Analysis of crack extension in anisotropic materials based on local normal stress

Theoretical and Applied Fracture Mechanics 11 (1989) 27-46

27

ANALYSIS OF CRACK E X T E N S I O N IN A N I S O T R O P I C MATERIALS BASED ON LOCAL NORMAL STRESS

J.L. BEUTH, Jr. and C.T. H E R A K O V I C H Department of Civil Engineering, University of Virginia, Charlottesville, VA 22903, U.S.A.

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.

1. Introduction

2. Normal stress ratio theory

Classical fracture mechanics methods can be used to obtain estimates of strength in isotropic materials containing crack-like flaws. For composite materials, however, classical fracture mechanics principles can, at best, only be applied to a small class of problems. As a result, there is currently a great need for a theoretical method that is applicable to and accurate in predicting notched composite strength behaviour. Previous work by these and other authors [1-6] suggests that application of the normal stress ratio theory within a macroscopic-level elastic stress analysis can provide accurate predictions of notched unidirectional composite crack extension behavior. In this study, the theory is applied within analytical stress solutions for two types of crack-like flaws. Analysis predictions are compared with results from a number of different types of crack growth experiments on unidirectional graphite-epoxy. From this comparison, both the normal stress ratio theory and its method of application are critically assessed.

Buczek and Herakovich [7] proposed the normal stress ratio as a crack growth direction criterion for composite materials. They define the normal stress ratio as

0167-8442/89/$3.50 © 1989, Elsevier Science Publishers B.V.

R(rs, ¢e)= %~ T~"

(1)

In this expression, %~0 corresponds to the normal stress acting on the radial plane defined by % at a specified distance, r~, from a sharp crack tip. Tq0r is the tensile strength on the ¢p plane (Fig. 1). The model assumes that the direction of crack extension corresponds to the radial direction having the maximum value of the normal stress ratio, independent of the far-field applied stress state. In the current study, the normal stress ratio theory will also be used to predict critical applied far-field stresses. Critical stresses are defined as far-field stresses causing the maximum value of the normal stress ratio (along the predicted crack extension direction) to equal 1.

28

J.L. Beuth, Jr., C. 72 Herakovich / Analysis of crack extension in anisotropic materials

T~,~2 ~ i ~'z-/7,

=

-X

Fig. 1. Normal stress ratio theory parameters for a sharp crack analysis.

Although the normal stress ratio theory was originally proposed assuming that a sharp crack analysis would be used, subsequent work has also involved application of the theory within an elliptical flaw analysis [8]. In [8], the stresses used within the theory are evaluated along the elliptical boundary. Figure 2 illustrates the procedure used in [8] and apphed in this study. At each point on the boundary of the elliptical hole, defined by the angle 12, the normal stress ratio is evaluated over a range of radial directions, defined by the angle ~. The angle ¢p is varied over a 180 o range bounded by the tangent to the elliptical boundary. Crack extension is predicted to occur in the direction of maximum normal stress ratio, at the point on the ellipse defined by the corresponding value of 12.

In this way, not only a direction of crack extension, but also a point of origin of crack extension along the elliptical boundary is obtained from the analysis. In applying the elliptical flaw analysis to model experiments on specimens with a cut center notch, the shape of the cut notch tip is modelled by specifying the ratio of major to minor axis dimensions so that the radius of curvature of the tip of the ellipse equals the cut notch tip radius. In the current study, the normal stress ratio theory will also be used within the elliptical notch solution to predict critical stresses. The normal stress ratio theory is a direct extension of the maximum normal stress theory [9], formulated to make it applicable to anisotropic fracture problems. Like the maximum normal stress theory, it is based upon the assumption that local mode I crack displacement controls crack extension. The difference between the two theories is that the normal stress theory is able to account for a directional dependence of material strength through the strength parameter T~r. Because no experimental method currently exists for measuring the tensile strength on an arbitrary plane, T~, must be defined mathematically, in a manner consistent with tests that can be performed. Such tests require that T ~ satisfy the following conditions: (1) for an isotropic material, Tr~ must not depend on % (2) for crack growth parallel to the fibers in a composite material, T~ 0 must equal the transverse tensile strength, Yt; (3) for crack growth perpendicular to the fibers in a composite material, T~r must equal the longitudinal strength, Xt. The definition of T ~ used in [7] and all subsequent work to satisfy these conditions is T ~ = Xt sin 2 13 + Yt cOS2 8,

(2)

where fl is the angle from the ~p plane to the fiber axis (Fig. 1).

3. Analytical procedure

1 Fig. 2. Normal stress ratio theory parameters for the elliptical flaw analysis used in [8].

The problem studied analytically is depicted in Fig. 3. The figure shows an infinite, two-dimensional, center-cracked, unidirectional homogeneous anisotropic plate, subjected to arbitrary far-field loading, with arbitrary crack and prin-

J.L Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials ~ry

29

3.1. Step 1." Far-field applied stresses In analyzing crack growth experiments for this study, values for far-field stresses must be used that approximate those applied to actual specimens. The methods used to obtain appropriate far-field stress values for each type of test analyzed are detailed below. °'x

~---X

x~

l

Fig. 3. Problem modelled analytically.

cipal material axis orientations. This analytical problem is used in the current study to investigate the behavior of the normal stress ratio theory, and to compare predictions from the analysis with observed crack growth behavior. A three-step analytical procedure is used to accomplish these goals.

l O"x

l

o x

"t"x y

f

1

J~

Grip Region

1

Nidtt

J~

o~

Region 2h ~Vidth 4-

=

.=J

t~

0

.[

II

01 ~t

J

0 ° Coupon

...

IOff Axis

90 ° Coupon

y¢

"NN,

On-Axis Tensile Coupon

Off-Axis Tensile Coupon

Fig. 4. Unidirectional tensile coupons.

"

3.1.1. On -axis tension tests An on-axis tension test is defined in this study as a tension test on a composite material with one of its principal material axes oriented along the tensile axis (Fig. 4(a)). For such tests, a composite behaves as a macroscopically orthotropic material, with compliance coupling terms $16 and $26 equal to zero. It is assumed in this study that for such tests the transverse normal stress caused by the coupling term $12 and the grip constraint on the specimen ends is negligible. As a result, for these tests, the applied tensile stress is used as the far-field stress in the analysis.

3.1.2. Off-axis tension tests An off-axis tension test is defined in this study as a tension test on a composite material with its principal material axes oriented at an angle to the tensile axis (Fig. 4(b)). Specimens are designated by the orientation of the fibers with respect to the tensile axis (x-axis). For such tests, a composite behaves macroscopically as a fully anisotropic material, with nonzero values of the compliance terms $16 and $26. For an off-axis tension test, where the ends of the specimen are constrained by grips, these compliance terms cause a shear stress and bending moment to be applied to the specimen in addition to the applied axial stress. As a result, some method of accounting for the shear stress applied to such specimens must be used. The method chosen in this study makes use of a two-dimensional elasticity solution applied to the problem of an unnotched off-axis coupon. A plane stress elasticity formulation by Pagano and Halpin [10] yields a closed-form solution approximating the stress state for the unnotched off-axis tensile coupon problem. In addition to satisfying the plane stress elasticity equations, their solution satisfies end displacement boundary conditions applied along the specimen centerline. The

J.L. Beuth, Jr. ('. 7~ Herakori~h / A nalvmv ~ff crack extensum in ant*'otropic mater*aZ~

3()

stresses for an unnotched off-axis coupon that result from the analysis are of the form:

S[6

o, = - 2CO~v - 2~-~ Coy- + C~y + Q . ~,, = 0 .

(3)

~,, = C ~ ( y :

- h:),

r~:o

where Co=

C1

=

6h2( $11S66 - $26) + $2112 " Col,

C,,

(6S66h 2 + Sl112).

The S,j are the plane stress lamina compliance coefficients in the x-y coordinate system. It is important to note that this solution predicts that the transverse normal stress, ¢!~" equals zero throughout the specimen. This is a direct consequence of the displacement boundary conditions used, which specify that no constraint is applied to the specimen in the y-direction. The goal of using the stresses in eq. (3) is to obtain an accurate value of applied far-field shear stress for a given applied axial stress. This was done in this study by first integrating the stresses o, and r~, across the specimen width to obtain applied normal and shear force resultants. The normal stress resultant, designated as P, that resuits is h

t'='f'2'

(

dy= 2th C2

2s16C°h2

•

(4)

The corresponding shear force resultant, designated as T, is given by h

,?-,, dy = -

~ [ 2CO k3

(5)

If the applied shear force is divided by the applied normal force, the expression that results simplifies to

T P

T

~'- = P

o'~/-

(7)

SI6

3S66

311/2

Sll

2S1~,

4S1~h 2

6S16eo

C, - 6S1~'

r=,f

eq. (3). In this study, it is assumed that this ratio also defines the ratio of applied far-field shear stress to applied far-field axial stress. Tile applied far-field shear stress is therefore obtained by multiplying the applied axial stress by the ratio of T / P , resulting in the expression:

1

(6)

$16

3S66

$1112

Sit

2Si6

4S16h2

where the definition of CO has been taken from

The value of r :¢ for a given value of o "£ is a function of the compliance coefficients in the x-v coordinate system and ( / / 2 h ) , the specimen aspect ratio. It is important to note the values that r < assumes for two special cases of the off-axis problem. For cases where S ~ equals zero (on-axis tensile tests), the value of the constant CO (eq. (3)) becomes zero. This causes T to equal zero, while P becomes equal to 2 t h e o / S ~ . As a result, o ' ¢ = e o / E 1, and eq. (7) gives r ~ = 0, agreeing with the method used in this study of using the applied axial stress as the far-field stress for on-axis tests. Another interesting case is when the aspect ratio of the specimen is very large. As indicated by eq. (7), ~-~ approaches zero for large values of ( l / 2 h ) . Thus, the effect of the end constraint is less significant for specimens having large aspect ratios.

3.1.3. Iosipescu tests The Iosipescu shear test was originally developed as a method for the determination of isotropic material shear properties [11]. It has more recently been applied to determine composite material shear properties [12 15] using a flat specimen. In this study, it is used as a crack growth test by locating a vertical notch in its center test section and allowing an arbitrary fiber orientation. Figure 5 provides a diagram of the notched specimen. Under idealized loading conditions in an unnotched specimen, the specimen center is subjected to a state of pure shear stress equal to P / A , where P equals the downward force applied to the specimen and A equals the crosssectional area of the specimen at its center. Determination of appropriate far-field stresses to use in analyzing the center-notched version of the Iosipescu test is not a simple task. This is because, even in unnotched specimens, boundary effects and the method of loading the specimen result in a state of stress in the center of the

31

J.L. Beuth, Jr., C T. Herakovieh / Analysis of crack extension in anisotropic materials specimen that is not pure shear. In attempting to model the near-crack-tip stress state, a method which can account for these factors is preferred. In the current study, however, end effects and the method of loading are not accounted for and the net shear stress applied to the Iosipescu specimen is used as the far-field applied stress in the infinite plate analysis. This represents a first-step approach at analyzing a center-notched Iosipescu specimen.

3.2. Step 2: Near-crack-tip stresses

o~v/~ Re f

S1S2

t

r v/a

+

(

S2

1

S2

Re t ( S1 -- S2 ) l V2

~1 = COS fp + S 1 sin ~;

~2 = cos ~ + S 2 sin qp.

The angle ~ is measured with respect to the crack and a is one-half of the crack length (Fig. 3). These expressions are used in this study to model the near-crack-tip stress field for the problem illustrated in Fig. 3 by using the crack angle, a, to rotate the far-field stresses into the crack coordinate system.

3.2.2. Infinite anisotropic plate with an elliptical

3.2.1. Infinite anisotropic plate with a sharp crack In [16], Lekhnitskii outlines a complex variable plane elasticity solution for an infinite homogeneous anisotropic plate with an elliptical flaw at its center. By reducing the minor axis dimension to zero and evaluating the stress potential functions in the neighborhood of the crack tip, the solution can be used to analyze the center-cracked infinite homogeneous anisotropic plate problem of this study. This procedure models a crack in an actual material as a line crack with an infinitely sharp tip. In [17], Sih, Paris and Irwin use the complex variable approach of Lekhnitskii to derive expressions for the near-crack-tip stresses which are of the form:

Ox -

roots of the characteristic equation for a plane linear elastic anisotropic material, and

sl } ~V 2

flaw The stresses given in eq. (8) are for the special case of an elliptical flaw having a zero minor axis dimension. In this study, the more general case of an elliptical flaw of arbitrary shape in an infinite anisotropic plate will also be studied as a model for near-notch-tip stresses. The stress solution is taken from Savin [18]. The analyses in [18] yields stresses of the form:

o,~ = o~7 + 2

(9)

r×~ = rx~ - 2 R e [ S w o ( z l ) + $2+o(Z2) ] , where i(a - iSlb )

~o(Zl)

Sl])

~1/2

-]- $221~0(z2)], Re[wo(Zl) + +o(Z2)],

Oxx = axx + 2 Re[ S2q%(z1)

2($1-$2)

×

ox×b + o,(iaS2) + T×~(bS2 + i a ) 1

Z1 q- ~Z2 - (0 2 + S2b 2)

/"

'

i(a - iS2b )

1 IS1 S ]I

Xo(Z2) =

2(S 1 - $ 2 )

×

oxxb + o . ( i a S , ) + rx~(b & + i a ) z2+~z2-(a2+S2b2)

+ ~

Re ($1- $2 ) +(/2

o~v/S Re ( TXe

~

SiS 2

1

I (S7 7 52 ) @1/2

r~v~ Re ( "}-~

1

$1

/ ( S1 -- S2 ) ~1/2

+]/2

'

(8)

1)

~V 2

G/2

In these expressions, a and b are, respectively, the major and minor semi-axes of the ellipse (Fig. 2), Oxx, o., and rx~ are the far-field stresses in the crack coordinate system, z I = x + Sly, and z 2 = x

+ S2y. ,

where o ~ and r ~ are the applied far-field stresses in the crack coordinate system, S 1 and $2 are the

3.3. Step 3: Application of the normal stress ratio theory The normal stress ratio theory is applied in this study using each of the two near-notch-tip models

32

J.L. Beuth, Jr., C. 72 Herakotseh / Analysis of crack extension in anisotropic materials 0 . 1 0 0 " C e n t e r Notch

91 ,,oo

0.75"

0.45"

f

p21 3~0 R 0.050" pl~ PI and P2 =Distributed

Forces

Fig. 5. Iosipescu shear specimen.

just detailed. The stresses that result from the Lekhnitskii sharp crack analysis necessarily result in a maximum value of the normal stress ratio occurring at the crack tip. Because of this, through application of the Lekhnitskii solution, it is assumed that crack extension will occur at the notch tip. Because a predicted location of crack extension cannot be obtained from the analysis, only crack extension direction and critical stress predictions are obtained from the sharp crack analysis. The elliptical flaw analysis is used to obtain predictions for the direction of crack extension, the location of crack extension along the notch boundary, and critical applied far-field stresses.

4. A critical test of the analysis

4.1. Sharp crack model A substantial amount of effort has been expended by these and other researchers in attempting to identify critical tests of the normal stress ratio theory and its method of application within the sharp crack analysis of Lekhnitskii. Because it was originally proposed exclusively as a criterion for the prediction of crack growth direction, and because there is a strong tendency for crack extension in unidirectional graphite-epoxy to be along the fiber direction, efforts have centered around identifying analytical cases for which the theory predicts crack extension other than along the fibers. A number of potential cases have been identified. One specific case, however, provides the greatest insight into the possible limitations of the theory a n d / o r its use within a sharp crack analysis. This case is illustrated in Fig. 6. An infinite unidirectional graphite-epoxy plate with a

center crack along the fiber direction is subjected to pure shear loading in the coordinate system of the crack. For the positive shear case illustrated in Fig. 6, the direction of crack extension measured with respect to the crack predicted by the normal stress ratio theory is - 1 2 ° if the theory is applied within the sharp crack analysis of Lekhnitskii. Symmetry conditions result in a prediction of q)~ = 12 ° for a negative applied shear stress. The normal stress ratio theory prediction of crack extension at 12 ° from the fibers for this case does not appear to offer a severe contradiction with expected experimental crack extension behavior (crack extension along the fibers). A discrepancy of 12 ° between theory and experiment is almost reasonable. It is instead the reason why the theory does not predict crack extension along the fibers that is important. For all cases of pure shear far-field loading in the crack coordinate system, the Lekhnitskii solution predicts that o~, the normal stress along the q~ = 0 axis (Fig. 3), equals zero. This is due to the fact that at ep = 0, ~1 = ~b2 = 1 (eq. (8)). For the case illustrated in Fig. 6, the angle q~ = 0 corresponds to the fiber direction. Thus, no local normal stresses are predicted to exist perpendicular to the fibers. The normal stress ratio theory, if it is applied within the Lekhnitskii solution, is therefore incapable of predicting crack extension along the fibers for this loading case. This result is valid for any notched anisotropic material analysed using Lekhnitskii solution, regardless of its elastic a n d / o r strength

~xy

~c = -12 °

0= o °

Fig. 6. Infinite anisotropic plate with a crack along the principal material axis under pure shear far-field stress.

J.L. Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials 0.040

l

I

I Critical T~t Case - ~

Crick

4.2. Rounded notch tip model

[

I

0.020

(in.)

.0.020

,

-0.040 0.0

I 0.020

I 0.040

I 0.060

33

0.080

X (in.) Fig. 7. Elliptical flaw analysis result for the case of a crack subjected to pure shear far-field stress taken from [8] (see Fig.

6).

properties. Thus, the apparent discrepancy between the theory's prediction and expected experimental behavior for this case is not simply due to potential inaccuracies, but is instead due to a fundamental error either in the theory itself or in its method of application within a sharp crack analysis. One possible explanation for why the normal stress ratio appears to break down for the case illustrated in Fig. 6 is that this is a theoretical case where a theory based solely on local normal stress cannot yield a prediction of crack extension along the fibers. This would suggest the need for the inclusion of stress components other than the normal stress defined with respect to the crack within a crack growth theory. Another explanation for the discrepancy between theory and expected behavior is that for some cases use of the stress solution by Lekhnitskii is incorrect. The Lekhnitskii stress solution models an actual notch as a line crack with an infinitely sharp tip. Clearly, a discrepancy exists between the geometry of an actual rounded notch tip and that of sharp crack model. For a crack growth theory such as the normal stress ratio, which is formulated to depend entirely upon near-crack-tip stresses, it is possible that this discrepancy is significant. Application of the normal stress ratio theory within the elliptical flaw analysis of Savin suggests that this is the case.

In order to assess the influence of notch tip shape on the analysis, the normal stress ratio theory has been applied to the case illustrated in Fig. 6 using the stress solution for an elliptical flaw in an infinite homogeneous anisotropic plate. An illustration of the results of the analysis is provided in Fig. 7, taken from [8]. In the figure, the ellipse used to model the rounded shape of the notch tip is indicated by a solid line. The geometry of an actual cut notch is indicated by the dashed line. The solid line extending from the elliptical boundary represents the predicted location and direction of crack extension. The figure shows that, when this stress solution and application procedure are used to model this case, the normal stress ratio theory predicts crack extension along the fiber direction. The location of crack extension is also important to note. Instead of predicting crack extension from the tip of the rounded notch, crack extension is predicted to occur just away from the notch tip. It appears that by using a rounded notch tip solution and by allowing crack extension to occur away from the tip of the notch, a more realistic crack growth direction prediction can be obtained.

5. Experiments analyzed All of the specimens tested for this study were made of 16-ply unidirectional AS4/3501-6 graphite-epoxy. Table 1 provides strength and stiffness properties for this material obtained from material property (unnotched specimen) tests performed by the authors. These properties were used in all of the analytical work of this study. Each crack growth specimen had a notch cut in its

Table 1 Summary of strength and stiffness properties obtained from experiment, AS4/3501-6 graphite-epoxy Stiffness properties

Strength properties

E 1 = 18.3 msi (126. GPa) E 2 = 1.45 msi (10.0 GPa) GI2 = 0.814 msi (5.61 GPa)

X t = 210. ksi (1.45 GPa) Yt = 7.75 ksi (53.4 MPa) S = 14.4 ksi (99.3 MPa)

PIE = 0.305

E--L= 12.6

X---3= 27.1

E2

Yt

34

J.L. Beuth, Jr.. C. T Herakovich / Analysis of crack extension in anisotropic materials

center using a wire saw. Before any crack growth experiments were performed, each specimen was examined for damage by taking an X-ray radiograph of its notched region. No damage was detected in any of the 0 °, 45 °, or 90 ° coupons or Iosipescu specimens. As outlined in [3], however, some of the 15 o off-axis specimens originally designated for testing did show extensive damage in the region of their cut notches. These specimens were withdrawn from the test matrix.

~ded Cr~

L

5.1. Fifteen degree off-axis tensile coupons

± Tension tests on center-notched 15 ° off-axis coupons are outhned in [3]. Specimens with aspect ratios of eight, four, and one were tested with the center notch having one of two orientations. Specimens with 90 ° notches had their notches cut perpendicular to the specimen loading axis. Specimens with 105 ° notches had their notches cut perpendicular to the material fiber axis. All of the twelve specimens tested (two of each aspect ratio and notch angle configuration) were 1.00 in. (25.4 mm) wide and had a 0.200 in. (5.1 mm) long and 0.005 in. (0.127 mm) wide notch. Each specimen was loaded incrementally and monitored visually through a microscope for evidence of initial crack growth. From each test, the direction of initial crack growth, the stress at initial crack growth, and the stress at fracture were obtained. In each of the experiments, crack extension occurred along the fiber direction, yielding experimental values of • c = - 7 5 ° for the coupons with 90 o notches and q~c = - 9 0 ° for the coupons with 105 ° notches. The majority of the specimens exhibited slow crack growth. Two of the twelve specimens exhibited fast fracture only. Figure 8 provides typical photographs of crack extension occurring during testing. Cracks appeared as dark lines extending from the cut notch. As with the stability behavior of the specimens, the location of crack extension along the cut notch boundary varied in the tests. In some specimens, cracks initiated from the tip of the notch. In other specimens, cracks clearly initiated away from the tip, near the point of tangency between the straight and semicircular notch tip regions. These two cases, each illustrated in Fig. 8, represent extremes in behavior, with other specimens experiencing crack initiation at these locations and between them. This behaviour was interpreted as indicating

Fig. 8. Microscopic photographs of crack extension in off-axis tensile coupons.

15 °

a weak tendency for crack extension to initiate away from the notch tip. The critical stresses obtained from the experiments are provided in Table 2.

5.2. Zero, forty-five, and ninety degree coupons Center-notched, 0 °, 45 °, and 90 ° coupons with notches oriented at 90 ° to the loading axis were tested under far-field uniaxial tension. Each specimen had a gage section of 5.0 in. (127.0 ram) and was 1.00 in. (25.4 mm) wide. Specimens had a center notch of length 0.100 in. (2.54 ram) and width 0.005 in. (0.127 ram). The notched area of each specimen was monitored visually during testing using a microscope in conjunction with a videotape machine. Videotaping allowed any visible crack growth events to be studied thoroughly after each test was completed.

5.2.1. Zero degree tests In all of the 0 ° notched coupon tests, crack extension occurred along the fiber direction, yield-

35

J.L. Beuth, Jr., C.T. Herakooich / Analysis of crack extension in anisotropic materials Table 2 Comparison of theoretical and experimental critical stresses, 15 o off-axis tests Spec. number

Axial stress at initial crack extension ksi (MPa)

1 2

8.49 (58.5) 7.55 (52.1)

1 2

7.88 (54.3) 7.55 (52.1)

1 2

10.1 (69.6) 7.70 (53.1)

1 2

6.76 (46.6) 6.76 (46.6)

1 2

10.1 (69.6) 7.94 (54.7)

1 2

7.59 (52.3) 7.58 (52.3)

Averages ksi (MPa)

Axial stress at fracture ksi (MPa)

Averages ksi (MPa)

Predicted critical axial stress Sharp Elliptical crack notch a ksi (MPa) ksi (MPa)

8.02 (55.3)

90 o notch A.R. = 1 8.49 (58.5) 8.79 (60.6) 8.64 (59.6)

12.2 (84.1)

18.3 (126.0)

7.72 (53.2)

105 o notch A.R. = 1 11.7 (80.7) 8.42 (58.1) 10.0 (68.9)

10.3 (71.0)

19.1 (132.0)

8.91 (61.4)

90 o notch A.R. = 4 12.8 (88.3) 9.61 (66.3) 11.2 (77.2)

7.17 (49.4)

14.6 (101.0)

6.76 (46.6)

105 ° notch A.R. = 4 10.3 (71.0) 8.92 (61.5) 9.59 (66.1)

6.64 (45.8)

16.1 (111.0)

9.00 (62.1)

90 o notch A.R. = 8 10.1 (69.6) 9.30 (64.1) 9.68 (66.7)

6.06 (41.8)

12.9 (88.9)

7.59 (52.3)

105 ° notch A.R. = 8 9.32 (64.3) 10.1 (69.6) 9.73 (67.1)

5.74 (39.6)

15.0 (103.0)

a As it is applied in this study, the elliptical notch solution predicts that the stresses on the flaw boundary are independent of flaw size.

ing experimental values of ~c = + 90 °. Figure 9 provides a typical photograph of crack extension occurring during testing. Cracks appeared as white lines, with crack initiation occurring at the very tip of the cut notch in all cases. Of the experiments performed for this study, these were unique in that cracks extended in two directions from each notch tip. All of the specimens experienced slow and stable crack extension. In these tests, crack extension typically occurred in an unsymmetric manner. Because of this, two crack initiation stress values were obtained from each test, one for each notch tip. As extended cracks grew in length, they eventually extended into the specimen grip areas. At this point in the tests, the middle of the specimens became entirely separated from the specimen sides. This caused the specimen to behave essentially as two intact coupons with a single fractured coupon between. Specimens thus did not experience true fractures, presenting the need to define specimen failure. Failure stress was defined as the axial stress causing the first noticeable displacement of the specimen center with respect to its sides. Table

3 provides a summary of all of the critical stress values obtained from the 0 o tensile coupon tests. The average crack initiation stress value was obtained by averaging all of the crack initiation stress values (from both tips) together. A total of five 0 ° tests were run because of the scatter exhibited in the results.

5.2.2. Forty-five degree off-axis tests In all of the 45 ° notched coupon tests, crack extension occurred along the fiber direction, yielding an experimental value of ~c = - 4 5 ° . All of the specimens experienced unstable crack extension. Figure 10 provides a photograph of the notched region of a typical specimen after fracture. As the figure indicates, crack extension occurred away from the notch tip in all cases. The location of crack initiation along the cut notch is best characterized as occurring at an angle of - 4 5 o measured along the semi-circular notch tip. This location of crack initiation was consistent throughout the tests, indicating a strong tendency for crack extension to occur away from the notch

36

J.L. Beuth, Jr., C. 72 Herakovich / Analysis of crack extension in anisotropic materials

Table 3 Comparison of theoretical and experimental critical stresses, 0 o, 45 o and 90 o notch coupon tests Specimen number a

Axial stress at initial crack extension ksi (MPa)

ksi (MPa)

(first tip) 38.6 (266.0) 37.8 (261.0) 42.4 (292.0) 29.7 (205.0) 28.1 (194.0)

(second tip) 48.0 (330.) 41.6 (287.0) 45.8 (316.0) 33.3 (230.0) 36.9 (254.0)

Averages ksi (MPa)

Axial stress at failure ksi (MPa)

Averages ksi (MPa)

Predicted critical axial stress Sharp crack ksi (MPa)

Elliptical notch ksi (MPa)

0 o tests 1 2 3 4 5 1 2 3 4 1 2 3

3.85 4.19 4.24 4.19

(26.5) (28.9) (29.2) (28.9)

2.72 (18.8) 2.95 (20.3) 2.75 (19.0)

64.0 67.9 55.1 53.0 54.1

(441.0) (468.0) (38O.O) (365.0) (373.0)

58.8 (405.0)

20.9 (144.0)

22.8 (157.0)

4.12 (28.4)

45 o tests 3.85 (26.5) 4.19 (28.9) 4.24 (29.2) 4.19 (28.9)

4.12 (28.4)

5.02 (34.6)

4.83 (33.3)

2.81 (19.4)

90 o tests 2.72 (18.8) 2.95 (20.3) 2.75 (19.0)

2.81 (19.4)

(baseline test)

38.2 (263.0)

a All 0 o, 45 o, and 90 o coupons had notches at 90 o to the specimen loading axis (horizontal).

Fig. 9. Microscopic photographs of crack extension in a 0 o tensile coupon.

Fig. 10. Microscopic photographs of the near-notch region in a 45 o off-axis tensile coupon after fracture.

J.L Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials

37

built at the University of Wyoming for the determination of material shear properties [13,14]. As illustrated in Fig. 5, specimens measured 3.00 in. (76.2 mm) in length and 0.75 in. (19.1 mm) in width. V-shaped notches with a notch root radius of 0.050 in. (1.27 mm) and an included angle of 110 o were ground into the sides of each specimen to a depth of 0.15 in. (3.81 mm), using a diamond-impregnated grinding wheel. Each specimen had a cut center notch at 90 ° (vertical). Notch dimensions were 0.100 in. (2.54 mm) in length and 0.005 in. (0.127 mm) in width. As with the 0 o, 45 o, and 90 ° coupon tests, the notched area of each specimen was monitored during testing using a microscope in conjunction with a videotape machine. 5.3.1. Zero degree Iosipescu tests

Fig. 11. Microscopicphotographs of the near-notch region in a 90 o tensilecoupon after fracture.

tip. The critical stresses obtained from the 45 ° off-axis tensile coupon tests are given in Table 3. 5.2.3. Ninety degree tests

In all of the 90 ° notched coupon tests, crack extension occurred along the fiber direction, yielding an experimental value of tpc = 0 °. All of the specimens experienced unstable crack extension. Figure 11 provides a photograph of the notched region of a typical specimen after fracture. Crack initiation occurred at the very tips of the notches in these specimens. The critical stresses obtained from the 90 ° tensile coupon tests are provided in Table 3. As for the 45 ° tests, because only unstable crack extension was exhibited in these tests, the stresses at crack initiation and failure are identical for each specimen.

In all of the 0 o notches Iosipescu tests, crack extension occurred along the fiber direction, yielding an experimental value of cpc = - 9 0 °. All of the specimens experienced stable crack extension. Figure 12 provides a typical photograph of crack extension occurring during testing. Cracks appeared as dark lines extending from the cut notch. Crack extension occurred far from the notch tip in all cases. Crack extension consistently occurred in a symmetric manner, yielding a single critical crack initiation stress value for each test. Extended cracks did not continue to grow throughout the tests. Instead, extended cracks stopped growing at a distance of approximately one-half of the total notch length (0.050 in.) from the original notch tip. As more displacement was applied to the specimen, extended cracks continued to open

5.3. Iosipescu shear tests

Center-notched Iosipescu shear specimens were also tested to study their crack growth behavior. Specimens were loaded in a fixture designed and

Fig. 12. Microscopic photograph of crack extension in a 0 ° Iosipescu shear specimen.

J.L. Beuth, Jr., 6". T. Herakotrich / A na(vsis of crack extenmon m anisotropic matemal.~

38

Table 4 Comparison of theoretical and experimental critical stresses, Iosipescu shear tests Specimen number ~

Shear stress at initial crack extension ksi (MPa)

1 2 3 4

4.32 3.92 3.94 3.95

1 2 3

3.69 (25.4) 3.34 (23.O) 3.46 (23.9)

1 2 3

4.17 (28.8) 4.02 (27.7) 4.19 (28.9)

1 2 3

3.96 (27.3) 4.01 (27.6) 4.28 (29.5)

(29.8) (27.0) (27.2) (27.2)

Averages ksi (MPa)

Shear stress at failure ksi (MPa)

Averages ksi (MPa)

0 ° tests (84.9) (85.8) (83.3) (80.5)

Predicted critical shear stress Sharp crack ksi (MPa)

Elliptical notch ksi (MPa)

12.1 (83.6)

3.10 (21.4)

6.51 (44.9)

4.03 (27.8)

12.3 12.4 12.1 11.7

3.50 (24.1)

15 ° tests 9.89 (68.2) 10.2 (70.3) 9.45 (65.2)

9.85 (67.9)

2.88 (19.9)

5.12 (35.3)

4.13 (28.5)

30 o tests 7.18 (49.5) 7.67 (52.9) 7.70 (53.1)

7.52 (51.8)

3.01 (20.8)

4.56 (31.4)

4.08 (28.1)

45 ° tests 3.96 (27.3) 4.01 (27.6) 4.28 (29.5)

4.08 (28.1)

3.80 (26.2)

4.70 (32.4)

a Each Iosipescu test specimen had a cut notch at 90 ° (vertical).

without extending. These specimens did not experience final fracture. Specimen failure stress was defined as the maximum shear stress applied to the specimen. The critical stresses obtained from the 0 o Iosipescu tests are given in Table 4. The highly stable crack extension exhibited by these specimens is reflected in the significant difference between crack initiation and failure stresses in all of the tests.

5.3.2. Fifteen degree Iosipescu tests

grow throughout these tests, instead stopping after reaching a length of approximately 1.5 original notch lengths (0.150 in.) and then opening as more displacement was apphed to the specimen. Table 4 provides a summary of the critical stresses obtained from the 15 ° Iosipescu tests. Because these specimens did not experience a true failure event, specimen failure stress was defined as the maximum shear stress applied to the specimen. As Table 4 indicates, specimens continued to acquire a significant amount of applied shear stress

In all of the 15 ° notched Iosipescu tests, crack extension occurred along the fiber direction, yielding an experimental value of ~vc = - 7 5 o. All of the specimens experienced highly stable crack extension before failure. Figure 13 provides a typical photograph of crack extension occurring during testing. Cracks appeared as dark lines extending from the cut notch. As the figure indicates, crack extension occurred away from the notch tip in all cases. The location of crack extension was not as far from the notch tip as it was for the 0 ° notched Iosipescu tests, however. Crack extension consistently occurred in a symmetric manner, yielding a single critical crack initiation stress value for each test. The extended cracks did not continue to

Fig. 13. Microscopic photograph of crack extension in a 15 ° Iosipescu shear specimen.

J.L. Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials beyond that causing crack initiation. The difference between crack initiation and failure stress levels is clearly less than it was for the 0 o tests however. Both the average stress causing crack initiation and that causing failure were significantly less than those obtained from the 0 ° Iosipescu tests. 5. 3. 3. Thirty degree Iosipescu tests In all of the 30 ° notched Iosipescu tests, crack extension occurred along the fiber direction, at 9~c= - 6 0 °. All of the specimens experienced stable crack extension. Figure 14 provides a typical photograph of crack extension occurring during testing. Cracks appeared as dark lines extending from the cut notch, initiating away from the notch tip in all cases. The distance away from the notch tip at which crack initiation occurred was less than that observed for the 15 ° Iosipescu tests. Crack extension was symmetric, yielding a single crack initiation stress value for each test. Extended cracks did not continue to grow throughout these tests, instead stopping after reaching a length of approximately two original notch lengths (0.200 in.) and then opening as more displacement was applied to the specimen. Table 4 provides a summary of the critical stresses obtained from the 30 ° Iosipescu tests. Unlike the notched Iosipescu specimens described thus far, these specimens did experience a fracture-type failure event. As Table 4 indicates, specimens continued to acquire a significant amount of applied shear stress beyond that causing crack initiation. The difference between crack initiation and failure stress levels is less than it was for the

Fig. 14. Microscopic photograph of crack extension in a 30 o Iosipescu shear specimen.

39

I Fig. 15. Microscopic photograph of crack extension in a 45 ° Iosipescu shear specimen. 0 ° and 15 ° notched Iosipescu tests, however. Although the failure stress continued the decreasing trend exhibited in going from the 0 ° to the 15 o fiber orientations, the average stress at crack initiation was clearly higher than that observed from either the 0 ° or the 15 o notched Iosipescu tests. 5.3.4. Forty-five degree Iosipescu tests In all of the 45 o notched Iosipescu tests, crack extension occurred along the fiber direction, at q0c = - 4 5 °. Crack initiation caused a significant drop in the shear stress applied to the specimen. As more displacement was applied to the specimen after crack initiation, the specimen acquired more stress; however, the stress the specimen could withstand never reached the value that caused crack initiation from the original cut center notch. As a result, crack initiation occurred at the maxim u m stress that the specimen could withstand, and the stress causing crack initiation was also designated as the failure stress for each specimen. Figure 15 provides a typical photograph of crack extension occurring during testing. Crack extension occurred away from the notch tip in all cases, at a location close to - 4 5 ° around the rounded notch tip. The location of crack extension was essentially identical to that observed for the 45 ° off-axis tension tests. Compared to the other notched Iosipescu tests performed as a part of this study, this test experienced a location of crack initiation that was the closest to the notch tip. As in the other notched Iosipescu tests, crack extension occurred at both notch tips simultaneously, yielding a single crack initiation stress value for each test.

40

J.L. Beuth, Jr., C.T. Herakooich / Analysis of crack extension in anisotropic materials

As with the other notched Iosipescu specimens, extended cracks did not continue to grow throughout the tests on the 45 ° specimens. In fact, almost all of the crack extension occurred at crack initiation. At initiation, cracks immediately extended almost to the specimen edges. After testing, each specimen was either in two pieces or close to it. This contrasted significantly with other notched Iosipescu specimens, which remained intact throughout their tests. Table 4 provides a summary of the critical stresses obtained from the 45 ° Iosipescu tests. The failure stress continued the decreasing trend exhibited in going from the 0 ° to the 15 ° and 30 ° fiber orientations. The average stress at crack initiation was essentially the same as that obtained from the 30 o tests.

Table 5 C o m p a r i s o n of theoretical a n d e x p e r i m e n t a l crack g r o w t h directions, all tests Experimental direction

Specimen

type a

of c r a c k extension b

6.1. Directions of crack extension

Table 5 provides a direct comparison between the directions of crack extension observed in the experiments of this study and those predicted by the normal stress ratio within the sharp crack and elliptical flaw analyses. A comparison of the experimental and analytical values provided in Table 5 show relatively strong agreement between theory and experiment for both solution approaches. The sharp notch solution provides the best agreement overall, and has a maximum disagreement with experiment of 3 ° . The elliptical notch solution appears to work well except for the case of a 15 ° off-axis specimen of aspect ratio 1 with a 90 ° notch. Its prediction of - 6 7 ° is in error by 8 o. This error is not severe in magnitude, however. The ability to accurately predict crack extension direction in these cases may seem somewhat trivial, considering the strong tendency for graphite-epoxy to exhibit crack extension along the fiber direction. Previous work by these and other authors [1-4] clearly indicates, however, that the ability to consistently predict crack extension direction in graphite-epoxy is not shared by other approaches and theories. 6.2. Locations of crack initiation

The locations of crack initiation for the experiments performed, as predicted using the elliptical

Sharp crack

Elliptical notch

0 0, 45 o, a n d 90 ° n o t c h e d c o u p o n tests 0 o Specimens 45 o S p e c i m e n s 90 o S p e c i m e n s

_+90 ° -45 ° 0o

_+87 ° -43 ° 0o

_+86 ° -44 ° 0o

15 o off-axis n o t c h e d c o u p o n tests 90 o n o t c h A.R. = 1

A.R.=4 A.R. =8 A.R. = 1 A.R. = 4

6. Comparison of theory and experiment

Predicted crack extension direction b

A.R. = 8

0 o Specimens 15 o S p e c i m e n s 30 ° S p e c i m e n s 45 o S p e c i m e n s a

-75 ° -72 ° -75 ° -73 ° -75 ° -73 ° 105 o n o t c h

-67 ° -73 ° -73 °

-90 ° -90 ° -90 °

-86 ° -86 ° -86 °

-87 ° -88 ° -88 °

n o t c h e d I o s i p e s c u shear tests -90 ° -89 ° -75 ° -75 ° -60 ° -45 °

-60 ° -45 °

-88 ° -74 ° -59 ° -45 °

All 0 o, 45 0, a n d 90 o c o u p o n s h a d n o t c h e s at 90 o to the

s p e c i m e n l o a d i n g axis (horizontal). E a c h Iosipescu test specim e n h a d a cut n o t c h at 90 o (vertical). b All angles are m e a s u r e d w i t h respect to the cut notch, with a n e x t e n s i o n direction of 0 o c o r r e s p o n d i n g to c r a c k g r o w t h collinear to the notch. See the angle q~ in Fig. 3.

flaw analysis, are illustrated graphically in Figs. 16-19. Because there was not an apparent crack initiation location trend with respect to the different 15 o off-axis specimen configurations, only one configuration, that having a horizontal notch and an aspect ratio of 8, was analyzed. As a group, the figures indicate that the crack initiation location predictions of the analysis provide excellent qualitative correlation with those observed experimentally. Both of the graphical representations provided in Fig. 16 indicate crack initiation from the tip of the notch, as observed in the 0 o and 90 o tensile coupon specimens. In Fig. 17, both modelled test cases show crack initiation away from the notch tip, as observed experimentally in the 15 ° and 45 ° off-axis tensile specimens. Figures 18 and 19 provide graphical representations of the predicted locations of crack initiation for the Iosipescu tests. As observed experimentally, the predicted location of crack initiation is away from the notch tip for

41

J.L. Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials 0.040

I

I

I

I 'r..,.C.p,,.""'"I 0.020

e.(in.)

I

.0.020

D J

-0.040 0.0

I 0.020

I

I

0.040

0.060

0.080

Z (in.) 0.040

I

I

I

i

9 0 " Tensile Coupon - Extended

t.=O"

Crl,ck

]

modelled sharp crack tip. In order to accomplish this, a value for r s must be obtained from a baseline test. Once this is done, rs is treated as a material constant. U n d e r this assumption, the theory can then be applied to obtain a critical stress prediction for a test with any fiber or notch orientation or applied stress state. Because all predictions are dependent on the calculated value of r S, it is important to choose a baseline test carefully. In the current study, the 90 ° tensile c o u p o n test is used as the baseline test. This test was chosen for a n u m b e r of reasons. First, by orienting the n o t c h along the fiber direction, crack extension was m a d e to be collinear with the original cut notch (along ¢p = 0 ° ) . F o r such crack extension in a composite, a macroscopic stress analysis (one using stress intensity factors [19]) 0.080

0.020

I

I

l

I 15° Off-AxisC o u p o n - I~tended Crack t. =-?3"

e(in.)

0.040

1

-0.020

(in.)

[] J

-0.040 0.0

I 0.020

I 0.040

I 0.060

.0.040 0.080

% (in.) Fig. 16. Predicted locations of crack initiation for 0 o and 90 o tensile coupons.

l -0.080 0.0

I

J

I

0.040

0.080

0.120

0.160

(in.) 0.040

all of the tests analyzed. The observed trend in location of crack initiation with respect to fiber orientation is also well represented in the analysis predictions. The elliptical flaw analysis predicts that the location of crack initiation will move closer to the notch tip with an increase in fiber angle, up to a fiber angle value of 30 o. A n identical location of crack initiation is predicted for the 30 ° and 45 ° Iosipescu specimens.

I

I

I

45° Off-AxisCoupon ExtendedCrack Oo=-44"

-

0.020

e (in.) -0.020

6.3. Critical stresses

L -0.040

6. 3.1. Sharp crack analysis In using a sharp crack analysis to predict critical stresses, the normal stress ratio theory is applied at a specific radial distance, rs, from the

0.0

I 0.020

I 0.040

I 0.060

0.080

Z (in.) Fig. 17. Predicted locations of crack initiation for 15 o and 45 o off-axis tensile coupons.

J.L. Beuth, Jr.. C.T. Herakovich / Analysis of crack extension in anisotropic materials

42 0.040

I

length of 0.100 in., the normal stress ratio equals 1 at a value of r~ = 0.006573. The theory was then applied to the other tensile coupon and the Iosipescu tests to obtain the applied far-field stresses to provide a normal stress ratio of 1 at r , = 0.006573, along the predicted direction of crack growth. By using the same value of r~ to analyze all tests, a proportional relationship between critical stresses and the square root of the notch length is obtained directly (see eq. (8)). The predicted critical stress values are given in Tables 2+ 3+ and 4.

4 0* I~ipescu Specimen

I

- Extended Crack

*.

= _Uo

0.020

(in.)

i+i:

-0.020

l[]r

-0.040 0.0

t 0.020

I

I

0.040

0.060

0.080

In the current study, the elliptical notch analysis is used to predict critical stresses by applying

X (in.) 0.040

I

I

6.3.2. Elliptical flaw analysis

I

1 ++++.+] - ExtendedC r a c k <,.= -74"

0.020

0.040

I

I

I 30* l a e l p e s c u S p e c i m e n - ExtendedCrick *,=-59"

]

0.020

(in.)

(in.)

-0.020

-0.040 0.0

I 0.020

I 0.040

+ 0.060

-0.020

l[]t

0.080

X (in.) Fig. 18. P r e d i c t e d l o c a t i o n s o f c r a c k i n i t i a t i o n f o r 0 ° a n d 1 5 " losipescu shear specimens.

-0.040 0.0

I 0.020

I 0.040

I 0.060

0.080

X (in.) 0.040

I

I 45

I I~lpc~cU Specimen 1

- Extended Crack

has been shown to be applicable. Second, the test is clearly normal stress dominated. Therefore, application of the normal stress ratio theory should be valid for this test, even if it is not for others. Third, experimentally, this test did not exhibit stable crack extension. As a result, the critical stress was easily determined as the fracture stress. Finally, the tests themselves yielded consistent fracture stress values. Using the average value of fracture stress for the three baseline tests (2.81 ksi) as the applied far-field stress ( o ~ ) , the normal stress ratio theory was applied within the sharp crack analysis along the predicted direction of crack extension (¢p = 0 o ). Along this direction, for a modelled crack

0.020

(in.) -0.020

-0.040 0.0

E 0.020

I 0.040

I 0.060

0.080

(in.) Fig. 19. P r e d i c t e d l o c a t i o n s o f c r a c k i n i t i a t i o n for 30 ° a n d 45 ° Iosipescu shear specimens.

J.L. Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials

the normal stress ratio along the elliptical notch boundary. This procedure is chosen to parallel the work in [8] on predicting the direction and location of crack initiation. The physical rationale behind this approach is that cracking initiates along the flaw border and then extends into the material. Evaluating stresses along the flaw boundary, as it is done in this study, is not well suited to modelling notches of different sizes. As long as the ratio of minor axis to major axis dimensions is not altered (a value fixed to model the actual notch tip radius), changing the size of the modelled flaw does not change the magnitude of the stresses along the flaw border. As a result, a flaw size effect is not predicted. Use of elliptical notch boundary stresses in applying the normal stress ratio is not the only analytical procedure that is physically reasonable. An equally admissible approach would be to use a "critical distance" approach following that of Whitney and Nuismer [20]. This approach is analogous to the approach used in the sharp crack analysis of this study. Stresses are consistently evaluated at a specified distance from the modelled flaw and flaw size effects are obtained directly. The physical rationale behind this approach is that crack extension occurs as the result of the nucleation of micro flaws ahead of the macroscopic flaw border. Whether crack initiation is truly controlled by events at the flaw border or slightly away from it is not clear and analyses assuming each type of phenomenon are needed. Because the elliptical flaw analysis does not model flaw geometry exactly, a baseline test is required to predict critical stresses. As in the sharp crack analysis, the 90 o tensile coupon test was chosen as the baseline test. This test was used to obtain a geometric scaling factor for the determination of critical stresses for other tests. Application of the theory within the elliptical flaw model yielded a predicted critical stress value of 0.732 ksi for this test. The scaling factor that would yield a correct critical stress value of 2.81 ksi was thus 2.81/0.732 = 3.84. All elliptical flaw predictions for other tests were thus multiplied by 3.84 to obtain the predicted critical stress. The geometric scaling factor acted to fit the analysis of an elliptical flaw in an infinite anisotropic plate to the tests on center-notched specimens using a single baseline test.

43

6. 3.3. Fifteen degree off-axis tests Based on the values given in Table 2, a number of comparisons can be made between the experimentally observed 15 ° off-axis specimen critical stresses and those predicted by the normal stress ratio within each type of analysis. First, the magnitudes of the critical stress predictions provided by the theory applied within both stress solution approaches agree reasonably well with experimental values. The sharp crack analysis provides the best agreement with experiment. Its predictions for the specimens of aspect ratio 4 agree well with the observed stresses at crack initiation. The predictions for specimens of aspect ratio 8 and 1 are slightly lower and higher, respectively, than observed values. All of the predictions of the elliptical notch analysis are higher than the observed values. This is due primarily to the inability of the analysis to model flaw size effects. As it is applied in this study, the elliptical flaw analysis is unable to account for the fact that the 15 o off-axis specimens had notch lengths twice that of the 90 ° (baseline test) specimens. Overall, these tests and the predictions for them do not provide a clear indication of which experimental stress is more predictable by theory, the stress at crack initiation or that at failure. The predictions of the normal stress ratio theory provided in Table 2 show trends with respect to both notch orientation and specimen aspect ratio. Within both analyses, the theory predicts that observed critical stresses will increase as specimen aspect ratio is decreased. The experiments, however, fail to show any trends in critical stresses with a change in specimen aspect ratio. The trend in each of the analysis predictions is due entirely to the changes in far-field applied stresses predicted by the Pagano and Halpin solution. The solution predicts a significant increase in applied far-field shear stress with a decrease in aspect ratio, especially between aspect ratio values of 4 and 1. The degree to which the Pagano and Halpin solution can be used to model the far-field stress state in notched coupons having small aspect ratios is questioned by the authors. It is felt that the boundary conditions may not be valid for specimens of small aspect ratios. Also, end effects, which the solution does not account for, may play a significant role in notched specimens having smaller aspect ratios. It is also important to note

44

J.L. Beuth, Jr., C.T. Herakovich / Analysis of erack extension in anisotropic materials

the scatter exhibited in the experimental crack initiation stresses. It is possible that a subtle trend in the experiments with respect to aspect ratio was lost in the experimental scatter. The normal stress ratio predictions also show distinct trends with respect to notch orientation for the 15 ° off-axis tests. For specimens of each aspect ratio, the sharp notch analysis predicts that a specimen with a 90 ° notch will have a higher critical stress than a specimen of the same aspect ratio with a 105 ° notch. The elliptical notch analysis predicts just the opposite, with the 105 ° specimen having a higher value. The experiments consistently show the 90 ° specimens having higher average crack initiation stresses than the 105 ° specimens. Thus, the sharp crack solution offers stronger correlation with the experimental values. Because it is exhibited independent of aspect ratio, this correlation is not affected by the applied stress predictions of the Pagano and Halpin solution.

6.3.4. Zero, forty-five, and ninety degree tensile coupon tests Table 3 provides comparisons of predicted and experimentally obtained critical stress values for the 0 °, 45 °, and 90 ° tensile coupon tests. Because the 90 ° test has been used as the baseline test for both analyses, a comparison between the predictions of the theory and experiment cannot be made for it. The normal stress ratio theory predictions of 5.02 ksi and 4.83 ksi for the 45 ° off-axis test agree well with the average critical stress value of 4.12 ksi obtained from experiment. For the 0 ° tests, the normal stress ratio predictions of 20.9 ksi and 22.8 ksi agree best with the crack initiation stresses obtained from the experiments. Both predictions are significantly less than the experimental values, however. As with the 15 ° off-axis specimens, it is felt that end effects may be playing a significant role in these experiments. In applying an infinite plate solution to model tensile coupon tests, it is assumed that stress is applied uniformly to the coupon ends. Preliminary finite element analyses indicate that this may not be the case for the 0 ° specimens of this study. Instead, a region of reduced stress along the specimen centerline may extend into the grip area. It is believed that this is due to small values in the ratio of G~,/Ex exhibited by tensile coupons with small off-axis angles. Further comparisons between theory and experiment with specimens having small

off-axis angles, including finite element analyses, are needed to better define this effect.

6.3.5. losipescu tests Table 4 gives comparisons of the predicted and experimental critical stresses from the Iosipescu tests. The magnitudes of the critical stress predictions from each type of analysis agree well with the experimental stresses at crack initiation. The predictions of the sharp crack analysis appear slightly lower than experiment, and the elliptical flaw analysis predictions appear slightly higher. The agreement between theory and experiment is even more evident with respect to their trends as a function of specimen fiber orientation. The experimental crack initiation stresses do not continue to decrease with an increase in fiber angle. Instead, the average values increase in going from 15 ° to 30 ° and then remain about the same going from 30 ° to 45 °. The normal stress ratio predicts similar behavior within each stress analysis. The sharp flaw analysis predicts an increase in critical stresses for fiber orientations of 30 ° and higher. The elliptical flaw analysis predicts an increase at 45 °. The trends in the predictions are not exact, however, the approximations made in analyzing the Iosipescu test, especially the assumption of a pure shear far-field stress state, necessarily limit the accuracy of the theory predictions. It is also important to note that the normal stress ratio critical stress predictions do not agree with the experimentally observed failure stresses in magnitude or with respect to the trends exhibited as a result of changing specimen fiber angle. Stresses at crack initiation appear to be the critical stresses predictable by theory. The agreement between critical stress values from the Iosipescu tests and theory is especially significant in that the normal stress ratio is entirely ignorant of the shear strength of the material. As eq. (2) indicates, only the longitudinal and transverse tensile strengths of the material are included in the theoretical formulation. The agreement of the theory with shear test results suggests that the strength of a notched specimen under far-field shear loading can be predicted using an analysis based solely on local normal stress and normal strength. 6.3.6. Overall stress prediction results The comparisons between theoretical and experimental critical stress values for all of the tests

J.L. Beuth, Jr., C.T. Herakovich / Analysis of crack extension in anisotropic materials

in this study indicate that, for an analysis based on original flaw geometry, stresses causing crack initiation are most comparable to theory. This is most strongly indicated in the comparison between analysis and experiment for the Iosipescu specimens. This is expected for any analysis based on original specimen geometry. For the tests analyzed in this study, there was generally good agreement in critical stress value magnitudes and trends. One exception to this is the trend in predicted crack initiation stresses with respect to aspect ratio for the 15 o off-axis specimens, which was not observed experimentally. The other significant exception to this was the prediction of the elliptical flaw analysis of a trend in critical stresses with respect to notch orientation opposite of that observed. Finally, the lower than observed normal stress ratio predictions for the 0 ° tensile coupon test warrant further investigation. The most impressive correlation between theory and experiment occurred for the Iosipescu shear tests. It is important to recognize that a large number of variables and assumptions that can affect critical stress predictions are involved in both of the analyses applied presented in this study. These assumptions, include an idealized crack geometry, approximate far-field stresses, and an infinite plate assumption. Both analyses also assume that critical stresses can be predicted using a single baseline test, either through the parameter rs or by scaling subsequent theory predictions. The number of potential sources of inaccuracy in predicted critical stress values highlights the significance of the agreement with experiment observed in this study.

45

Because the theory does not account for shear stresses or shear strength, the agreement with resuits from the Iosipescu shear tests are particularly encouraging. The correlation between theory and experiment noted in this work offers support for the validity of the normal stress ratio theory and, in general, for the concept that crack growth phenomena are controlled by local normal stress. Use of the elliptical notch solution to model near-notch-tip stresses shows promise because it more accurately models notch tip shape. This may be important in applying a theory based on nearnotch tip stresses. It also can model crack initiation away from a rounded notch tip, agreeing with experimental observations. It provided reasonable predictions for the direction and location of crack initiation and critical stresses. Its predictive ability does not appear to be greater than the sharp notch solution, however, with respect to crack extension direction and critical stress predictions. As is expected from an analysis based on original notch geometry, crack initiation stresses, not failure stresses, were the analytically predictable critical stresses in this study. The noted exceptions in the correlation between theory and experiment suggest two topics of interest in future work. First, the role of finite geometry effects should be studied. In some notched tests, these effects may significantly alter results. Second, the application of a critical distance approach to the elliptical notch solution is warranted in order to address the minor inconsistencies in some of its predictions and to incorporate notch length effects.

Acknowledgment 7. Conclusions Application of the normal stress ratio theory within each stress solution has yielded critical stress and crack extension direction predictions that correlate reasonably well with experimentally obtained values. Exceptions do exist, but are not unreasonable given the wide range of tests analyzed and the number of assumptions in the analysis. The elliptical flaw analysis has also shown the ability to qualitatively predict the location of crack initiation along the periphery of a rounded notch tip. Specimens subjected to a wide range of far-field applied stresses were tested and analyzed.

The authors gratefully acknowledge the support of Hercules Incorporated for this research. Both authors were affiliated with Virginia Polytechnic Institute and State University while completing a significant portion of the work.

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