Pergamon Engineering Fracture Mechanics Vol. 58, No. I/2, pp. 87-96, 1997 ~', 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain PII: S0013-7944(97)00070-2 0013-7944/97 $17.00 + 0.00

STRESS INTENSITY FACTOR AND COMPLIANCE SOLUTIONS FOR AN ECCENTRICALLY LOADED SINGLE

EDGE C R A C K E D G E O M E T R Y

REJI JOHNf

Materials Directorate, Wright Laboratory (WL/MLLN), Wright-Patterson Air Force Base, Ohio, OH 45433-7817, U.S.A.

Abstract--Eccentrically pin-loaded single edge cracked geometries are an excellent alternative to com- pact tension type geometries for testing some advanced composite materials. This paper discusses the development of stress intensity factor and compliance solutions for an eccentrically pin-loaded single edge cracked geometry, ESE(T). ~© 1997 Elsevier Science Ltd

Keywords---compliance, eccentric load, fracture, fatigue crack growth, single edge crack, stress intensity factor, test geometry.

1. INTRODUCTION

THE SCHEMATIC of an eccentrically pin-loaded single edge cracked geometry, ESE(T) is shown in Fig. 1. In this geometry, P = applied load, CMOD = crack mouth opening displacement, a = crack length, W = width of plate, d = distance of load-line from edge of plate, e = eccen- tricity of load-line = 0 . 5 W - d , B = thickness and H = distance between points of load appli- cation. This geometry was used by John et al. [1] for analyzing thermally induced crack growth in a steel plate with H = 2.9W and d = 0.33W. Recently, Piascik et a/.[2,3] introduced a geome- try identified as the Extended Compact Tension, EC(T). The EC(T) geometry [2, 3] is similar to the ESE(T) geometry shown in Fig. 1 with H = 3 W and d = 0.2 W. Piascik and Willard [4] used the EC(T) geometry for investigating environmental fatigue crack propagation of small surface cracks and long through-the-thickness cracks in aluminum alloys. Underwood et al. [5] used the EC(T) geometry for characterizing the translaminar fracture toughness of carbon/epoxy lami- nates. Richardson and Goree[6] used a geometry similar to ESE(T) for fracture testing of poly- methyl methacrylate (PMMA) to study the size effect on fracture toughness and evaluate a two- parameter fracture model.

John et al.[1] calculated the stress intensity factor, K for the ESE(T) geometry using tension and bending solutions [8] for a pin-loaded single edge cracked geometry for a / W ~ 0 . 3 and d/ W = 0.33. Richardson and Goree [6] used a finite element alternating method [7] to calculate K for the ESE(T) geometry with d/W ranging from 0.13 to 0.5 and H / W from 0.6 to 3.3. Piascik et al. [2, 3] used the boundary force method [9] to develop K and CMOD solutions for the EC(T) geometry with d = 0.2W applicable in the range 0 < a/W < 1. This paper discusses the develop- ment of K and compliance solutions for the ESE(T) geometry with d/W ranging from 0 to 0.5. The solutions obtained during this study were also compared with the results from Piascik et al. [3] for d = 0.2 W.

2. ANALYTICAL PROCEDURE

The proposed procedure is based on the superposition of the results for a single edge cracked geometry subjected to tension and bending loads as shown in Fig. 2. The stress intensity factor the ESE(T) geometry, KI,ESET can be calculated as,

KI,ESET = KI,Tension + KI,Bending. (1)

fUniversity of Dayton Research Institute, Dayton, OH 45469-0128, USA.

87

88 R. JOHN

CMOD-- ! =

a

W

d e W/2

Fig. 1. Schematic of an eccentrically loaded single edge cracked geometry.

K solutions are available [8] for the single edge cracked geometry subjected to tension and bend- ing loads, as given by eqs (2) and (3).

and

KI,Tension -- a T e - a F T (2)

KI,aending = O'BV/-~FB, (3)

where o T and aB are the far-field tension and bending stresses along the edges, respectively, and the equations for FT and FB are given in Appendix (A). The terms FT and FB in Appendix (A) are valid for any a/W and H/W> 3.0 within +0.5% [8]. In eqs (2) and (3) the far-field stresses O" T and o'a are calculated as,

P CrT - - BW (4)

P (~P ~ O'T=BW ~ M=Pe

d e W/2

M=Pe

Fig. 2. Schematic highlighting the principle of superposition used to calculate the stress intensity factor and compliance solutions for the eccentrically loaded single edge cracked geometry.

Eccentrically loaded single edge cracked geometry 89

5.0 ' I ' ' ' I ' ' ' I ' ' ' I ' '

• Boundary Force Method d=0.0 [Piascik et al.]

~ / • Finite Element 4 .0 , ~ [This study]

\ ' , , , 77 °''w a.o \ d:O2W

I-- m

2.0 U_ i

1.0 ~'-~=0.SW, Tension t r , , , I , , , , . . . , , , , i , , i 1

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

Crack Length / Width, aNV

Fig. 3. Normalized stress intensity factors for the eccentrically loaded single edge cracked geometry.

and

6Pe cra - B W 2" (5)

Combining eqs (1)-(5) we obtain KI for the ESE(T) geometry as,

P KI,ESET = ~ X/~-dFEsET, (6)

where

6e F ( 6d'~ F FESE'r = FT + - ~ B = FT + 3 ----~ ) B. (7)

Similar to the solution for KI,ESET, the CMOD for the ESE(T) geometry can be calculated as,

CMODEsET = CMODTension + CMODBending. (8)

The solutions for CMOD are also available [8] for the single edge cracked geometry subjected to tension and bending loads, as given by eqs (9) and (10).

O'wa GT (9) CMODTension = E

2 5 . 0 ' ' I ' ' ' I ' ' ' I ' ' ' I ' '

d=0.0 • Boundary Force Method ~,~// [Piascik et al.]

20.0 " ~ . . . . • Finite Element ~ d~U,lW [This study]

I,u lo.o

5.0 \ d=0.4W Tension

. , . I , , , I , , , t , , , I , ,

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 .0 Crack Length / Width, a/W

Fig. 4. Normalized crack mouth opening displacements for the eccentrically loaded single edge cracked geometry.

90 R. JOHN

and

O'Ba CMODBending ---- ~ - GB, ( lO)

where the equations for GT and G~ are given in Appendix (A). GT and GB in Appendix (A) are valid for any a/W and H/W> 3.0 within +1.0% [8]. Combining eqs (8)-(10), we obtain the CMOD for the ESE(T) geometry as,

where

Pa CMODEsET -- GESET, ( 1 1)

BWE

6e GESET : G T -t- --~ GB

6d = GT + (3 - - -~)GB. (12)

Equations (7) and (12) were used to generate FESET and GESET solutions for the ESE(T) geome- try as discussed later.

Limited finite element analysis was also conducted during this study to verify the results from the above procedure. The finite element analysis was conducted for a single edge cracked geometry with H/W = 3.0, a/W = 0.5 and O<d/W<0.5. Eight-node quadrilateral elements were used for the finite element mesh and collapsed quadrilateral singular elements with quar- ter-point nodes were used at the crack tip. These results were used to verify the effect of d/W on FESET and GESET as discussed later.

I-.- u.I ~0 I..U

LI-

e / W 1 =0"5u 0.4 0.3 0.2 0.1 0.0

~ / W - O . ' 7 . . . . . • ' 'Fi'ni'te' E,;ment' J

0 i i i i I i i L I [ I I ' ' I . . . . I . . . . I

0.0 0.1 0.2 0.3 0.4 0.5 d / W

e / W 0 5 04 03 02 01 0.0

2 0 0 ' , , , , ; . . . . ; . . . . ; . . . . ; . . . .

(b) • Finite Element . ' ~ a/W=0.7 [This study]

150

iii lOO

CO

50'

0 ' n n l l l l o l l a n a I I I ' ' ' t ' ' ' ' ~

0.0 0.1 0.2 0.3 0.4 0.5 d / W

Fig. 5. Effect of eccentrically of loading on the normalized stress intensity factor and CMOD.

Eccentrically loaded single edge cracked geometry

3. RESULTS AND DISCUSSION

91

3.1. Effect of eccentricity on K and CMOD

Using eqs (7) and (12), FESET and GESET were generated for a/W ranging from 0 to 1, and d/W from 0 to 0.5. These values of FESET and GESET are shown as solid lines in Figs 3 and 4, respectively. Note that d = 0.5 W corresponds to a single edge cracked geometry subjected to tension, and d = 0 to a specimen with the load-line coincident with the edge of the plate. When d = 0.5W, FESET = FT and GESET = GT. As d/W decreases, the load-line moves from the center towards the edge of the specimen. The decrease in d~ W corresponds to an increase in the bending component of the loading. Consequently, for a given crack length, FESET and GESET increase with decrease in d/W.

The effect of eccentricity of loading on FESET and GESET is shown in Fig. 5 for a~ W = 0.2, 0.5 and 0.7. FESET and GESET decrease linearly with increase in d/W consistent with eqs (7) and (12). This implies that higher K and CMOD can be achieved for the same load as the load-line is moved towards the edge of the specimen. For example, at a/W = 0.5, FESET increases by ~85% when d/W decreases from 0.5 to 0.2. Such large increases in FESET significantly decreases the load required to conduct fracture and fatigue crack growth tests. Thus, a single specimen and test system configuration can be used to study the effect of tension and bending loading on the fracture and fatigue crack growth behavior of materials by varying the location of the load- line.

During compliance-based fatigue crack growth tests, the changes in crack length can be monitored through changes in the compliance (C) of the specimen[10]. The normalized specimen compliance is defined as EBC, as given by,

CMOD a EBC = E B - - - - GESET. (13)

P W

Since the compliance can range from 0 to c~, a convenient form of expressing EBC is achieved by defining a new variable U[3, 10] as given by,

1 U -- (14)

1 + e,/VYC

Figure 6 shows a plot of the normalized crack length, a/W vs U for various values of d/W. Knowing U through measurement of the load-CMOD response, the corresponding a/W can be calculated using the equations developed later.

1.0

d/W = 0.5, 0.4, 0.3, 0.6 0.2, 0.1, & 0.0,

~ 0.2

0.0 ' 0.0 0.2 0.4 0.6 0.8 1.0

U=l/ [I+4(EBC)] Fig. 6. Normalized crack length ( a / W ) as a function of the specimen compliance (EBC) for the eccentri-

cally loaded single edge cracked geometry.

92 R. JOHN

3.2. Comparison with other solutions

As discussed earlier, Piascik et al. [2, 3] determined FESE'r and GESEX for d/W = 0.2 and 0.1 < a / W 0.92 using the boundary force method. The solutions obtained during this study for d = 0.2W correlates well with the results from Piascik et al. [2, 3] as shown in Figs 3 and 4. The percentage difference between the results from this study and that from refs [2, 3] is within +1.3% for FESET and GESET. Figures 3-5 also compares the solutions obtained using the prin- ciple of superposition with finite element results for d/W ranging from 0 to 0.5 and a/W = 0.5. The linear dependency of FESEX and GESET on d/W is confirmed by the finite element results.

3.3. Expressions for normalized K, CMOD and crack length

Using the results for d/W = 0 and 0.5, new expressions were developed for FESET as a func- tion of a~ W and d/W. FESET is given as,

where

7

FESET = Z i=0

a )3/2 ' 1--~

(15)

P0 = 1.1215 + 6.7328(0.5 - ~dw), (16a)

Pl =--1.3923 - 17.518(0.5 - --~dw), (16b)

P2 = 4.316 + 41.1028(0.5 - ~dw), (16c)

P3 = -5.1009 - 64.082(0.5 - -~dw), (16d)

P4 = 0.6692 + 57.9826(0.5 - d ) , (16e)

P5 = 5.1684 - 23.3734(0.5 - ~dw), (16f)

and

e6--- -5 .4702-1.7092(0.5 --~-~) (16g)

P7 = 1.8135 + 3.114(0.5 - --~dw). (16h)

The limiting solutions at a~ W = 0 and a/W = 1 were satisfied automatically because the sol- utions FT and Fa[8] contained the limiting solutions for tension and bending loads, respectively. When d = 0.5W, eqs (15) and (16) are identical to the solution for SE(T) geometry. The con- stants, Pi, are linear with d/W consistent with the linear dependence of the stress distribution ond/W. Equations (15) and (16) are valid in the range O<a/W< 1, O<d/W<0.5 and H/ W>_ 3.0.

Eccentrically loaded single edge cracked geometry

Similarly, the expression for GESET is given as,

7 qi ( - -~) i

GESET = Z 2 '

i=° (1 ---~W)

where

and

q0 = 5.8399 + 35.0392(0.5 - --dw),

q l - - - - - 11.5467 - 79.0168 (0.5 - --dw),

q2 = 34.3928 + 155.4186(0-5 - -~dw),

q3 = - 4 2 . 4 5 4 4 - 138.2746(0.5 - --~),

q4 = 20.5012 + 7.3128 0.5 ---~ ,

q5 = 24.3695 + 73.4932(0.5 -- d ) ,

q6 = -38.1176 - 53.3828 0.5 ---~

q7 = 14.9641 + 15.2504(0.5 - --dw).

93

(17)

(18a)

(18b)

(18c)

(18d)

(18e)

(18f)

(18g)

(18h)

The limiting solutions at a / W = 0 and a / W = 1 were satisfied automatically because the sol- utions GT and GB[8] contained the limiting solutions for tension and bending loads, respectively. When d = 0.5W, eqs (15) and (16) are identical to the solution for the SE(T) geometry. Similar to Pi in eq. (15), the constants qi are linear with d/W. Equations (17) and (18) are valid in the ranges 0 < a / W < 1, 0 <_ d / W < 0.5 and H / W > 3.0.

Compliance-based fatigue crack growth testing requires the equations to calculate a /W knowing the measured compliance of the specimen. As discussed earlier, the normalized compli- ance of the specimen is given by U, i.e. eq. (14). The expression for a /W is given as,

4 ° Z - - = riU i, (19) W i=0

where

r0=0.99146+0.042148(~dw)-0.091724(~dw) 2, (20a)

r~ = - 3 . 8 1 8 3 + 1.1826 +2.5002 ~ , (20b)

94 R. J O H N

' ' ' I ' * ' I " ' ' I ' ' ' I ' ' '

0.4 g ~ 0.2

LULL 0.0 . . . . . . . . . . . . ..,~l

~ -0.2 LU

-0.4 I I I I I I I I I ' ' I , , , I , ,

0.0 0.2 0.4 0.6 0.8

= 0 . 4 i | ' ' ' ' ' ' ' ' ' ' ' ' " ' ' ' ' ' '11 ._o ~ 0.2

I I I t n 0 . 0 I I ~ m h . . . . . . . . . . . . . . . . . . . . . . . l i i l l U . . . . . I

~ .... n . . . . . . . . . . . ,,,m, . . . . . . . . qq l l p , =

-0.2 f u.I -0 .4

I I I I I I I I . . . I , , . I * , ,

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.0

¢ -

.9 ~ 0.5

.w=_~ 0.0

-0.5 uJ

-1.0 0.0 0.2 0.4 0.6 0.8 1.0

Crack Length / Width, a/W

Fig. 7. Er ror between proposed equations for F. G and a/14/, and corresponding superposit ion solutions for O.O<d/W<_ 0.5.

() 1 d ", r 2 = 4.9522 + 0.0215 8 ~ - 5.9794 - ~ (20c)

r3 = -5.2085 - 6.163 +0.11636 +21.967 - ~ (20d)

and

r4 = 6.7559 + 0.035694 + 5.2758 - 28.294 . (20e)

Equations (19) and (20) are valid in the ranges 0.1 < a/W< 0.95, 0 < d/W< 0.5 and H/W> 3.0. Equations (15)-(20) were developed based on values generated at discrete values of d/W.

Equations (15)-(18) were developed using results obtained for d/W = 0.0 and 0.5. Equations (19) and (20) were developed using results for d/W = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5. The accuracy

Eccentrically loaded single edge cracked geometry 95

of eqs (15)-(20) at intermediate values of a/W and d/W was evaluated by comparing these equations with those calculated using eqs (7), (12) and (14). During this comparison, d/W was varied from 0.0 to 0.5 in increments of 0.01 and a/W from 0.0 to 1.0 in increments of 0.01. The difference between the proposed eqs (15)-(20) and those obtained using the principle of superpo- sition is plotted as a function of a/W in Fig. 7. At each value of a~ W, the range of results corre- sponds to 0 < d/W< 0.5. The error is within +0.2%, 0.4% and 1.0% for FESET , GESET and a/ W, respectively. Hence, eqs (15)-(20) can be used for fracture and fatigue crack growth testing with acceptable accuracy.

4. CONCLUSIONS

Stress intensity factor (K) and crack mouth opening displacement (CMOD) solutions were developed for an eccentrically pin-loaded single edge cracked geometry, ESE(T). The expressions for K and CMOD are valid over a wide range of crack length to width ratio and eccentricity of loading. An expression was also developed to calculate the crack length from the specimen com- pliance for a practical range of a/W(=O.1 to 0.95). The proposed equations can be used for fracture and fatigue crack growth testing.

Acknowledgements--This research was partially conducted at Materials Directorate, Wright Laboratory (WL/MLLN), Wright-Patterson Air Force Base, OH 45433-7817 under on-site contract number F33615-94-C-5200. The author is grate- ful to Dr J. C. Newman, Jr and Dr J. H. Underwood for providing the results prior to the publication of their papers.

REFERENCES

1. John, R., Hartman, G. A. and Gallagher, J. P., Crack growth induced by thermal-mechanical loading. Experimental Mechanics, 1992, 32, 102-108.

2. Piascik, R. S. and Newman, J. C., Jr, An extended compact tension specimen for fatigue crack propagation and frac- ture. NASA-TM-110243, NASA Langley Research Center, Hampton, VA, U.S.A., 1996.

3. Piascik, R. S., Newman, J. C., Jr and Underwood, J. H., The extended compact tension specimen. In Fatigue and Fracture t~f Engineering Materials and Structures, 1997, accepted for publication.

4. Piascik, R. S. and Willard, S. A., The growth of small corrosion fatigue cracks in alloy 2034. Fatigue and Fracture of Engineering Materials and Structures, 1994, 17, 1247-1259.

5. Underwood, J. H., Kortschot, M. T., Lloyd, W. R., Eidinoff, H. L., Wilson, D. A. and Ashbaugh, N. E., Translaminar fracture toughness test methods and results from interlaboratory tests of carbon/epoxy laminates. Fracture Mechanics: 26th Volume, ASTM STP 1256, ed. W. G. Reuter, J. H. Underwood and J. C. Newman, Jr. American Society for Testing and Materials, Philadelphia, PA, 1996, accepted for publication.

6. Richardson, D. E. and Goree, J. G., Experimental verification of a new two-parameter fracture model. In Fracture Mechanics: Twenty Third Symposium, ASTM STP 1189, ed. R. Chona. American Society for Testing and Materials, Philadelphia, PA, U.S.A., 1993, pp. 738-750.

7. Raju, I. S. and Fichter, W. B., A finite element alternating method for two-dimensional mode I crack configurations. Engineering Fracture Mechanics, 1989, 33, 525-540.

8. Tada, H., The Stress Analysis of Cracks Handbook, 2nd edn. Paris Productions, St Louis, MO, U.S.A., 1985. 9. Tan, P. W., Raju, I. S. and Newman, J. C., Jr, Boundary force method for analyzing two-dimensional cracked

plates. ASTM STP 945, ed. D. T. Read and R. P. Reed, 1988, pp. 259-277. 10. Standard Test Method for Measurement of Fatigue Crack Growth Rates. ASTM E647-95, Annual Book of ASTM

Standards, American Society for Testing and Materials, Philadelphia, PA, U.S.A., Vol. 03.01, 1995, pp. 578-614.

APPENDIX A

The expressions for Fx, FB, GT and GB used in eqs (2), (3), (9) and (10) are given in the following equations[8].

3] 12W zra L " " \ w , ] +0.37 1 . Jra

FT = x / ~ t a n ~-7~ 7 r a l J / t / . ( . ~g

C O S - -

2W

~ f0 923 +0.199(1 - ' zra 4 /2W rra[." s m ~ ) ]

FB = ~/-~-~a tan~-ff cos zra 2W

1 (cos )

(A1)

(A2)

(A3)

96

and

R. JOHN

7 a q 0.66 ] . GB=4 0.8-I. (~)-I-2.04(--~) 2 (i___~)2 ] (A4)

(Re,rived 21 Februao' 1997, in.final form 15 May 1997, accepted 17 May 1997)