Residual-stress Measurement in Orthotropic Materials Using the Hole-drilling Method

by G.S. Schajer and L. Yang

ABSTRACT--The hole-drilling method is used here to v~, vy., measure residual stresses in an orthotropic material. An ~rx, ~y existing stress-calculation method adapted from the iso- (]'max tropic case is shown not to be valid for orthotropic ma- terials. A new stress-calculation method is described, (rmi" based on the analytical solution for the displacement field around a hole in a stressed orthotropic plate. The validity "r~y of this method is assessed through a series of experi- ~b mental measurements. A table of elastic compliances is provided for practical residual-stress measurements in a t~l, +2 wide range of orthotropic materials.

List of Symbols

A, B, C = calibration constants c** = orthotropic strain relief compliances

Ex, Ey = elastic moduli along x and y (elastic sym- metry) axes

G~y = x -y shear modulus m = orthotropic elastic modulus ratio [eq (10)] r~ = hole radius r,, - mean radius of strain-gage rosette

u, v = displacements in x and y directions x, y - coordinates along elastic symmetry axes

W1, W2 = geometrical parameters [eqs (14) and (15)] XI, X2 = geometrical parameters [eqs (18) and (19)] Y1, I12 = geometrical parameters [eqs (20) and (21)]

a, 13 = orthotropic elastic material constants [eqs (12) and (13)]

~,:y = x - y Cartesian shear strain 8 r = measured relieved strain

ex, e r = x - y Cartesian normal strains 0 = counterclockwise angle measured from the

x direction to the axis of the strain gage K = orthotropic elastic material constant [eq

(11)]

G.S. Schajer (SEM Member) is Associate Professor and L. Yang is Graduate Student, University of British Columbia, Department of Mechanical Engineering, Vancouver V6T 1Z4, British Colum- bia, Canada.

Original manuscript submitted: May 9, 1993. Final manuscript received: January 24, 1994.

= x -y Poisson's ratios = x -y Cartesian normal stresses = maximum (most tensile) principal stress = minimum (most compressive) principal

stress = x - y Cartesian shear stress = angle measured counterclockwise from the

x direction to the direction of O'max = geometrical parameters [eqs (16) and (17)]

Introduction

The hole-drilling-method 1 5 is a well-established, popular technique for measuring residual stresses in a wide range of engineering materials. The method is easy to use, reliable in operation, and involves only limited damage to the specimen.

The conventional hole-drilling method can be used only with isotropic materials. However, many mod- em materials, such as fiber-reinforced composites, have distinctly anisotropic elastic properties. Bert et al. 6'7

and Prasad et al. 8 have generalized the computational procedure for the hole-drilling method to extend the use of the method to orthotropic materials. However, this generalization is shown here not to be valid. This paper presents a different solution method that can be used for materials of any degree of elastic orthotropy. An experimental example is presented to illustrate the use and applicability of the method.

Isotropic Case

Residual stresses are measured by the hole-drilling method using a strain-gage rosette of the type shown in Fig. 1. The positive x direction defining the x-y Cartesian stress system lies along the axis of strain gage 1. For the 'clockwise' rosette pattern 9 shown in Fig. 1, the negative y direction lies along the axis of gage 3. With a 'counter-clockwise' rosette, the positive y direction would lie along the axis of strain gage 3.

324 ~ December 1994

ty

X

Fig. 1 - - A S T M strain-gage rosette used for hole-dril l ing measurements 4

When making residual-stress measurements, a cir- cular hole is drilled at the geometrical center of the rosette to a depth slightly greater than the hole di- ameter. This hole locally relieves the stresses in the surrounding material, and the associated strain reliefs are measured by the three strain gages. For an isotro- pic material, the relieved strain measured by a strain gage whose axis is inclined at an angle 0 from the x direction is

er = A(~r~ + %) + B(% - %) cos 20

+ C %y sin 20 (1)

where the symbols are defined in 'List of Symbols' above. In this study, it is assumed that the residual stresses %, % and "rxy do not vary with depth from the specimen surface. The calibration constants A, B and C depend on the material properties, the rosette geometry, the hole diameter and the hole depth. For an isotropic material, C = 2B. 7 The calibration con- stants can either be determined experimentally' or nu- mericallyfl

Equation (1) can be rewritten in matrix form to re- late the three measured strains gl, 1~2 and ~3 in Fig. 1 to the Cartesian stresses ox, % and "rxy.

A - A T x y = E 2

- B 0 A + LO-yj 13 3 (2)

For a 'counter-clockwise' rosette, 9 the quantity - C in eq (2) becomes C.

The Cartesian stresses calculated by solving eq (2) can be used to determine the principal stresses, using

O" . . . . O'min = (~x + % ) / 2

_ _ 2 • V ( ( O " x 0" , ) /2) 2 -'1- "rxy (3)

and

1 [ 2"rxy ] 6 = - arctan - 2 L~x - %J (4)

Orthotropic Material Properties

For the two-dimensional case, five elastic constants are required to relate the Cartesian stresses and strains in an orthotropic material, m'u When the x and y axes lie along the principal elastic directions of the mate- rial, Hooke's Law generalizes to

= x/E. - % V , x / E ,

~, = % / E , - ~ x v ~ / E .

% = ~ / 6 ~ (5)

Only four of the five elastic constants are independent because of the elastic symmetry relationship

V~y/E~ = Vy~/Ey (6)

In general, the shear modulus Gxy is independent of all other elastic constants. However, in the isotropic case, Ex = Ey = E, Vxy = Vyx = v, and G,~ = G = 0.5E/(1 + v). There are then only two independent elastic constants and eqs (5) reduce to their more fa- miliar forms.

An interesting special case occurs when the shear modulus of an orthoU-opic material happens to be re- lated to the other elastic constants as follows.

1 l + v ~ l + v y ~ - - _ _ - ] - - -

G,~ Ex Ey (7)

In this particular case, the shear modulus is the same in all directions. The material has isotropic shear be- havior, but orthotropic axial behavior. An approxi- mately opposite case occurs when Ex = Ey but G # 0.5E/(1 + v). The latter material has an orthotropic shear modulus, with equal (but not isotropic) princi- pal axial moduli. These two types of orthotropy are considered in subsequent sections.

Orthotropic Hole-drilling Solution

A simple approach to hole-drilling in an orthotropic material is to assume that the relieved strain response has a similar trigonometric form to that in an isotropic material. Equations (1) and (2) are assumed still to apply, providing that gages 1 and 3 are aligned along

Exper imental Mechanics . 325

the elastic symmetry directions of the orthotropic ma- terial. In this case, C is an independent calibration constant, not related to A or B. The use of eq (1) was suggested for hole-drilling applications by Bert e t a l . 6'7

and subsequently by Prasad e t a l . 8 However, it is shown here that eqs (1) and (2) are not valid for hole drilling in an orthotropic material because the displacement field around a hole in a stressed orthotropic plate does not have a simple trigonometric form.

A mathematical solution for the displacements around a hole in a stressed orthotropic plate is used here to determine the relationship between the resid- ual stresses and the hole-drilling relieved strains�9 It is assumed that the orthotropic material under study has a sufficiently fine microscopic structure that it can be approximated as a homogeneous continuum�9 Follow- ing the method described by Smith,~4 the relieved dis- placement field around a hole in a stressed orthotropic plate (plane stress case) for x 2 + y2 _ r ] can be shown to be

oL2m 2 + 1)xy /2=

m ( a - ~ ) E x ( 1 - a m )

�9 [Y~(1 + f3m) "rxy - X ~ ( % - 13mCry)]

2m2 + Pxy +

m(13 - oO E x ( 1 - f3m)

�9 [Y2(1 + o tm ) "r~ - X2(O'x - cxm~ry)] (8)

V

1 + (x2m 2 l)y x

c~m 2 (et -- [3) Ey(1 - c~m)

�9 [XI(1 + [3m)Txy+ Y l ( f f x - ~m%)]

1 + [32rn 2 V~y +

f3m 2 (f3 - o 0 E y ( 1 - 13m)

�9 [ X 2 ( 1 + e~m) %y + I12(% - e tm%)] (9)

where

m = 4VEx/Ey (I0)

K = ~/-ExEy (I/Gx:y - 2Vxy/Ex)/2 (Ii)

o~ = ~V/K + ~v/(K 2 -- I) (12)

13 = ~,/K - V(K 2 - 1) (13)

W 1 = 4 V ( x 2 _ r 2 _ o ~ 2 m 2 ( y 2 - ra2)) 2 -[- (2~mxy) 2

= 2 132mZ(y: r2))2 + ( 2 1 3 m x y ) 2 W 2 4V(X2 -- r a --

04)

(15)

t~, = arctan [ 2 o t m x y / ( x 2 - ~ - e~2m2(y 2 - rZ~))]/2 (16)

~J2 = arctan [ 2 1 3 m x y / ( x 2 - r 2 - 132m2(y2 = r]))]/2 (17)

X, = x - W, cos ~, (18)

X2 = x - W2 cos ~2 (19)

Y , = oLmy - W1 sin lt,ll 1 (20)

}12 = 13my - W 2 sin q*2 (21)

The above solution is valid only for K > 1. This requirement puts a maximum limitation on the allow- able size of the shear modulus Gxy relative to the other elastic moduli. Except for composites that are spe- cially designed for high shear stiffness, most ortho- tropic materials have elastic properties for which K > 1. A solution for the case K --< 1 is presented by Schimke e t a l . u The angles +1 and Ill 2 in eqs (16) and (17) both lie in the same quadrant as 0 = arctan (y/ x). However, they do not in general each equal 0.

Deviation from Trigonometric Behavior

Figures 2 and 3 schematically show the calculated strain response versus angle 0 for an ASTM strain gage in a uniaxial tensile stress field�9 The curves are cal- culated by integrating the relieved displacement field described by eqs (8) and (9) over the strain-gage grid.15 To focus on curve shape rather than curve size, sche- matic vertical scales are used in the two figures�9 All the curves are scaled to a uniform size and are dis- placed vertically so that they coincide at 0 = 0 deg and 0 = 90 deg. This plotting procedure emphasizes the deviations from the trigonometric relationship in eq (1), independent of the actual sizes of the strains involved. Absolute strain values cannot be inferred from Figs. 2 and 3.

The two figures illustrate the similar effects of the two different types of elastic orthotropy. Figure 2 shows

(b Q:

0

"1

-2 0 ~

A ex = ~--~ A A

E y = 1 ~ x y = 0 . 3 G x y = i s o t r o p i c

i , i i i i i T

90 o 1800 270 ~ 3600

Angle, e

Fig. 2- -Angular variation of relieved strain in materials of varying degrees of axial orthotropy. ASTM strain- gage geometry 4 with hole radius ra = 0.464 rm. All curves are scaled so they coincide at e = 0 deg and e = 90 deg

326 �9 December 1994

.~2 03

o

0

-2 0 ~

' 7

Ex= 1 E y = l Vxy=0.3

900 1800 2700 3600

Angle, O

Fig. 3--Angular variation of relieved strain in materials of varying degrees of shear orthotropy. ASTM strain- gage geometry 4 with hole radius ra = 0.464 rr,. All curves are scaled so they coincide at 0 = 0 deg and 0 = 90 deg

the effect of having unequal principal axial elastic moduli Ex and Ey, with an isotropic shear modulus G~y. Figure 3 shows the effect of an orthotropic shear modulus G~y, with equal principal elastic moduli Ex and Ey. Increasing orthotropy of either kind causes increasingly large deviations from the trigonometric strain response predicted by eq (1). Figures 2 and 3 together confirm that any deviation from isotropic elastic behavior, either in terms of axial or shear mod- uli, causes eq (1) to be violated. The calculation method using eq (1) is therefore not seen not to be valid.

An important practical exception to the above ob- servations occurs during bore-hole measurements of rock stresses. 12'13 For such measurements, eq (1) al- ways applies exactly, even for highly orthotropic ma- terials. This is because the bore-hole technique uses displacement measurements at the curved boundary of the hole, rather than strain measurements beyond the hole boundary, as in the hole-drilling method. The stress and strain solutions presented by Leknitskii a6 directly illustrate the trigonometric relationship at the hole boundary. Unfortunately, this convenient result does not apply to hole drilling because the strain mea- surements are made beyond the hole boundary.

Numerical Results

Assuming only linear elasticity, the matrix ap- proach used in eq (2) can be generalized so that it accurately applies to an orthotropic material. In the orthotropic case, eq (2) generalizes to

x c12c137E x j E ll IC2, C22C231

E~/~Ey Lc31 c32 c33 J O-y ~3

(22)

where the elastic compliances c11-c33 are in general not related in any way to trigonometric-based con- stants such as A, B or C. The factor 1/V"-E~Ey is in- cluded so that the compliances c11-c33 are dimension- less constants. For the case of interest here, where the x and y directions of the rosette coincide with the prin- cipal elastic directions of the orthotropic material, the compliances c12 and c32 both equal zero.

The values of the elastic compliances in eq (22) depend on the orthotropic elastic properties of the specimen, the hole diameter and the strain-gage ro- sette geometry. Hole depth is also an important fac- tor. Practical experience with isotropic materials 1-3 and finite-element calculations 5 show that the elastic com- pliances for the blind-hole case converge at large hole depths to the results calculated from the plane-stress through-hole solution. Thus, for an orthotropic ma- terial, the plane-stress solution, eqs (8) and (9), can be used for the blind-hole case, providing the hole is made deep enough for the limiting state to be reached. This plane-stress solution applies, even when working with thick materials, because the strains are measured on the specimen surface, not within the plane-strain regime in the interior of the material. For an isotropic material, the limiting hole depth approximately equals the mean radius r,, of the hole drilling rosette. For an orthotropic material, the limiting depth depends on the ratio of the out-of-plane shear moduli to the in-plane axial moduli. The lower this ratio, the more rapidly the limiting hole depth is reached.

Table 1 lists numerical values of the compliance values to be used in eq (22) for a range of elastic constants. These numerical values are calculated us- ing eqs (8) and (9) and the method described in Ref 15. They apply to the case of a deep hole with an ASTM strain-gage rosette of the type shown in Fig. 1. The hole radius specified in Table 1 corresponds to a 3/32-in. hole in a standard ASTM strain-gage rosette of 1/16-in. nominal size, or a 3/16-in. hole in a 1/8-in. nominal rosette. Linear or polynomial interpolation 17 can be used to determine compliance values for materials with elastic properties between the tabulated values. For hole diameters slightly dif- ferent from the specified values, the compliances c11- c33 can be assumed to be proportional to the square of the hole diameter. When working with materials for which Ex > Ey, the first row of headings for the columns in Table 1 should be used. When Ey > Ex, the second row of headings should be used instead.

Table 1 illustrates how the compliance values in eq (22) for an orthotropic material deviate from the trig- onometric values expected from eq (2). For example, when Ex = 2, Ey = 0.5, V~y = 0, G~y = 0.4, the com- pliance matrix is

Cll C12 C13 ~ [ - . 2 9 1 0 .1831 Cal c22 c23| = - .073 .728 - . 1 9 6 ] C31 C32 C33 J ,228 0 - . 6 5 9 ] (23)

If eq (2) were obeyed, then Ctl = c33, c13 = C31 and c21 -" c23 = (Cll -~- C31. ) /2 = (C13 n L C 3 3 ) / 2 . Equation

Experimental Mechanics �9 327

TABLE 1--DIMENSIONLESS COMPLIANCES FOR HOLE DRILLING INTO AN ORTHOTROPIC MATERIAL i in

Ex/ Ey Vxy Gxy/ Ey c,1 c,3 c2, c22 c23 c3, c33

E~/Ex ~'yx G~y/E~ c33 c31 c23 c22 c2~ c~3 c~1

1 .00 .10 -.591 1 .25 .10 -.583 1 .50 .10 -.575 1 .75 .10 -.568 1 .00 .20 -.514 1 .25 .20 -.503 1 .50 .20 -,491

1 .75 .20 -.478 1 .00 .30 -.474 1 .25 .30 -.460 1 ,50 .30 -.445 1 .00 .40 -.450 1 .25 .40 -.433 1 .00 .50 -.433 2 .00 .15 -.453 2 .25 .15 -,448 2 ,50 .15 -.443 2 .75 ,15 -.438 2 .00 .30 -.403 2 .25 .30 -.396 2 .50 .30 -.389 2 .75 ,30 -.382 2 .00 .45 -.377 2 .25 .45 -.369 2 .50 .45 -.360 2 .75 .45 -.350 2 .00 .60 -,361 2 .25 .60 -.351 4 .00 ,20 -.350 4 .25 .20 -.347 4 .50 .20 -.344 4 .75 .20 -.341 4 .00 .40 -.318 4 ,25 .40 -.314 4 .50 .40 -.310 4 .75 .40 -.306 4 .00 .60 -.301 4 .25 .60 -.296 4 .50 .60 -.291 4 .75 .60 -.286 4 .00 ,80 -.291 4 .25 .80 -.285 4 .50 .80 -.279 4 .00 1.00 -.283 8 .00 .30 -,263 8 .25 .30 -.262 8 .50 .30 -.260 8 ,75 .30 -.258 8 .00 .60 -.244 8 .25 .60 -.242 8 .50 .60 -.239 8 .75 .60 -.236 8 .00 .90 -.234 8 .25 .90 -.231 8 .50 .90 -.228 8 .75 .90 -.224 8 .00 1.20 -,227 8 .25 1.20 -.224 8 .50 1.20 -.220

16 .00 .40 -.199 16 .25 ,40 -.198 16 .50 .40 -.196 16 .75 .40 -.195

.169

.123

.076

.030

.193

.146

.098

.049

.205

.157

.109

.213

.164

.218

.156

.122

.088

.053

.180

.145

.110

.074

.193

.158

.122

.085

.201

.165 ,137 .112 .088 .063 .162 .136 .110 .084 .175 .149 .122 .096 .183 .157 .130 .189 .123 .105 .087 .068 .146 .127 .108 .089 .158 .139 .120 .100 .166 .147 .127 .105 .092 .079 .065

-.291 -.314 -.336 -.357 -.188 - .205 -.221 -.235 -.146 -.160 ".172 -.123 -.135 -.108 - .226 -.244 -.263 -.281 -,144 -.159 -.174 -.188 -.111 -.124 -.137 -.149 -.092 --.105 -.186 -.201 -.216 -.232 -.116 -.130 -.143 -,156 - .088 -.101 -.112 -.124 -.073 - .084 -.095 -.063 -.138 -.150 -.162 --.174 - .084 -.095 -.105 -.116 -.062 -.072 -.082 -,092 -,050 -.060 - ,069 -.109 -.118 -.127 -.136

1.180 -,291 .169 -.591 1.097 -,314 .123 -.583 1.013 -,336 .076 -.575 .930 -,357 .030 -.568 .877 -,188 .193 -.514 .788 -,205 .146 -.503 .699 -,221 .098 --.491 .609 -,235 .049 -.478 .757 -.146 .205 -.474 .665 -.160 .157 -.460 ,572 -.172 .109 -.445 ,692 -.123 .213 -.450 .598 -.135 .164 -.433 .650 -.108 ,218 -.433

1.156 -.345 .185 -,743 1.097 -~357 .153 -.737 1.038 -.369 .122 -.730 .978 -.380 .090 -.723 ,865 -.225 .207 -.631 .802 -.233 .175 -.621 .739 --.240 .143 --.611 .675 -.247 .110 -.600 .751 -.177 .217 -.576 .686 -.182 .185 -.563 .620 -.187 .152 -.550 .554 -.191 .119 -.536 .689 -.150 .223 -.541 .623 -.154 .191 -.527

1,212 -.435 .195 -.949 1.171 -.441 .174 -.944 1.129 -.447 .152 -.938 1.087 -.453 .131 -.932 .910 -.288 .215 -.787 .866 -.291 .193 -.778 .821 -.294 .171 -.770 .776 -.296 .149 -.761 .792 -.229 ,223 -.707 .746 -.230 .201 -.696 .700 -.231 .179 -,685 .653 -.231 .157 -.674 ,728 --.196 ,228 --.659 .681 -.196 .206 -.646 .634 -.196 .184 -.633 .687 -.175 .231 -.626

1.224 -.508 .207 -1.160 1.194 -.510 .192 -1.155 1.164 -.513 .177 -1.149 1.135 -.515 ,163 -1.144 .935 -.343 .222 -.939 .903 -.343 .207 -.931 .871 -.342 .192 -.923 .839 -.342 .177 -,914 .823 -.278 .228 -.834 .790 -.276 .213 -.824 .757 --.275 .198 --.814 .724 -.273 .183 -.803 .763 -.242 .231 -.771 .729 -.240 ,216 -.760 .696 -.237 ,201 -.748

1.323 -.634 .213 -1.450 1.301 -.634 .203 -1.446 1.280 -.634 ,193 -1.441 1.259 -.634 .182 -1.437

328 ~ December 1994

TABLE 1--Continued

Ex/Ey Vxy G~/Ey ~ qa c2i cea c23 c3~ c33

Ey/Ex v~ G~y/Ex caa C31 C23 022 ~1 ~3 ~1

16 .00 .80 -,187 .126 -.062 1.019 -.437 .225 -1.147 16 .25 .80 -.186 .113 -.071 .997 -.435 .214 -1.140 16 .50 .80 -.184 .099 -.079 .974 -.433 .204 -1.133 16 .75 .80 -.182 .085 -.088 .952 -,432 .193 -1.126 16 .00 1.20 -.181 .138 -.044 .903 -,360 .228 -1.006 16 .25 1.20 -.179 .124 -.052 .880 -.357 .218 -.997 16 .50 1.20 -.177 .109 -.060 .856 -,354 ,207 -.988 16 .75 1.20 -.175 .095 -.068 .833 -,351 .197 -.979 16 ,00 1.60 -.177 .145 -.034 .840 -,318 .230 -.923 16 .25 1.60 -.175 .131 -.042 .816 -,315 .220 -.913 16 .50 1.60 -.173 .116 -.049 .792 -,311 .209 -.902 16 .75 1.60 -.171 .102 -.057 .768 -.308 .198 -,892 16 .00 2.00 -.175 .150 -.027 .801 -.292 .231 -.867

ASTM rosette with gage 1 aligned along the Ex principal elastic direction, and with hole radius ra = 0.464 rm. The first row of column headings applies when Ex > Ey. The second row of column headings applies when Ey > Ex.

(23) clearly shows that these relationships are not obeyed in this orthotropic example.

Equation (23) also illustrates a fundamental prob- lem when trying to evaluate the constants A, B and C for eq (1). I rA and B were determined from a uniaxial tension test where crx = 1, O'y = "r,:y = 0, the two con- stants would be evaluated as A = (q l + c31)/2 = - . 0 3 2 , and B = (Cll - c31)/2 = - . 2 6 0 . However, quite different results would be obtained from a uni- axial tension test along the perpendicular axis. In that case, ~ry = 1, o'x = "r~y = 0, and the two constants would be evaluated as A = (c33 -~- c 1 3 ) / 2 = - . 2 3 8 , and B = (c33 - c13)/2 = - . 421 . The difference be- tween these two sets of 'constants' is substantial.

Experimental Measurements

A series of experimental measurements was under- taken to test the effectiveness of the proposed resid- ual-stress measurement method. The test procedure was adapted from a combination of the orthotropic material property measurement method described by Lineback 18 and the hole-drilling calibration method described by Rendler and Vigness.1 The test method measures the material and hole-drilling constants in- dependent of any residual stresses that may be pres- ent. The test material was a graphite-epoxy laminate, 3.5-ram thick, composed of 24 unidirectional graphite fiber layers in an epoxy matrix. The lay-up was sym- metrical, with two layers in the 0-deg direction, six layers each in the two 45-deg directions, and 10 lay- ers in the 90-deg direction. Three tensile test speci- mens, 250 x 38.1 m m in size and oriented at 0, 45 and 90 deg, were cut from a 300-mm square panel, as shown in Fig. 4.

Measurements Group 125-RE residual-stress ro- settes were attached to one surface of each tensile specimen. Additional single gages were attached to opposite sides of each specimen, as shown in Fig. 4,

90~ hole-drilling / / ~ rosette / 4 0 ,2

(one side) / / \ / _ /

\ / /~ ing legages

/ both sides) l ~ / /I

" - 0 o

Fig. 4--Test specimens cut from a sheet of graphite- epoxy laminate. The rosettes are attached on one side only. The single gages are attached on both sides

to check for the possible presence of bending strains. When making strain measurements, corresponding strain gages in pairs of specimens were connected to- gether in half-bridge circuits. During measurements, one specimen was loaded while the other was kept undisturbed. This procedure minimized any thermal strain effects. Thermal drift of the strain gages was a potential problem due to the low thermal conductivity of the laminate.

Each of the three test specimens was secured using wedge grips in a lO-kN capacity Instron tensile test-

Experimental Mechanics o 329

ing machine. Strain readings were taken from all seven gages on each specimen as the axial load was incre- mentally increased and then decreased. Fine sand- paper was inserted within the wedge grips to improve the grip on the smooth-faced specimens. The posi- tions of the pieces of sandpaper were adjusted so that the strain readings on the four single gages were all equal within two percent.

Two series of measurements were made with each specimen, one before and one after drilling a 4.8-mm diameter hole at the geometric center of the hole-drill- ing rosette. From the 0-deg specimen, strain readings were taken after 1.5-kN increments in load up to 9.0 kN. For the 45-deg specimen, the readings were taken in 1.0-kN increments up to 6.0 kN, and for the 90- deg specimen, in 0.8-kN increments up to 4.8 kN. The measured strains were in the range - 5 0 0 txe to +1200 Ixe, with a scatter (deviation from linearity) less than 10 ixe. To maximize measurement accuracy, the gradients of the graphs of measured strain versus applied load were used to determine the elastic strain responses (compliance), i.e., the strain per unit ap- plied stress.

Column 3 of Table 2 lists the strain responses of the three test specimens, measured before hole drill- ing. These measurements, together with the equations presented by Lineback, TM were used to determine the orthotropic elastic properties of the laminate. Table 3 lists the results together with the values calculated from laminate theory based on the properties and allgn- ments of the 24 component layers. Two different sets of experimental data can be used for calculating the elastic properties of the laminate. One data set con- sists of the strain readings from the 0-deg and 45-deg specimens, and the other data set from the 45-deg and 90-deg specimens. The results from the two data sets are up to 10-percent different. This discrepancy can be seen directly in the measured strain responses in column 3 of Table 2. The elastic symmetry require- ment in eq (1) implies that the transverse responses (gages 3 and 1) of the 0-deg and 90-deg specimens should be identical. However, the measured values, - 7.1 and - 6 . 5 ~x~/MPa, are about 10-percent different.

The 10-percent discrepancy in elastic property mea- surements from the various specimens is much greater than the likely experimental error. Variation in the ac- tual material properties of the test specimens is sus- pected to be the cause. Figure 4 shows that the three specimens were cut from different parts of the orig- inal square sample. Of necessity, the 0-deg and 90- deg specimens had to be cut close to the sample edges. Unfortunately, these are areas where material non- uniformity is most likely.

In Table 2, column 5 lists the differences between the strain responses measured before and after hole drilling (columns 3 and 4). These values correspond to the strain reliefs that would be measured during a hole-drilling residual-stress measurement. Column 6 lists the theoretically expected strain responses cal- culated from eq (22) using the elastic properties listed in Table 3. To minimize the effect of material prop- erty variation, the elastic properties measured from the 0-deg and 45-deg specimens were used for the 0- deg specimen. The measurements from the 45-deg and 90-deg specimens were used for the 90-deg specimen, and the average of both sets was used for the 45-deg specimen. The given laminate had a very high shear stiffness because of the large number of 45-deg lay- ers. As a result, the shear modulus slightly exceeds the limiting value for a valid mathematical solu- tion (e.g. G~y = 17.6 GPa for the 45-deg specimen when K = 1). The results in column 6 have been extrapolated to reach the actual 18.6 GPa shear modulus.

The measured and calculated strain-response values listed in columns 5 and 6 of Table 2 are almost all within 1 txe/MPa. This agreement is certainly close enough to be convincing. However, some of the dif- ferences are larger than might be hoped. Several rea- sons for these differences were identified. One likely error source is the variation in the material properties of the three test specimens. The 0.6 Ixe/MPa differ- ence in the e3 values for the 0-deg and 90-deg spec- imens, which is mostly due to material property vari- ation, is comparable to the differences up to 1.0 txe/ MPa between columns 5 and 6.

TABLE 2--MEASURED AND CALCULATED STRAIN RESPONSES OF THE GRAPHITE-EPOXY LAMINATE USED IN THIS STUDY

Strain Response, ~e/MPa

Strain Measured Measured Difference Calculated Specimen Gage -no hole with hole due to hole due to hole

0 deg 1 15.3 7.0 -8.3 -7.9 0 deg 2 4.1 1.5 -2.6 -2.7 0 deg 3 -7.1 -4.0 3.1 4.0

45 deg 1 3.9 1.4 -2.5 -2.3 45 deg 2 22.1 10.5 -11.6 -9.9 45 deg 3 13.8 8.4 -5.4 -4.4

90 deg 1 -6.5 -4.1 2.4 2.8 90 deg 2 14.2 9.2 -5.0 -4.0 90 deg 3 32.7 21.4 - 11.3 - 11.7

330 �9 December 1994

TABLE 3--ELASTIC CONSTANTS OF THE GRAPHITE-EPOXY LAMINATE USED IN THIS STUDY i

Elastic Modulus, GPa

Ex Ey Gxy Vxy

Calculated from Laminate Theory 73.6 34.5 21.3 0.45

Measured from 0-deg and 45-deg specimens 72.0 30.6 18.6 0.46

Measured from 45-deg and 90-deg specimens 65.2 28.9 18.6 0.46

Average measurement from all specimens 68.6 29.7 18.6 0.46

The strain responses in column 5 of Table 2 are particularly sensitive to measurement errors because they derive from the difference between the much larger values in columns 3 and 4. In most cases, the values in column 5 are less than half of those in column 3. Thus, any errors in columns 3 and 4 have magnified relative effects on column 5.

Rosette angular misalignment was found to be an additional source of error in Table 2. For the 0-deg and 90-deg specimens before hole drilling, the read- ings from gage 2 are expected to be the average of the readings from gages 1 and 3. This is true for the 0-deg specimen but not for the 90-deg specimen. The discrepancy was traced to a 1.5-deg error in the ro- sette alignment. This alignment error was taken into account when computing the values in column 6.

Comparisons with Other Published Data

Prasad et al . 8 did a series of hole-drilling measure- ments, similar to those reported here, using a graph- ite-polyimide laminate. They also made calculations of the strain responses expected during their experi- mental measurements. Their mathematical method differs from that presented here, notably by their choice of working in terms of strains. To simplify their strain- based computations, Prasad et al. approximated each strain gage in the hole-drilling rosette as being con- centrated at a point. In the present study, displace- ment-based calculations were chosen because this ap- proach lends itself very conveniently to computin~ the response of practical strain gages of finite area.

Table 4 lists the measured strain responses of Pra- sad et al . and the corresponding theoretical values, calculated in three different ways. Column 3 lists the measured strain responses, and column 4 lists Pra- sad's corresponding calculated values. Column 5 lists the strain responses calculated from eqs (8)-(21) as- suming a strain gage concentrated at a point. These results are mostly the same as those of Prasad et al . The differences between columns 4 and 5 for the 45- deg specimen are due to a suspected numerical error by Prasad et a l . in calculating shear strain response. In their study, they worked in terms of constants A, B and C in eq (1). Their value of C is smaller than

expected from c~z in eqs (2) and (22). A numerical error is suspected because their calculated C values do not approach 2B in the two nearly isotropic ex- amples that they examine. The results in columns 5 and 6 are believed to be reliable because the calcu- lation method gives C = 2B for an isotropic material and also gives displacement fields identical to those reported by Schimke e t al . 11

Column 6 of Table 4 lists the predicted strain re- sponse when the finite areas of the hole-drilling ro- sette gages are taken into account. The calculation method is the same as used for column 6 of Table 2. In some cases, the difference between the 'point' and finite-area gage calculations are quite significant. In all cases, the relieved strain responses listed in col- umn 6 of Table 4 correspond more closely with the measured values in column 3 than do the 'point' gage values in columns 4 and 5. These results demonstrate the significance of using the finite-area strain-gage calculation.

Residual-stress Measurement Accuracy

The discussion so far has focused on how well the proposed calculation method predicts the measured strain responses in calibration tests. This assessment provides an important measure of the theoretical method. However, in practice, the question of interest is "What level of accuracy can be expected when ma- trix eq (22) and theoretical compliance values are used to evaluate residual stresses from experimental strain measurements?" A related question is "What is the consequence of using the trigonometric assumption, eq (1), rather than the matrix eq (22)?" These ques- tions are examined here.

Table 5 lists the residual stresses that would be cal- culated for the five tensile specimens whose measured strain data are reported in Tables 2 and 4. Each spec- imen supports a purely longitudinal stress, which for simplicity of comparison has been normalized to 1.00 MPa. The table lists the stresses that would be cal- culated from the measured strains in column 5 of Ta- ble 2 and column 3 of Table 4, using the finite-area compliance values in columns 6. The actual applied stresses are shown in parentheses. In general, the cal-

Experimental Mechanics ~ 331

TABLE 4--MEASURED AND CALCULATED STRAIN RESPONSES OF THE GRAPHITE-POLYIMIDE LAMINATE STUDIED BY PRASAD ET AL. 8

Strain Response, ixe/MPa

Strain Strain Prasad 8 'Point' Finite Specimen Gage due to Hole Calculated Gage Area Gage Area

Tension a 2.76 3.33 3.32 3.04 Tension b - 1.28 - - 0.37 -0.36 Tension c -4.14 -5.17 -5.16 -3.76

Shear a -19.1 -15.62 -16.78 -16.93 Shear b 0.8 0.00 0.00 0.00 Shear c 18.7 15.62 16.78 16.93

TABLE 5--COMPUTED STRESSES FOR A 1.00 MPa APPLIED LONGITUDINAL STRESS APPLIED STRESSES IN PARENTHESES)

Computed Stress, MPa

Specimen cr,~ "rxy %, ~rr,~ ~rm,n s

0 deg 1.09 .04 .11 1.09 .11 3 deg (1.00) (.00) (.00) (1.00) (.00) (0 deg)

45 deg .57 -.58 .61 1.17 .01 46 deg (.50) (-.50) (.50) (1.00) (.00) (45 deg)

90 deg .06 -.07 1.00 1.00 .05 96 deg (.00) (.00) (1.00) (1.00) (.00) (90 deg)

Tension 1.12 -.04 .03 1.12 .03 -2 deg aef. 8 (1.00) (.00) (.00) (1.00) (.00) (0 deg)

Shear - . 19 1.12 .04 1.04 - 1.20 48 deg aef. 8 (.00) (1.00) (.00) (1.00) (-1.00) (45 deg)

culated stresses are somewhat higher than the actually applied stresses, up to 20 percent higher in extreme cases. This over-estimation of the actual stresses is believed to be a consequence of the laminar structure of the test material.

When using eqs (8)-(21) to calculate the strain re- sponses in Tables 2 and 4, the assumption is made that the graphite-epoxy laminate is a homogeneous continuum. However, in reality, the test material con- sists of 24 discrete layers, each about 0.15-mm thick. In such a case, the continuum assumption is reason- able only for macroscopic features that extend over regions significantly larger than one layer thickness. For the tests done in this study, the hole is 4.8 mm in diameter, which is much larger than the 0.15-mm layer thickness. Thus, the hole can be expected to be a 'macroscopic' feature. However, St. Venant's prin- ciple suggests that the laminar structure is likely to disturb the continuum assumption within about one layer thickness from the hole boundary. Thus, the ef- fective hole diameter could be expected to be slightly larger than the actual hole diameter. If the enlarge- ment in effective hole radius is one layer thickness, then the expected increase in strain response is about 12 percent. (The strain response is approximately pro- portional to the square of the hole diameter). This per-

centage increase corresponds well with the results in Table 5.

Table 6 compares the calculated stresses for the 0- deg and 90-deg specimens used in this study, deter- mined using matrix eq (22) and also using the trigo- nometric assumption, eq (1). Column 2 of Table 6 lists the A, B and C values that would be used for each specimen, determined from longitudinal tension calibration tests. Column 3 lists the corresponding A, B and C values from hypothetical transverse tension calibration tests. The values in columns 2 and 3 sig- nificantly differ. The longitudinal A, B and C values for one specimen should normally equal the trans- verse values for the other specimen. They are not ex- actly equal here because of the 10-percent difference in elastic properties of the two specimens.

Column 4 of Table 6 lists the stresses calculated using eq (22) for a nominal longitudinal tension of 1.00 MPa. These values are the same as those listed in Table 5. Column 5 lists the stresses calculated us- ing eq (2) with the 'longitudinal' A, B and C values from column 2. These calculated stresses are similar to those from eq (22) in column 4. Column 6 lists the stresses calculated using eq (2) with the 'transverse' A, B and C values from column 3. These calculated stresses greatly differ from the expected values, and

332 �9 December 1994

TABLE 6--STRESSES COMPUTED USING EQS (2) AND (22) FOR A 1.00-MPa APPLIED LONGITUDINAL STRESS

ABC, I~E/MPa Computed Stress, MPa

eq (2) eq (2) Specimen Longitudinal Transverse eq (22) Long. ABC Trans. ABC

A = -1 .9 -4 .8 trx = 1.09 1.15 .63 0 deg B = -6 .0 -8 .0 "rxy = .04 .00 .00

C = -13.5 -13.5 %, = .11 .20 - .08

A = -4 .4 -1 .7 Crx = .06 .03 .65 90 deg B = -7 .3 -5 .3 "rxy = - .07 - .04 - .04

C - - - 1 2 . 6 -12.6 %, = 1.00 .98 1.95

are seriously in error. The results in Table 6 clearly demonstrate that the trigonometric assumption, eq (1), gives useful results only when the calibration stresses are similar to the stresses to be measured. Serious er- rors develop when the calibration stresses differ sig- nificantly from the stresses to be measured.

Conclusions

(1) When used with eq (22), the hole-drilling method can successfully measure uniform residual stresses in orthotropic materials. In calibration tests, stress-mea- surement errors were in the 10-20-percent range. This error range is expected to be typical of hole-drilling measurements in orthotropic laminates. The likely major error sources are the laminar structure of most orthotropic materials, uncertainty of local elastic properties and angular misalignment of the strain-gage rosette relative to the material principal elastic direc- tions.

(2) Equation (22) provides theoretically exact re- sidual-stress solutions for a wide range of linear-elas- tic orthotropic materials. The compliances Cll - c33 required for eq (22) can be determined using the plane- stress solution, eqs (8)-(21), for the displacement field around a hole in a stressed orthotropic plate.

(3) The compliances Cll - c 3 3 required for eq (22) are more accurately calculated when the finite areas of the hOle-drilling strain gages are taken into account using the method described in Ref. 15. Table 1 lists computed Cll - c33 values for a wide range of material properties.

(4) The relieved strain versus angle relationship at any radius beyond the boundary of a hole in a stressed orthotropic material does not have a simple trigono- metric form. An existing stress calculation method based on the trigonometric assumption in eq (1) is not valid for hole-drilling residual-stress calculations with orthotropic materials.

Acknowledgments

This work was supported by a research grant from the Natural Sciences and Engineering Research Coun- cil of Canada (NSERC). Drs. Sheldon Green, Anoush Poursartip and Bruce Lehmann kindly reviewed the manuscript. Mr. Alan Russell of the Defense Re- search Establishment Pacific generously provided the laminate sample used in this study.

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