrix d. The columns of the matrix correspond to the strain relaxations caused by the stresses within a fixed incre­ment, for holes of increasing depth. The rows of the matrix correspond to the strain relaxations caused by stresses within successive increments of a hole of fixed depth. The combina­tion of all the coefficients in each row corresponds to uniform stress field over the entire hole depth. Therefore, the row sums of a equal the strain relaxation coefficients d in equations (2) and (6) for uniform stress fields in holes of corresponding depth. With this in mind, the Incremental Strain and Average Stress methods can be seen as approximations of the Integral Method. The former two procedures seek to estimate the co­efficients dy given only the row sums from uniform stress field calibrations. In the Incremental Method, all the coefficients in each column are assumed to be the same. In the Average Stress

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i^ar i_# ±n

d32 ^33

AU\ %Z d43 a44

Fig. 4 Stress loadings corresponding to the coefficients aVj of matrix a

method, the coefficients in each row are assumed proportional to the size of the hole depth increments. Equations (20), (21), and (22) show the a matrices corresponding to each method for the case of four equal hole depth increments, Az = 0.1 rm, and a hole radius r„ = 0.4 rm.

Incremental Strain Method:

- .0490

a = .0490 - .0580

.0490 -0.580 - .0433

.0490 - .0580 -.0433 • .0257

(20)

Average Stress Method:

- .0490

.0535 - .0535

.0501 -0.501 - .0501

..0440 - .0440 - .0440 - .0440

(21)

Integral Method:

- .0490

- .0671 - .0399

- .0754 -0.507

- .0792 -.0547

.0242

.0305 - .0116

(22)

The numerical values of the coefficients a,-, derive from the finite element calculations described in Part II of this paper.

For conceptual simplicity, the discussion so far has been limited to a simple equal biaxial stress field. The in-plane stresses at a given depth were the same in all directions, and the three measured relaxed strains were all equal. For the general case, the three stress components a,, <r3, and T13, and the three strains e,, e2, and e3 vary independently throughout the hole depth. For calculations with such general non­uniform stress fields, it is mathematically convenient to work in terms of incremental transformed stress and strain vectors, P, Q, T and p, q, t, analogous to the scalar quantities defined in equations (4) and (5). Equations (6), (7), and (8) then become

b Q = £ q

bT =Et

(24)

(25)

In the equal biaxial stress field example, only equation (23) was needed because the stresses and strains in equations (24) and (25) were all zero.

To summarize, the stress calculation procedure for non­uniform stress fields is as follows. First the transformed strains after each hole depth increment are calculated from the measured strains using the vector equivalent of equations (5). Then the transformed stresses within each hole depth incre­ment are determined by solving matrix equations (23), (24), and (25). Finally, the Cartesian stresses within each increment are recovered from the transformed stresses using equations (9). Alternatively, the principal stresses and principal stress angle can be found from the transformed stresses using equa­tions (10) and (11).

The incremental stress calculation procedure described by Flaman and Manning [12] is mathematically equivalent to that described here. They work in terms of Cartesian stresses and strains, and do not decouple their equations. In that case, it is necessary to solve for all stress components simultaneously, using an enlarged strain relaxation matrix which combines the elements of both a and b. The matrix is block lower-triangular, where each block is a 3 x 3 submatrix containing the elements shown in equation (3). Because of this structure, the matrix has nine times more coefficients than either a or b, and has a much bulkier and less revealing solution.

Comparison of Methods

Experimental data from hole-drilling measurements into residual stress fields with steep gradients are not readily available in the literature. Here, a theoretical example is con­sidered, where a hole of radius r„ = 0.4 rm is drilled in four equal depth increments, Az = 0.1 rm into an equal-biaxial stress field. The theoretical stress field a = 1 - H - 2H2, where H = Z/rm, is chosen so as to be both non-uniform and nonlinear. The numerical results tabulated in Part II of this paper are used to compute the corresponding strain relaxations.

Figure 5 shows a comparison of the results of all four stress calculation methods. Calculations for the Incremental Strain, Average Stress, and Integral Methods use the a matrices in

100

Incremental Strain Method

Average Stress Method

Power Series Method

Integral Metiiod

Actual Stress

^ - 4 . v;

a P =Ep/(l + v) (23) Fig. 5 Comparisons of the results from the four stress calculation methods

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equations (20), (21), and (22). The Power Series Method calculation uses the coefficients tabulated in Figs. 2 and 3. The Integral Method gives a good stepped approximation to the actual stress variation with depth, and the Power Series Method gives a close straight-line fit. The Incremental Strain and the Average Stress methods give much less satisfactory results. Both of these methods do not work well with stress fields that are significantly non-uniform. This is because they are "calibrated" using uniform stress field data. They main­tain the correct row sums in their corresponding a matrices, but the individual coefficients are not correct.

For smoothly varying stress fields, such as the one il­lustrated in Fig. 5, the Power Series Method with many small hole depth increments is probably the best choice. The least-squares procedure used by the Power Series Method tends to average out the effects of random measurement errors. This increases the robustness of the calculation. The Integral Method is better suited to the case where the residual stress field varies abruptly, and where strain relaxations are measured after only a few hole depth increments. Meticulous experimental technique using a drilling procedure that does not introduce additional localized stresses is essential whatever stress calculation method is used. Small strain measurement errors can cause significant variations in calculated stresses, particularly for stresses remote from the surface. This characteristic of the hole drilling method is explored further in Part II of this paper.

All calculation methods assume linearity of the specimen material. When the residual stresses exceed about 50 percent of the yield stress, small deviations start to occur in the measured strain relaxations due to localized yielding at the stress concentration created by the presence of the hole [2]. These strain deviations can have a significant effect on the calculation of non-uniform stresses because of the sensitivity to strain measurement errors.

The calibrated constants for all calculation methods are calculated assuming homogeneity of the specimen material. Where this is not true, for example with some case-hardened materials, a decrease in accuracy can be expected. If the depth variations of £"and v are known, specific calibration constants could be calculated.

Conclusions

Four calculation procedures are available to determine non­uniform residual stress fields from incremental strain relaxa­tion data from the hole drilling method. The two traditional procedures, the Incremental Strain and the Average Stress methods are the simplest to apply because they use experimen­tally measured calibration data. However, they have some theoretical shortcomings, and should be avoided if possible. The Power Series Method is suitable for use with smoothly varying stress fields. It is relatively robust numerically because the least-squares procedure used tends to smooth out the ef­fects of random errors in the experimental strain data. The In­tegral Method is the most general of the four stress calculation procedures, and is suitable for calculations with irregular stress fields.

It is shown that the Incremental Strain and Average Stress methods are simple approximations of the Integral Method. For slightly non-uniform stress fields, all stress calculation

methods give satisfactory results. For more steeply varying stress fields, the Incremental Strain and Average Stress methods become increasingly unreliable, particularly for the stresses more remote from the specimen surface.

The simple transformations of stress and strain variables, equations (4) and (5), decouple the stress/strain equations (3). This greatly simplifies the stress calculations, particularly for non-uniform residual stress fields.

Acknowledgments This work was supported by Weyerhaeuser Company,

Tacoma, WA, and by Kimura Knife & Saw Mfg. Co. Ltd. and Kanefusa Knife & Saw Co. Ltd., Nagoya, Japan. Their kind support is much appreciated. Sincere thanks are due especially to Dr. Shiro Kimura of Nagoya University, without whose help and encouragement this study would not have been possi­ble. Dr. Will Fohrell and Ms. Sharon Geffen kindly reviewed the manuscript.

References

1 Rendler, N. J., and Vigness, I., "Hole-Drilling Strain-gage Method of Measuring Residual Stresses," Experimental Mechanics, Vol. 6, No. 12, 1966, pp. 577-586.

2 Beaney, E. M., "Accurate Measurement of Residual Stress on any Steel Using the Centre Hole Method," Strain, Vol. 12, No. 3, 1976, pp. 99-106.

3 Soete, W., and Vancrombrugge, R., "An Industrial Method for the Determination of Residual Stresses," Proceedings SESA, Vol. 8, No. 1, 1950, pp. 17-28.

4 Kelsey, R. A., "Measuring Non-Uniform Residual Stresses by the Hole Drilling Method," Proceedings SESA, Vol. 14, No. 1, 1956, pp. 181-194.

5 Bathgate, R. G., "Measurement of Non-Uniform Bi-Axial Residual Stresses by the Hole Drilling Method," Strain, Vol. 4, No. 2, 1968, pp. 20-29.

6 Scaramangas, A. A., Porter Goff, R. F. D., and Leggatt, R. H., "On the Correction of Residual Stress Measurements Obtained Using the Centre-Hole Method," Strain, Vol. 18, No. 3, 1982, pp. 88-97.

7 Nickola, W. E., "Practical Subsurface Residual Stress Evaluation By The Hole-Drilling Method," Proceedings of the Spring Conference on Experimental Mechanics, New Orleans, June 3-13, 1986, pp. 47-58. Society for Experimental Mechanics.

8 Bijak-Zochowski, M., " A Semidestructrve Method of Measuring Residual Stresses," VDI-Berichte, Vol. 313, 1978, pp. 469-476.

9 Schajer, G. S., "Application of Finite Element Calculations to Residual Stress Measurements," ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 103, No. 2, 1981, pp. 157-163.

10 Niku-Lari, A., Lu, J., and Flavenot, J. F., "Measurement of Residual-Stress Distribution by the Incremental Hole-Drilling Method," Experimental Mechanics, Vol. 25, No. 6, 1985, pp. 175-185.

11 Flaman, M. T., Mills, B. E., and Boag, J. M., "Analysis of Stress-Variation-With-Depth Measurement Procedures for the Center-Hole Method of Residual Stress Measurement," Experimental Techniques, Vol. 11, No. 6, 1987, pp. 35-37.

12 Flaman, M. T., and Manning, B. H., "Determination of Residual Stress Variation with Depth by the Hole-Drilling Method," Experimental Mechanics, Vol. 25, No. 9, 1985, pp. 205-207.

13 Hu, C. N., "Recent Developments Achieved in China About the Centre Hole Relaxation Technique for Residual Stress Measurement," Strain, Vol. 22, No. 3, 1986, pp. 119-126.

14 "Measurement of Residual Stresses by the Hole-Drilling Strain-Gage Method," Tech Note TN-503-2. Measurements Group, Inc., Raleigh, NC, 1986.

15 "Determining Residual Stresses by the Hole-drilling Strain-Gage Method," ASTM Standard E837-85.

16 Dahlquist, G., Bjork, A., and Anderson, N., Numerical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1974, Chapter 4.

17 Schajer, G. S., "Measurement of Non-Uniform Residual Stresses Using the Hole-Drilling Method. II. Practical Application of the Integral Method," ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, published in

this issue pp. 344-349.

Journal of Engineering Materials and Technology OCTOBER 1988, Vol. 110 / 343

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