Rs In Orthotropic Using Hole Drilling Schajer

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 measure residual stresses in an orthotropic material. An existing stress-calculation method adapted from the isotropic case is shown not to be valid for orthotropic materials. A new stress-calculation method is described, based on the analytical solution for the displacement field around a hole in a stressed orthotropic plate. The validity of this method is assessed through a series of experimental measurements. A table of elastic compliances is provided for practical residual-stress measurements in a 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 symmetry) 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 8r 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)]

v~, vy., = x - y Poisson's ratios

~rx, ~y = x - y Cartesian normal stresses (]'max = maximum (most tensile) principal stress = minimum (most compressive) principal (rmi" stress = x - y Cartesian shear stress "r~y ~b = angle measured counterclockwise from the x direction to the direction of O'max t~l, +2 = 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 modem 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.

=

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

324 ~ December 1994

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.

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

ty

O". . . .

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

• V((O" x

__

(3)

2 0",)/2) 2 -'1- "rxy

and [ 2"rxy ]

1

6 = - arctan 2

-

L~x -

(4)

%J

X 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 material, Hooke's Law generalizes to Fig. 1 - - A S T M strain-gage rosette used for hole-drilling measurements 4

=

x/E. -

~, = % / E ,

%V,x/E,

- ~xv~/E.

(5)

% = ~/6~

When making residual-stress measurements, a circular hole is drilled at the geometrical center of the rosette to a depth slightly greater than the hole diameter. This hole locally relieves the stresses in the surrounding material, and the associated strain reliefs are measured by the three strain gages. For an isotropic 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 (1)

+ C %y sin 20

where the symbols are defined in 'List of Symbols' above. In this study, it is assumed that the residual stresses %, % and "rxydo 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 constants can either be determined experimentally' or numericallyfl Equation (1) can be rewritten in matrix form to relate the three measured strains gl, 1~2and ~3 in Fig. 1 to the Cartesian stresses ox, % and "rxy.

Only four of the five elastic constants are independent because of the elastic symmetry relationship (6)

V~y/E~ = Vy~/Ey

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 familiar forms. An interesting special case occurs when the shear modulus of an orthoU-opic material happens to be related to the other elastic constants as follows. 1

G,~

--

l+v~

_

_

Ex

-]-

l +- v y ~

-

(7)

Ey

In this particular case, the shear modulus is the same in all directions. The material has isotropic shear behavior, but orthotropic axial behavior. An approximately 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) principal axial moduli. These two types of orthotropy are considered in subsequent sections.

Orthotropic Hole-drilling Solution A - B

-

A 0

A +

Txy

LO-yj

=

E 2

133

(2)

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

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

Experimental Mechanics

.

325

the elastic symmetry directions of the orthotropic material. 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 al. 6'7 and subsequently by Prasad e t al. 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 residual stresses and the hole-drilling relieved strains9 It is assumed that the orthotropic material under study has a sufficiently fine microscopic structure that it can be approximated as a homogeneous continuum9 Following the method described by Smith,~4 the relieved displacement field around a hole in a stressed orthotropic plate (plane stress case) for x 2 + y2 _ r ] can be shown to be oL2m2 + 1)xy /2= m(a

-

~) Ex(1 -

9[Y~(1 +

am)

f3m) "rxy - X ~ ( %

-

13mCry)]

o t m ) "r~ - X2(O'x -

cxm~ry)]

2m2 + Pxy

+ m(13 -

oO E x ( 1 -

9[Y2(1 +

f3m)

(8)

1 + (x2m2 l)yx V

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

9 [XI(1 + [3m)Txy+

Yl(ffx-

~m%)]

X, = x - W, cos ~,

(18)

X2 = x - W2 cos ~2

(19)

Y , = oLmy -

W1

sin

}12 = 13my -

W2

sin q*2

(21)

The above solution is valid only for K > 1. This requirement puts a maximum limitation on the allowable size of the shear modulus Gxy relative to the other elastic moduli. Except for composites that are specially designed for high shear stiffness, most orthotropic 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 field9 The curves are calculated 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, schematic vertical scales are used in the two figures9 All the curves are scaled to a uniform size and are displaced 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

1 + [32rn2 V~y

+

f3m 2 (f3 -

o0 E y ( 1

-

13m)

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

etm%)]

(9) A

where

ex = ~--~ A

A

(I0)

m = 4VEx/Ey K =

(20)

lt,ll1

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

(Ii)

0 (b

o~ = ~V/K + ~v/(K2 -- I)

(12)

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

(13)

W 1 =

4V(x2

_

r2 _

o~2m2(y 2 -

ra2)) 2 -[-

Q: "1 Ey = 1

(2~mxy) 2

04)

-2 0~

W 2

= 4V(X2 -- r 2a -- 132mZ(y: r2))2 +

(213mxy)2

(15) t~, = arctan

[2otmxy/(x

~J2 = arctan

[213mxy/(x 2 -

2 -

~ - e~2m2(y 2 -

r2

-

rZ~))]/2 (16)

132m2(y2 = r]))]/2

(17)

326 9 December 1994

i ,

i

90 o

~xy i

= 0.3

Gxy

i

i

1800 Angle,

= isotropic i

270 ~

T

3600

e

Fig. 2 - - A n g u l a r variation of relieved strain in materials of varying degrees of axial orthotropy. ASTM straingage geometry4 with hole radius ra = 0.464 rm. All curves are scaled so they coincide at e = 0 deg and e = 90 deg

.~2

'

7

03

o

Ex= 1

0

-2 0~

900

Ey=l

Vxy=0.3

1800 Angle,

2700

3600

O

Fig. 3--Angular variation of relieved strain in materials of varying degrees of shear orthotropy. ASTM straingage geometry4 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 moduli, 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 observations occurs during bore-hole measurements of rock stresses. 12'13 For such measurements, eq (1) always applies exactly, even for highly orthotropic materials. 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 measurements are made beyond the hole boundary.

Numerical Results Assuming only linear elasticity, the matrix approach 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 j xEll IC2, C22C231

E~/~Ey Lc31 c32 c33 J

O-y

~3

where the elastic compliances c11-c33 are in general not related in any way to trigonometric-based constants such as A, B or C. The factor 1/V"-E~Ey is included so that the compliances c11-c33 are dimensionless constants. For the case of interest here, where the x and y directions of the rosette coincide with the principal 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 rosette geometry. Hole depth is also an important factor. Practical experience with isotropic materials 1-3 and finite-element calculations 5 show that the elastic compliances 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 material, 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 using 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 different from the specified values, the compliances c11c33 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 trigonometric values expected from eq (2). For example, when Ex = 2, Ey = 0.5, V~y = 0, G~y = 0.4, the compliance matrix is

Cll C12 C13~ Cal c22 c23| =

C31 C32 C33J (22)

[-.291 -.073 ,228

0 .728 0

.1831 -.196] - . 6 5 9 ] (23)

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

Experimental Mechanics 9 327

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

Ex/ Ey E~/Ex

Vxy ~'yx

Gxy/ Ey G~y/E~

c,1 c33

c,3 c31

c2, c23

c22 c22

c23 c2~

c3, c~3

c33 c~1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 16 16 16 16

.00 .25 .50 .75 .00 .25 .50 .75 .00 .25 ,50 .00 .25 .00 .00 .25 ,50 .75 .00 .25 .50 .75 .00 .25 .50 .75 .00 .25 .00 .25 .50 .75 .00 ,25 .50 .75 .00 .25 .50 .75 .00 .25 .50 .00 .00 .25 .50 ,75 .00 .25 .50 .75 .00 .25 .50 .75 .00 .25 .50 .00 .25 .50 .75

.10 .10 .10 .10 .20 .20 .20 .20 .30 .30 .30 .40 .40 .50 .15 .15 .15 ,15 .30 .30 .30 ,30 .45 .45 .45 .45 .60 .60 ,20 .20 .20 .20 .40 .40 .40 .40 .60 .60 .60 .60 ,80 .80 .80 1.00 .30 .30 .30 .30 .60 .60 .60 .60 .90 .90 .90 .90 1.20 1.20 1.20 .40 ,40 .40 .40

-.591 -.583 -.575 -.568 -.514 -.503 -,491 -.478 -.474 -.460 -.445 -.450 -.433 -.433 -.453 -,448 -.443 -.438 -.403 -.396 -.389 -.382 -.377 -.369 -.360 -.350 -,361 -.351 -.350 -.347 -.344 -.341 -.318 -.314 -.310 -.306 -.301 -.296 -.291 -.286 -.291 -.285 -.279 -.283 -,263 -.262 -.260 -.258 -.244 -.242 -.239 -.236 -.234 -.231 -.228 -.224 -,227 -.224 -.220 -.199 -.198 -.196 -.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 1.097 1.013 .930 .877 .788 .699 .609 .757 .665 ,572 ,692 .598 .650 1.156 1.097 1.038 .978 ,865 .802 .739 .675 .751 .686 .620 .554 .689 .623 1,212 1.171 1.129 1.087 .910 .866 .821 .776 .792 .746 .700 .653 ,728 .681 .634 .687 1.224 1.194 1.164 1.135 .935 .903 .871 .839 .823 .790 .757 .724 .763 .729 .696 1.323 1.301 1.280 1.259

-,291 -,314 -,336 -,357 -,188 -,205 -,221 -,235 -.146 -.160 -.172 -.123 -.135 -.108 -.345 -~357 -.369 -.380 -.225 -.233 --.240 -.247 -.177 -.182 -.187 -.191 -.150 -.154 -.435 -.441 -.447 -.453 -.288 -.291 -.294 -.296 -.229 -.230 -.231 -.231 --.196 -.196 -.196 -.175 -.508 -.510 -.513 -.515 -.343 -.343 -.342 -.342 -.278 -.276 --.275 -.273 -.242 -.240 -.237 -.634 -.634 -.634 -.634

.169 .123 .076 .030 .193 .146 .098 .049 .205 .157 .109 .213 .164 ,218 .185 .153 .122 .090 .207 .175 .143 .110 .217 .185 .152 .119 .223 .191 .195 .174 .152 .131 .215 .193 .171 .149 ,223 .201 .179 .157 ,228 .206 .184 .231 .207 .192 .177 ,163 .222 .207 .192 .177 .228 .213 .198 .183 .231 ,216 ,201 .213 .203 ,193 .182

-.591 -.583 -.575 -.568 -.514 -.503 --.491 -.478 -.474 -.460 -.445 -.450 -.433 -.433 -,743 -.737 -.730 -.723 -.631 -.621 --.611 -.600 -.576 -.563 -.550 -.536 -.541 -.527 -.949 -.944 -.938 -.932 -.787 -.778 -.770 -.761 -.707 -.696 -,685 -.674 --.659 -.646 -.633 -.626 -1.160 -1.155 -1.149 -1.144 -.939 -.931 -.923 -,914 -.834 -.824 --.814 -.803 -.771 -.760 -.748 -1.450 -1.446 -1.441 -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 16 16 16 16 16 16 16 16 16 16 16 16

.00 .25 .50 .75 .00 .25 .50 .75 ,00 .25 .50 .75 .00

.80 .80 .80 .80 1.20 1.20 1.20 1.20 1.60 1.60 1.60 1.60 2.00

-,187 -.186 -.184 -.182 -.181 -.179 -.177 -.175 -.177 -.175 -.173 -.171 -.175

.126 .113 .099 .085 .138 .124 .109 .095 .145 .131 .116 .102 .150

-.062 -.071 -.079 -.088 -.044 -.052 -.060 -.068 -.034 -.042 -.049 -.057 -.027

1.019 .997 .974 .952 .903 .880 .856 .833 .840 .816 .792 .768 .801

-.437 -.435 -.433 -,432 -,360 -.357 -,354 -,351 -,318 -,315 -,311 -.308 -.292

.225 .214 .204 .193 .228 .218 ,207 .197 .230 .220 .209 .198 .231

-1.147 -1.140 -1.133 -1.126 -1.006 -.997 -.988 -.979 -.923 -.913 -.902 -,892 -.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 problem when trying to evaluate the constants A, B and C for eq (1). I r A and B were determined from a uniaxial tension test where crx = 1, O'y = "r,:y= 0, the two constants 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 uniaxial 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 = - . 4 2 1 . The difference between these two sets of 'constants' is substantial.

Experimental Measurements A series of experimental measurements was undertaken to test the effectiveness of the proposed residual-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 independent of any residual stresses that may be present. 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 symmetrical, with two layers in the 0-deg direction, six layers each in the two 45-deg directions, and 10 layers in the 90-deg direction. Three tensile test specimens, 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 rosettes 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 (one side)

/

\/_

/

,2

40

/

/

\

//~inglegages / bothsides) l

~

/

/I

" -

0o

Fig. 4--Test specimens cut from a sheet of graphiteepoxy 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 together 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 incrementally increased and then decreased. Fine sandpaper was inserted within the wedge grips to improve the grip on the smooth-faced specimens. The positions 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-drilling 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 90deg 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 applied stress. Column 3 of Table 2 lists the strain responses of the three test specimens, measured before hole drilling. 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 allgnments 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 consists 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 requirement 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 measurements from the various specimens is much greater than the likely experimental error. Variation in the actual material properties of the test specimens is suspected to be the cause. Figure 4 shows that the three specimens were cut from different parts of the original square sample. Of necessity, the 0-deg and 90deg specimens had to be cut close to the sample edges. Unfortunately, these are areas where material nonuniformity 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 calculated from eq (22) using the elastic properties listed in Table 3. To minimize the effect of material property variation, the elastic properties measured from the 0-deg and 45-deg specimens were used for the 0deg 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 layers. As a result, the shear modulus slightly exceeds the limiting value for a valid mathematical solution (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 differences are larger than might be hoped. Several reasons 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 difference in the e3 values for the 0-deg and 90-deg specimens, which is mostly due to material property variation, 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 Measured Difference with hole due to hole

Strain Gage

Measured -no hole

0 deg 0 deg 0 deg

1 2 3

15.3 4.1 -7.1

7.0 1.5 -4.0

-8.3 -2.6 3.1

-7.9 -2.7 4.0

45 deg 45 deg 45 deg

1 2 3

3.9 22.1 13.8

1.4 10.5 8.4

-2.5 -11.6 -5.4

-2.3 -9.9 -4.4

90 deg 90 deg 90 deg

1 2 3

-6.5 14.2 32.7

-4.1 9.2 21.4

2.4 -5.0 - 11.3

2.8 -4.0 - 11.7

Specimen

330 9 December 1994

Calculated due to hole

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 readings 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 rosette alignment. This alignment error was taken into account when computing the values in column 6.

Comparisons with Other Published Data Prasad e t al. 8 did a series of hole-drilling measurements, similar to those reported here, using a graphite-polyimide laminate. They also made calculations of the strain responses expected during their experimental measurements. Their mathematical method differs from that presented here, notably by their choice of working in terms of strains. To simplify their strainbased computations, Prasad e t al. approximated each strain gage in the hole-drilling rosette as being concentrated at a point. In the present study, displacement-based calculations were chosen because this approach lends itself very conveniently to computin~ the response of practical strain gages of finite area. Table 4 lists the measured strain responses of Prasad e t al. and the corresponding theoretical values, calculated in three different ways. Column 3 lists the measured strain responses, and column 4 lists Prasad's corresponding calculated values. Column 5 lists the strain responses calculated from eqs (8)-(21) assuming 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 45deg specimen are due to a suspected numerical error by Prasad e t al. 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 examples that they examine. The results in columns 5 and 6 are believed to be reliable because the calculation 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 response when the finite areas of the hole-drilling rosette 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 column 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 matrix 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 questions are examined here. Table 5 lists the residual stresses that would be calculated for the five tensile specimens whose measured strain data are reported in Tables 2 and 4. Each specimen 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 calculated from the measured strains in column 5 of Table 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

Specimen

Strain Gage

Tension Tension Tension

a b c

Shear Shear Shear

a b c

Strain Response, ixe/MPa Prasad8 'Point' Calculated Gage Area

Strain due to Hole 2.76 - 1.28 -4.14 -19.1 0.8 18.7

Finite Gage Area

3.33 --5.17

3.32 0.37 -5.16

3.04 -0.36 -3.76

-15.62 0.00 15.62

-16.78 0.00 16.78

-16.93 0.00 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,~

0 deg

1.09 (1.00)

.04 (.00)

.11 (.00)

1.09 (1.00)

.11 (.00)

3 deg (0 deg)

45 deg

.57 (.50)

-.58 (-.50)

.61 (.50)

1.17 (1.00)

.01 (.00)

46 deg (45 deg)

90 deg

.06 (.00)

-.07 (.00)

1.00 (1.00)

1.00 (1.00)

.05 (.00)

96 deg (90 deg)

Tension aef. 8

1.12 (1.00)

-.04 (.00)

.03 (.00)

1.12 (1.00)

.03 (.00)

- 2 deg (0 deg)

Shear aef. 8

- . 19 (.00)

1.12 (1.00)

.04 (.00)

1.04 (1.00)

- 1.20 (-1.00)

48 deg (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 responses in Tables 2 and 4, the assumption is made that the graphite-epoxy laminate is a homogeneous continuum. However, in reality, the test material consists of 24 discrete layers, each about 0.15-mm thick. In such a case, the continuum assumption is reasonable 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 principle suggests that the laminar structure is likely to disturb the continuum assumption within about one layer thickness from the hole boundary. Thus, the effective hole diameter could be expected to be slightly larger than the actual hole diameter. If the enlargement in effective hole radius is one layer thickness, then the expected increase in strain response is about 12 percent. (The strain response is approximately proportional to the square of the hole diameter). This per-

332 9 December 1994

~rm,n

s

centage increase corresponds well with the results in Table 5. Table 6 compares the calculated stresses for the 0deg and 90-deg specimens used in this study, determined using matrix eq (22) and also using the trigonometric 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 significantly differ. The longitudinal A, B and C values for one specimen should normally equal the transverse values for the other specimen. They are not exactly 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 using 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

TABLE 6--STRESSES COMPUTED USING EQS (2) AND (22) FOR A 1.00-MPa APPLIED LONGITUDINAL STRESS ABC, I~E/MPa

Computed Stress, MPa

Transverse

eq (22)

eq (2) Long. ABC

eq (2) Trans. ABC

A = -1.9 B = -6.0 C = -13.5

-4.8 -8.0 -13.5

trx = 1.09 "rxy = .04 %, = .11

1.15 .00 .20

.63 .00 -.08

A = -4.4 B = -7.3 C---12.6

-1.7 -5.3 -12.6

Crx = .06 "rxy = - . 0 7 %, = 1.00

.03 -.04 .98

.65 -.04 1.95

Specimen

Longitudinal

0 deg

90 deg

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 errors develop when the calibration stresses differ significantly 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-measurement 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 directions. (2) Equation (22) provides theoretically exact residual-stress solutions for a wide range of linear-elastic orthotropic materials. The compliances Cll - c33 required for eq (22) can be determined using the planestress 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 trigonometric 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 Council of Canada (NSERC). Drs. Sheldon Green, Anoush Poursartip and Bruce Lehmann kindly reviewed the manuscript. Mr. Alan Russell of the Defense Research Establishment Pacific generously provided the laminate sample used in this study.

References 1. Rendler, N.J. and Vigness, I., "Hole-drilling Strain-gage Method of Measuring Residual Stresses," EXPERIMENTAL MECHANICS, 6 (12), 577--586 (1966). 2. Beaney, E.M. "Accurate Measurement of Residual Stress on any Steel Using the Centre Hole Method," Strain, 12 (3), 99-106 (1976). 3. "Measurement of Residual Stresses by the Hole-Drilling Strain-Gage Method," Tech. Note TN-503-4, Measurements Group, Inc. Raleigh, NC (1993). 4. "Determining Residual Stresses by the Hole-Drilling StrainGage Method," ASTM Standard E837-92, Amer. Soc. for Test. and Mat. (1992). 5. Schajer, G.S., "Application of Finite Element Calculations to Residual Stress Measurements," J. Eng. Mat. and Tech., 1 0 3 (2), 157-163 (1981). 6. Bert, C.W. and Thompson, G.L. "A Method for Measuring Planar Residual Stresses in Rectangularly Orthotropic Materials," J. Composite Mat., 2 (2), 244-253 (1968). 7. Lake, B.R., Appl, F.J. and Bert, C.W., "An Investigation of the Hole-drilling Technique for Measuring Planar Residual Stress in Rectangularly Orthotropic Materials," EXPERIMENTAL MECHANICS, 10 (10), 233--239 (1970). 8. Prasad, C.B., Prabhakaran, R. and Thompkins, S., "Determination of Calibration Constants for the Hole-Drilling Residual Stress Measurement Technique Applied to Orthotropic Composites," Composite Structures, Part 1: Theoretical Considerations, 8 (2), 105-118 (1987); Part H: Experimental Evaluations, 8 (3), 165-172 (1987). 9. Perry, C.C., "Data Reduction Algorithms for Strain-Gage Rosette Measurements," Exp. Tech., 13 (5), 13-18 (1989). 10. Hearmon, R.F.S., "An lntroduction to Applied Anisotropic Elasticity," Oxford Univ. Press (1961). 11. Schimke, J., Thomas, K. and Garrison, J., "Approximate Solution of Plane Orthotropic Elasticity Problems," Scholarly (1970). 12. Kawamoto, T., "On the State of Stress and Deformation Around Tunnel in Orthotropic Elastic Ground," Memoirs of the Faculty of Engineering, Kumamoto Univ., Japan, 10 (1), 1-30 (1963). 13. Becker, R.M., "An Anistropic Elastic Solution for Testing Stress Relief Cores," U.S. Bureau of Mines, Report of Investigations 7143 (1968). 14. Smith, C.B., "Effect of Elliptic or Circular Holes on the Stress Distribution in Plates of Wood or Plywood Considered as Orthotropic Materials," USDA Forest Products Lab., Mimeo 1510 (May 1944). 15. Schajer, G.S., "Use of Displacement Data to Calculate Strain Gauge Response in Non-Uniform Strain Fields," Strain, 2 9 (1), 9-13 (1993). 16. Lekhnitskii, S.G., Anisotropic Plates, Gordon and Breach, New York (1968). 17. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover, New York (1965). 18. Lineback, L.D. ESA Notebook, Measurements Group, Inc., Raleigh, NC 3 7-14, (May 1986).

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