Study of Pure and Mixed-Mode Fracture of a Brittle Material

by Jean Royer

ABSTRACT--In order to study the behavior of a crack in a linear-elastic material in plane mixed mode (modes I + ll), a specimen's shape and loading have been specially adapted. The specimens are first precracked in mode I and then sub- jected to monotonic loading until instability is reached by an original device which makes it possible to control the nature of the mixed mode applied and which is adjustable from pure mode I to pure mode I1. After the specimens are fractured, the lengths of the initial cracks and the kinking angles were measured in the plane-strain area. Then the stress-intensity factors /(i and KH at the moment when crack instability appeared were calculated. This made it possible to apply two criteria: maximum principal stress and maximum stress- intensity factor k~* at the onset of kinking. From comparing the calculated values with the experimental values we may note that there is good agreement with respect to the crack- kinking directions. However, for the limit load values con- siderable divergences have been recorded which are analyzed.

Introduction

Initially, in linear fracture mechanics, the behavior of cracks in mode I has been the main object of investigation since this mode is the most critical; its study is also easier since the crack propagates along its initial direction. However, the question of the effect of mode II super- imposed on mode I remains.

Unfortunately in this case, the nature of the problem is completely different. The mechanical symmetry of the problem with respect to the crack disappears because of the antisymmetry due to mode II. There is then kinking of the crack. Several experimental studies have been made involving simple shapes: compact-tension (CT) specimens with asymmetrical arms or plates with a crack that is inclined with respect to the direction of the traction. The mode II values thus obtained remain low with respect to that of mode I. To increase mode II it is possible to apply a biaxial load. ' However the investment is considerable, and due to the magnitude of the stresses to be developed, only plates under plane stress may be studied.

The difficulty of obtaining a crack in pure mode II is even greater since, in addition to the loading device which must not introduce mode I, the shape of the specimen must allow for precracking in mode I obviously.

Jean Royer (SEM Member) is Professor, Laboratoire de Mdcanique des Structures, Ecole Nationale Supe'rieure de M~eanique, 1 rue de la Nob~ 44072 Nantes Cedex 03, France. Original manuscript submitted: June 2, 1987. Final manuscript received: May 20, 1988.

In other words, the shape of the specimen must allow for loading in mode I and then in mode II (or a combination of both). Several general devices have already been proposed of which a review is given in Ref. 2. More specific types of devices exist which can be used in particular in pure mode II. 3 But they are not exactly adapted for use in fracture mechanics. For example the crack in the specimen recommended in Ref. 3 cannot be obtained by fatigue.

If we consider a simple criterion like that of the maximum normal stress of Erdogan and Sih 4 in a mode which is close to pure mode I, the kinking direction of the crack varies very rapidly with the value of mode I. How- ever, it is experimentally very delicate to apply a pure mode II to a crack since there are many causes which lead to a parasitic mode I, such as asymmetry of load and variation of the initial length of the crack depending upon thickness or anisotropy of the material. It is also advisable to add the effect of material behavior which is much greater for mode II than for mode I. Any plastic area in the material, however small, alters the behavior of the crack.

This has been confirmed by a study made involving fatigue in mixed mode using a tough steel. ~ The results show not only a high degree of variance of the kinking directions, but also differences in the overall behavior of the crack including double kinking. It was also observed that the behavior depended upon the load level. As load level increased the results deviated more and more from the predictions of linear fracture mechanics which may be explained by the influence of plasticity.

According to these results it was decided that a brittle material could be used for the study so as to minimize the complications due to yielding and obtain cleavage fracture under monotonic loading. In order to study the instability of a crack under mixed mode, and including up to pure mode II, it was necessary first of all to look for a suitable shape specimen; that is to say, a specimen that could not only produce a pure mode II in the crack but also a mode I, which could be easily controlled. It was at first considered that a symmetrical specimen be used which would make it possible to apply a double shearing action to two symmetrical cracks as suggested by Jones and Chisholm) Unfortunately, calculations and even simple intuition show that it isn't enough to exert a force parallel to the plane of the crack to obtain a pure mode II.

It is for this reason that the Y specimen (Fig. 1) which was already used for a fatigue study in mixed mode on 22M5 steep has been chosen and modified and its loading

382 �9 December 1988

conditions adapted to the study. This symmetrical speci- men consists of two cracks and two arms which are loaded by means of 20-mm diameter pins which transmit loads ~1 and ~ at an angle c~ with respect to the y axis (see Fig. 1). Each arm may be likened to a CT half specimen for which the parameter w is equal to two times the thickness of 25 mm. The length ao of the crack is determined as for a CT specimen: it is the distance along the x axis between the load pin A and the front of the crack. When the load corresponds to a pure mode I, the conditions at the front of the crack are the same as for a standard CT-25 specimen with ao/w = 0.6. The equili_.: brium of the specimen is assured by a third force F parallel to .~ applied through pin C.

Under these conditions, the loading of the cracks depends upon two independent parameters, c~ and F : a determines the loading mode whereas the level of load depends on F.

Experimental Procedure In order to insure a linear behavior of the material, this

study has been made using Plexiglas. A first series of specimens was made using exactly the same shape as that used in Ref. 6 for the fatigue tests but an additional difficulty appeared. During the precracking of the specimens in pure mode I according to the procedure

described in Ref. 7, even though all of the conditions for the determination of Kxc were satisfied, it was observed that a large difference in the velocity of crack propagation between the edges and the central part of the front of the crack existed. The thickness of the specimen (25 ram) makes it possible in the case of Plexiglas to assure plane- strain conditions at the center of the specimen whereas, given cracks of equal length, the crack-propagation velocity under these conditions is much greater than that observed under plane stress. The result of this was a curving of the front of the crack which was much too great to enable the conditions to be considered those of a two-dimensional crack problem; this was true even after correcting the load parameters with respect to the real length of the crack.

To offset this difficulty the precracking was carried out on 30-mm thick specimens with lateral grooves 2-ram deep on the front and back and in the plane of crack development. The added stress concentration on the front and back of the specimen made it possible to increase the crack speed at the ends of the crack front and to thereby obtain a straighter crack front. After cracking, the thick- ness of the specimen was reduced to 25 mm by machining which eliminated all traces of the grooves. The difference in the length of the initial crack between the edge and the center then became less than 2 mm on the finished specimen. Furthermore, the machining of the thickness of the specimens made it possible to eliminate the hetero-

Fig. 1--Y specimen geometry

'i

' r y~

{,95" d

I

i L,r

grooves for precracking

--..,

, 1 1 t E x p e r i m e n t a l M e c h a n i c s �9 383

geneity of the surface of the material due to the manufac- turing of the plates.

The calculation by FEM of the stress-intensity factors (SIF) developed on the crack front has shown' that in order to obtain a pure mode II load, the FI forces to be applied on the arms of the specimen (Fig. 1) had to be inclined ce = 64 deg with respect to the horizontal axis. Mode I being null the changing of the direction of the forces only amounts to changing the algebraic sign of the mode II applied to the specimen. The first test was made by introducing a wedge between the bearings mounted on the pins of the specimens at A and A ', which is equivalent to applying a force F1 directed toward the outside of the specimen. However, in this configuration the side opposite from the crack in the 1.5-mm radius connecting area is loaded more than the front of the crack and a fracture in this area results. In order to avoid this type of fracture it was necessary to reverse the direction of the forces applied to the bearings at A and A ' which are then directed toward the center of the specimen in the direction (c~ = - 116 deg).

Since this modification destabilized the equilibrium of the specimen, it was necessary to add between the lower edge of the specimen, below pin C and the bottom of the clevis, an adjustable wedge which makes it possible to replace the joint at C with a clamped end. In other words, instead of inserting a wedge between the bearings, it is the latter which are pressed against the sides of a V as shown in Fig. 2. In order to apply a mixed mode to the crack it is sufficient to choose another value for the angle

which determines the angle at the top of the wedge inserted between the bearings. The knowledge of what this angle is establishes the nature of the mixed mode defined by parameter ~ = Kx/Kzz.'

In this study the angles a have been selected so as to cover all of the plane mixed modes from pure mode I to pure mode II and 16 specimens have been tested and fractured. In the results, it should be noted that the specimens have not been equally distributed according to the load modes but have been assigned as the test was made so as to better confirm the results already obtained whose interpretation may have been uncertain. It is due to the anomalies in behavior, which will be explained when the results are examined, that several specimens have been loaded near pure mode II.

In order to eliminate the effect of temperature on the materials characteristics, the tests were conducted at a constant ambient temperature of 20~ The rate at which

Fig. 2--Plexiglas specimen under pure mode II

Fig. 3--Kinked crack after fracture

a

I Y

_i Y --I

i Fig. 4--Parameters of a kinked crack

- - . - - . 4 ~ . . X

= lim&~ k i

384 �9 December 1988

the specimen is loaded is not only dependent upon the speed at which the jack moves but also upon the angle ~t of the loading device. The speed of the jack was controlled so that maximum load would be attained in 20 to 30 minutes.

The recording of the resultant force applied to the specimen as a function of the displacement of the jack shows a linearity which confirms the linear behavior of the material. The instability of the crack generally appeared at the center of the specimen (under plane strain), which caused an immediate fracturing of the specimen. However, for two specimens (7 and 10) there was at first a stable propagation of the crack along an edge (under plane stress). The result of this was to slightly modify the crack front before sudden fracture at the center of the specimen occurred. These anomalies were not taken into considera- tion during the evaluation of the results.

Figure 3 shows a precracked specimen whose crack was then deflected during fatigue under pure mode II stress and whose final fracture was obtained by impact.

Results

After the tests the initial length of the fatigue crack was measured at six points spaced 5-mm apart along the front

of the crack and from the front and back of the specimen. Given the 25-mm thickness of the specimen and the fact that the onset of crack propagation always appeared at midthickness lead one to suppose the results below are characteristic of the behavior under plane conditions. Under these conditions the average length of the crack was calculated only from the values at the four inter- mediate points and excluding the values at the end points on both sides of the specimen.

For the kinking angle of the crack, /~, (Fig. 4) the average value under plane strain was obtained by the same procedure as that already used to determine the initial length of the crack. These various measurements were made optically with a profile projector. Using the maximum force (F,,,~) applied by the testing machine (MTS), the symmetrical forces F-~ or ~ (Fig. 1) supported by the specimen were calculated taking into consideration the value of the angle a . The known value of the com- ponents of ~, and of initial length ao of the crack made it possible to calculate the stress-intensity factors KI and Ku by interpolation at crack instability using the results shown in Ref. 7.

The values of K~ and KH make it possible to apply several mixed-mode failure criteria among which we have selected two. That of Erdogan and Sih 4 is based upon the

Fig. 5-- Kz, K . values at the onset of instability

K I

[8) 4C

~-+ (6)

~(15)

30"

20-

10-

(16) (5) -~-

+

31-(18)

%)

I

10

(17) +

(10)

11)

{.[9)

I

20

(7) (1%(1) J I " ~ KII

30 MPa

Exper imenta l Mechanics �9 385

p ..... de I I 6~ ~ kinking angle p ..... de II

75

(9),

(i0) (12)~ (7)

4"1( ( n

k 2 =0 (ii)~ ~ 8 8 ~ 2~r maximum

(4)X

(16)

(15)/(5)

. . . . . I O O . 2 5

2 5

O I I I

0.5 i _ . . o . ,

Fig. 6--Kinking angle under plane strain

i i I 0.5 0.25 0

maximum principal stress at the front of the crack accord- ing to the polar direction 0. So it is possible to predict the direction fl of kinking of the crack 8 and the value of the stress-intensity factor aoo 2~/~7 along this same direction from the deduced relationship

0 [K1cos 0 _ 3/(,i sin 0 o002 = cos 2 T T T ]

The values of this latter parameter have been calculated at the moment that crack initiation appeared.

The other criterion selected is based on the stress- intensity factors calculated at the end of the kinked crack and for a distance Ae along polar direction 0. These values that can be calculated by FEM depend upon the two parameters Ae and 0. But we indicate k*, the limit value of ki, when the length A/approaches zero. Accord- ing to this criterion 9 the crack propagates along the direction which corresponds to maximum k*, the pure mode I direction since k* is then zero. The values of k* have been calculated using the stress-intensity factors/<i and Kn applied to the crack and the results taken from Ref. 10.

Discussion Figure 5 shows that K1 and Kit have entirely different

effects upon the fracture. The degree of scattering of the experimental values is considerable but some information may be obtained. For fracture in pure mode I or pure

mode II the value of/<-i is greater than that of KH, a result which had already been predicted. However, the most interesting thing about this graph involves the inter- action of the modes which is not at all reciprocal because when beginning with a pure mode I we may observe that the addition of a mode II has almost no effect on the value of Kz when instability appears: KxM. This value, KxM, remains practically constant until Kxl = KJ3. Then the drop in this value K ~ is abrupt when K , reaches a value of approximately Kx/3. On the other side of the graph, the addition of a mode I, even relatively small, to a crack subjected to an almost pure mode II causes the values of Kzz which the material can withstand to drop rapidly. This could be made more apparent by increasing the number of experimental data points near pure mode II.

Figure 6 shows the experimental value of the kinking angle of the crack as a function of 6 = K/KH. We have also drawn the theoretical curves for this value deter- mined by the two previous criteria: maximum normal stress ao0 and maximum k*. These two curves which con- verge near the pure mode I area separate more and more towards pure mode II. Despite the scatter in experimental values, we observe that the latter are closer to the maximum k* curve. However, for pure mode II the experimental results do not agree with the two criteria. Even though the average experimental values remain near the theoretical curves, it does not seem that the latter fit the phenomenon studied perfectly. The experimental values seem to be distributed along a curve which is rounder and which has a tangent that is practically horizontal for pure mode II.

386 �9 December 1988

mode I put

.jr. (ll)

+ ~ 161 (3)

+ (5) (18) (8u + ,o

kl~ I MPa r

I

3O

2o

10

0 i i I i i - - m

0 o. 5 0 . 5 0 . 7 5 - - .

Fig. 7--Experimental values of k,*at the onset of instability

%

( 1 7 ~ i - _ i . _ ( 1 0 )

q- (13)

mode II put

i I i I

.-I~I]KII j --D- 0.75 0.5 0.25 O

(1% (1)

(7) +

Figure 7 shows the variation of k* as a function of the ratio ~ = K / K n . The scatter of the experimental values is greater than for propagation direction/3: this result was predictable by considering the very nature of the pheno- menon studied which is associated with the instability of a crack. In spite of this scatter, which is acceptable for this type of study, the results shown in Fig. 7 lead us to think that the value of k* is not independent of the ratio ~ even though the fluctuations seem to be of limited amplitude.

The maximum normal-stress criterion leads to values of aoo~/27rr which are very close to those calculated for k*. The differences between the values given by these two criteria are slight compared to the differences observed between the experimental values.

Conclusions The results obtained in this mixed-mode study which

was conducted on Plexiglas shows that the SIF K~ and K . applied to a crack have very different effects upon the behavior of the latter. If K . has very small effect upon the critical value of Kx in the vicinity of pure mode I, the critical values of Kz~ decrease very rapidly with/<I in the vicinity of pure mode H.

In linear fracture mechanics the maximum normal- stress criterion, as that of maximum k*, allows the direc- tion of kinking of the crack to be predicted rather well except in the vicinity of pure mode II where the divergence becomes considerable. Furthermore, the experimental values tend to validate the maximum k* criterion.

Finally, the limit load at crack initiation in mixed mode does not seem to be predicted by either of the two criteria considered. The variation of this limit load with the ratio 6 = K x / K H seems complex. This result confirms the observations made with respect to Fig. 6. The SIF Kz and K , must play very different roles in crack behavior and a criterion which is effective in mixed mode should make this last point apparent.

Refe rences

1. Truchon, M. Amestoy, M. and Dang Van, K., "Experimental Study of Crack Growth Under Biaxial Loading," Advances in Fracture Research, 4, ICF 5, Cannes (1980).

2. Abdallah, M.G., "Review of the Stqte-of-the-Art of Advanced Composites Interlaminar Shear Test Methods, "" Proc. 1986 SEM Spring Conf. on Exp. Mech., 301-310 (1986).

3. Arcan, M. and Banks-Sills, L., "'Mode H Fracture Specimen, Photoelastic Analysis and Results, "" Proc. 7th lnt. Conf. on Exp. Stress Analysis, Technion, Haifa, lsrael, 187-201 (Aug. 1982).

4. Erdogan, F. and Sih, G.C., "'On the Crack Extension in Plates Under Plane Loading and Transverse Shear, "" J. Basic Eng. (Dec. 1963).

5. Jones, D.L. and Chisholm, D.B., "An Investigation of the Edge- Sh'ding Mode in Fracture Mechanics, "' Eng. Fracture Mech., 7 (1975).

6. Royer, El., "'Fatigue en mode mixte de l'acier 22M5,'" Proc. lOth Can. Cong. of Appl. Mech. (June 1985).

Z Royer, J., "',4 Specimen Geometry for Plane Mixed Modes, "" Eng. Fract. Mech., 23 (4), 763-775 (1986).

8. Bui, H.D., Mecanique de la Rupture fragile, ed. Masson (1978). 9. Bilby, B.A., Cardew, G.E. and Howard, J.C., "'Stress Intensity

Factors at the Tips of Kinked and Forked Cracks, "" Fracture, 3, 1CF4, Waterloo, Canada (1977).

10. Amestoy, M., Bui, H.D. and Dang Van, K., "'Analytic Asymptotic Solution of the Kinked Crack Problem, "' Advances in Fracture Research, ICF5, Cannes (1980).

Experimental Mechanics �9 387