J

Available online at www.rilem.net Materials a n d Structures 38 (April 2005) 305-312

Mode I and mixed mode fracture studies in brittle materials using the Brazilian disc specimen

H. N. Atahan, M. A. Tasdemir, C. Tasdemir, N. Ozyurt and S. Akyuz Istanbul Technical University, Civil Engineering Faculty, 34469, Maslak Istanbul, Turkey Received." 3 February 2004; accepted." 14 June 2004

ABSTRACT High Strength Cement Mortar (HSCM) with very fine sand exhibits a typical brittle behavior. In the present work, a linear elastic fracture mechanics based model is used for the fracture studies conducted On this material. The experimental testing program is based on the diametral compression test of disc specimens containing an internal slant crack. Under the Mode I loading condition, the test method which has been previously used is applied to determine the critical value of the stress intensity factor, Klo for HSCM. The same disc specimen is also tested under combined Mode I and Mode II loading conditions. By changing the notch orientation angle with respect to the loading direction, the mode of fracture is varied from pure Mode I to Mixed-Mode. Based on the Mixed-Mode fracture envelope, it is shown that the disc specimen which is currently used for several brittle materials provides a wide range of IK iiI/KI ratios. In pure Mode I loading case, after determining Klo it is possible to obtain the graph of normalized critical load versus normalized crack length. For the purpose of comparison, some available experimental data on Mode I and/or Mixed Mode fracture of some other brittle materials such as glass, sintered carbide, and polymethyl methacrylate (PMMA) were also evaluated. It can be concluded that there is a good fit between the experimental results and the theory. 1359-5997 9 2004 RILEM. All rights reserved. RESUMI~ Les mortiers dl tr~s haute r~sistance (MTHR) pr@ar~s avec du sable tr~sfin ont exhibit~ un comportement (ypiquement fragile. Dans eel article un moddle de rupture Olastique lin~aire est utilisOpour ~tudier la rupture de ces mat~riaux. L "dtude exp&imentale consistait des essais de compression diamOtrale sur des @rouvettes en forme de disques contenant une fissure inclinde. Les essais conformes au Mode I utilis~s auparavant sont appliquOs pour ddterminer la valeur critique du facteur d'intensit~ de la contrainte Kic du MTHR. Les m~mes ~prouvettes ont OtOem'uite essayOes sous charge combinOe du Mode Iet Mode II. En changeant l "angle d "orientation de 1'entaille par rapport dl la direction de la force appliqude, le mode de rupture a dtO modifi~ du Mode I au Mode-MLrte. En se servant de l'enveloppe du Mode-Mixte de rupture, utilisde actuellement on a montr~ que l'@rouvette disque pouvait fournir une large plage pour les rapports IK , ]/ K I . Au cas de chargement simple de Mode I, aprOs avoir dOtermind Kxc, il ~taitpossible d"obtenir un graphique entre la charge critique et la longueur de la fissure normalisdes. D'autres r~ultats expOrimentaux obtenus sur d'autres matOriauxfragiles comme verre, carbide synthOris~ et m~thacrylate de polymethyl (PMMA) sont aussi ~valu~s afin de r~aliser une comparaison. On peut conclure qu 'il existe une bonne eonformit~ entre les rOsultatsexp~rimentaux et thOoriques.

1. I N T R O D U C T I O N High Strength Cement Mortar (HSCM) is a very fine mortar having enhanced mechanical properties. HSCM represents a new generation of cementitious materials with cube strengths above 150 MPa. These materials have the following advantages: i) an improved homogeneity of the mortar by reducing the maximum size of particles, ii) an optimum packing density by the use of free and ultra-fme particles, iii) a substantial reduction in the amount of water used, iv) very high strengths by hardening at an elevated temperature in order to reach [1, 2]. Since the grading curves 1359-5997 9 2004 RILEM. All rights reserved. doi:10.1617/14104

of these mortars are discontinuous, the maximum density is obtained by using different sizes of fine particles [3-6]. HSCM without fibers, however, exhibits an ideal brittle behavior. It is very difficult to record the post-peak response of test specimens without fibers [4, 6]. A limited amount of information is available on the fracture of these materials under Mode I and Mixed Mode loading conditions. The main objective of this work is to obtain the critical values of stress intensity factor, Kic, crack initiation angles, normalized crack initiation stress, and fracture envelope of HSCMs under mixed-mode loading conditions. For the purpose of comparison, the available test results on the fracture of glass,

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H.N. Atahan et al. /Materials and Structures 38 (2005) 305-312 p

sintered carbide, graphite and PMMA are also evaluated using an analytical approach given in the following paragraph. 2. T H E O R E T I C A L

2R

BACKGROUND

Structural engineering changes its scope to meet the requirements brought by the developments of new technology. Because of the progress in fracture mechanics, engineers and designers have gained an insight into crack propagation and residual strength of structures containing cracks [7]. Since HSCM is a new material used in structural engineering, basic information on crack initiation and propagation in HSCM with and without fibers is needed. Recently, a number of investigations were carried out on the failure of brittle materials under Mode I (opening) loading. In practice, structures are generally subjected to Mixed Mode type of loading. For a cracked specimen under Mixed Mode loading, the following three fracture criteria have been established; i) the Maximum Hoop Stress Criterion [8], ii) the Minimum Strain Energy Density Criterion [9], and iii) the Maximum Energy Release Rate Criterion [10]. These theoretical criteria are used to predict the direction of crack initiation and the stability of crack growth. Some applications and comparisons of these criteria can be found in [11] and [12]. The experimental testing program is based primarily on the compression test of the Brazilian disc specimens with center notch (CND) as shown in Fig. 1. A disc of radius R with an internal diameter crack length of 2a, has many advantages in terms of being an experimental sample which has been considered analytically in Russian literature [1315]. As seen in Figs. 2 and 3, a disc shaped specimen with an inclined crack subjected to diametral compression, offers the following advantages when testing an open crack: i. By changing the inclination of the notch with respect to the loading direction ([3), it is possible to change the mode of fracture from Mode I (tensile) to Mode II (shear), and to obtain a combined loading state with a wide range of Kl/Kic versus IKII I/Klc (where KII is the Mode II (sliding) stress intensity factor (SIF) and KI is the Mode I (opening) SIF at the notch tip) [16, 17]. ii. It is easy to obtain high positive and negative values of [KIll~ Kic ratio. Note that when [3=0, Kn is equal to zero, and when 13~30~ (the exact value depends on the a/R ratio), KI is equal to zero. It is not difficult to obtain higher Kn/K~ratios at about 30 ~ using this type of loading configuration (Fig. 3). In a linearly elastic, perfectly brittle material such as hardened cement paste or very high strength cement mortar with fine sand, an existing crack, flaw or notch begins to propagate in Mode I when the SIF, KI, at the tip of the crack reaches a critical value which is called fracture toughness (Kic) [18]. For concrete, Klc is assumed to be a material parameter, however, its determination based on LEFM revealed a significant dependence on the size of specimen. The difficulty of obtaining a valid fracture toughness from LEFM using laboratory-size specimens was emphasized by several researchers [19, 20]. The notched cylinder is a good test specimen with many advantages, and future

in.

P

p (a)

P (b)

Fig. 1 - Disc specimens:(a) Mode I case; (b) mixed mode loading case. U n i a x i a l c o m p r e s s i o n - s h e a r test

Uniaxial t e n s i o n - s h e a r o r C o m p a c t t e n s i o n - s h e a r test

IK~d/K,c

9 9

,~d~

1.5

cPr~pp~id' 2 ~

1.0 o,

t

9 I

!

-I .0

-0.5

0.5

L, 1.0

Fig. 2 - IKIII/K1cversus K1/KIccurve in mixed mode for different loading configuration. 10.0 i

.0 / : J ! -5.00 -10.0

20

40~ i

/

F

60~

80~

100~

Crack orientation angle, 13

Fig. 3 - Kn/KIversus crack orientation angle, 13. investigations in cracking of concrete may be carried out using this type of specimen; it also offers a strong potential for standardization. Recently, Atahan et al. [21] have applied the Size Effect Law proposed by Ba2ant [22, 23] to the notched disc specimens with different sizes under Mode I loading condition. All specimens were cast from the same batch of concrete. The sizes of the disc specimens were 600mm, 300mm, 150mm and 100ram in diameter and 300mm, 150mm, 60mm and 50mm in thickness, respectively. The ratios of 2a/2R considered were 0.0, 0.1, 0.3 and 0.5, where 2a is the length of the notch, and 2R is the diameter of disc specimen. It was shown that as the dimension of the specimen increases, the nominal strength decreases for both notched and unnotched specimens; this decrease is more noticeable in the latter ease. For the calculations of K~ and KH, a solution is available for plane stress condition. For a disc specimen subjected to

H.N. Atahan et aL / Materials and Structures 38 (2005) 305-312

307

mixed mode loading, ignoring terms of the order of (a/R) m and higher, according to Yarema et al. [13], the stress intensity factors at crack tips can be expressed as: [2A0 + L2(3A0 + A2) + ~4(3A2 + 3A4) + KI = - P I ~ I ~ 6 ( 3 A 0 -1A2 +3A4 +5A6) + [ ~ Ts [ Z L ~. 15 '8(~3 A 0+1-@1A2-~3 A 4 + ~ A 6+3-~5A8) 3Z 10 .SZ 10 Oq-

(1)

B0 +;L2(B0+lB2)+~4(-1B 0 + I B 2 +3B4) ] z 8 ~4 5 l KI[ =-2P.,~--_I+~6(B0-1B2 +--IB4+--5B6)+ 1 (2) t v~n I

16

s

16

I

/9~8(- 17 B0 +3B2 - ~ 5 B4 +~5B6 + 35 BS) / [ 64 8 128 64 128 J where, A0= 1 -Cos 2]3 2 A2=2[-cos413+cos213] A4=3 [-cos613+cos413] A6=4[-cos813+cos613] A8=5 [-cos 1013+cos8]3]

B0= Sin 213 B2=2[sin4]3-sin2]3] B4=3 [sin613-2sin413] B6=4[sin813-3sin613] B2=5 [sin 10]3-4sin8 ]3]

where R and t are the radius and thickness of the specimen, 2a is the crack length and )~=a/R. In the special case, when ]3--0 (i.e. pure Mode I loading condition), the Mode I SIF is given by +3

a 2

3 a 6

3

a 8

P rtRt Several solutions are available for calculating SIFs for diametrically compressed test specimens containing a central notch (CND) at an orientation angle of ]3 with respect to the loading axis. The solutions given in Fig. 4 were obtained by Yarema et al. by applying LEFM and approximation [13]. Fig. 4 shows how it is possible to change the mode of fracture from opening mode (mode I for 13=0~ to mixed mode (tensionshear for 13approximately less than 30 ~ and compression-shear for ]3 approximately greater than 30 ~ (depending on a/R)) by rotating the notch inclination angle with respect to the loading direction [24]. At the main crack tip, thus, K) is greater than zero for the crack initiation in tension-shear, and it is less than zero in compression-shear. After the crack initiation, KI is always dominant along the propagating crack as shown in [12] and [16]. The main crack faces do not come into contact during the failure mechanism. Yarema et al. [13, 15] used integral equation formulation to obtain KI and Ku and a comparison of the solutions proposed can be found in [17]. A close agreement was observed with the results of other researchers, such as Atkinson et al. [24], and Awaji and Sato [25]. For the closed crack case, the determination of the friction coefficient in concrete using disc specimens was theoretically studied by Tasdemir and Karihaloo [7]. where o =

Fig. 4 - Nondimensional SIFs under mixed mode loading condition.

3. D E T E R M I N A T I O N OF THE C R A C K INITIATION ANGLES AND THE CRITICAL VALUE OF SIF Disc specimens shown in Fig. 1 were made of HSCM with a constituent combination of 1:0.296 : 0.296:0.254 : 0.250 : 0.290 (cement : sand l (0-150 ~tm) : sand 2 (0-500pm) : sand 3 (0-1 mm) : silica fume : water) mix. The cement content in the mix was 1000 kg/m 3 and the silica fume was 25 percent by weight of cement. The water-binder ratio was 0.23, and a polycarboxylether type of superplasticizer was used. The specimens were cast in a specially designed steel mold, and a 2 mm thick and 60 mm long steel blade was inserted vertically through the thickness of the specimen during casting. Several different steel blades were used to obtain different a/R ratios. The steel blade was removed about 4 hours after casting for easy removal. The specimens were cured in a watertight environment for twenty-four hours and thereafter the molds were removed. The specimens were then cured in lime saturated water for 7 days. Further a water curing period of 2 days at about 90~ they were cured in lime saturated water again for 13 days. At least three specimens from each group were tested under Mode I or mixed Mode types of loading at 22 days. The average compressive strength of HSCM was 110 MPa, and the static modulus of elasticity calculated from the ascending part of stress-strain curve for stress below approximately 33% of the ultimate strength in compression was 31 GPa. For the fracture studies with HSCM subject to pure Mode I loading condition, six different notch ratios (a/R--0.1, 0.2, 0.3, 0.4, 0.5, and 0.6) were chosen. For Mixed Mode loading, specimens were tested with eight inclination angles (]3--0~ 10~ 20 ~ 30 ~ 40 ~ 50 ~ 60~ and 70~ Mixed Mode fractures were obtained by rotating the notch orientation

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H.N. Atahan et al. / Materials' and Structures' 38 (2005) 305-312

angle with respect to the loading direction. For HSCM with fine grain, the notch of 2 mm width was suitable for initiating crack propagation from the notch tip. In the range of inclinations used in this study, the cracks initiated from the notch tip. The crack initiation angle (0) versus crack orientation angle ([3) is plotted in Fig. 5 according to the Maximum Hoop Stress Criterion [8] which can be expressed as follows: Kz 3 0 ., K I I . 0 2 0 cr0 = ---~--cos - - - J - - s i n - - c o s -(4) 2 x]2rcr 2 2

This is an indication of the linearly elastic and brittle behavior of HSCM. The results obtained are very close to those reported in the literature [28]. In Fig. 8, the critical load, P~r, was normalized by using the critical value of SIF, Kin, using disc specimens of different initial notch lengths. As shown in this figure, there is an excellent correlation between the experimental and analytical results. The experimental results obtained for HSCM have

where

i=omfi( / KII = (sm

,

(6)

The determination of crack initiation angle 0o was achieved by maximizing ~0 (Equation (7)) with respect to 0 for a prescribed 13. Fig. 5 - Crack initiation angle (0~ versus notch orientation angle 4~0 2 ~ n r -

-

472=.

- k1(3 cos ~ + cos ~ ) - 3k2 (sin-~ + s i n - ~ )

(13~

(7) where k 1 = K I/c~/~-~a and k 2 = KII/~t~-7. As seen in Fig. 5, the maximum hoop stress theory overestimates the experimental results at values of 13 that are less or greater than 50 ~. The variation of crack initiation angle 0 with notch orientation angle 13for disc specimens is shown in Fig. 6 and a comparison is shown with the experimental results gained for sintered carbide [13] and glass [17]. There is a very good agreement between the theory and experimental result for sintered carbide; the results for glass, however, lie somewhat below the theoretical curve. Similar results were also reported by Shetty et al. [27]. It can be seen that this type of specimen provides a wide range of crack initiation angles. Based on the findings of the Russian researchers [13-15], the initial crack sizes were used to calculate K~and Kn. In pure Mode I loading case, the solution given for the normalized load at fracture as a function of the normalized crack can be expressed as follows;

Fig. 6 - Crack initiationangle (0) versusnotch orientationangle ([3).

3( a/2 4_3(a)6+ 3 ( a ~ 8 / . ~ -

{1+

2t, R )

4~RJ

tx]rtR 6 4 \ R ) JVR = K I ( 8Per )-

Equation (8) can be re-arranged into the linear form of Y=KIX in which Y=f(aJR) and X = t4r~-R-/ Por" As shown in Fig. 7, a linear regression analysis was used to determine the critical value of the SI1~, K~c, from the slope of the line with the correlation coefficient of 0.97. Here, Klc can be taken as a material property. If equation of the line takes the form ofY=Kl X + b, it can be seen that b will be very low.

Fig. 7 - Determination of Klc for HSCM.

309

H.N. Atahan et al. /Materials and Structures 38 (2005) 305-312

/

1

Fig. 8- Normalized crack load Per/~Kgc t4rcR) versus normalized crack length (a/R).

Fig. 9 - Determination of Kic for PMMA (Evaluated from Sanchez [29]). shown that the method of determining Kic described in the present paper can be used for material modeling and application purposes. Similar results have also been obtained for PMMA. In Figs. 9 and 10, the experimental results of Sanchez [29] were evaluated by Tasdemir et al. [30]. As can be seen in these figures, very good correlations were obtained in these results. To obtain the critical value of SIF (Kic) in glass, center notched disc (CND) specimens were used by Tasdemir et al. [17]. As shown in Fig. 11, a linear regression analysis was used to determine the critical value of the SIF, Klo using the slope of the line with the correlation coefficient of 0.98. Here, Kic can be taken as a material property. The results obtained are very close to those reported in the literature [7, 17, 30]. In Fig. 11, the critical load Per was normalized by using the critical value of SIF, K~c, of the disc specimens for different initial notch lengths. As shown in this figure, the experimental and the analytical results show excellent correlation. The experimental results obtained for glass show that the method of determining K~c described here can be used for application purposes.

4. M I X E D - M O D E F R A C T U R E

r""-/

Fig. 10-Normalized crack load Pcr/~KIct4nR ) versus normalized crack length (a/R) for PMMA (Evaluated from Sanchez [29]).

L_ r-:-_~ Fig. 11-Normalized crack load Per/[Kictx/nR) versus normalized crack length (a/R) for glass specimens [ 17]. which the kink-crack starts propagating from the main crack of HSCM specimen. According to the MHSC, the normalized kink initiation stress (Gm/GI) Can be calculated as described below: If Equations (5) and (6) are inserted into Equation (4), the following equation can be obtained: a G0 -

Gmfl(R '13) ~ ~/2nr

a

30

cos - - - 3 2

Gmf2(R -'13) ~ ~/2nr

0

20

sin--cos -- (9) 2 2

In Mode I case (13=0, and 0=0), G0 can be written as follows:

GO--

~xf~(R,O)

(lO)

By inserting Equation (10) into Equation (9), the following equation can be given as:

cYlfI (~-,0) : O'mfl(~-,~)COS a 3 O - 30"mf2 (~-,13) sin O cos20

4.1 Normalized kink initiation stress If the kink initiates when G0 reaches a critical value as postulated by the Maximum Hoop Stress Criterion (MHSC) discussed above, then it is possible to predict the stress at

(11) and then, for HSCM specimens, the normalized kink initiation stress (Om/OI) becomes;

H.N. Atahan et al. /Materials and Structures 38 (2005) 305-312

310

f(R0)

~m = (12) (5"I el(R, 133cos3~- - 3f2(R, 133sin 0~0-cos2 0; The length of pre-existing crack (2a) was 60 mm, and hence a/R=0.4. Eight different inclinations from 0 ~ to 70 ~ were used. As shown in Fig. 12, the comparison between the LEFM based criterion and the experiments is satisfactory for the lower inclination angles (i.e. 13<50~ The theoretical curve passes through a minimum at an inclination of about 30 ~ The experimental results however, exhibit almost a plateau. Further investigations are needed to explain this discrepancy. 4.2 M i x e d - m o d e f r a c t u r e e n v e l o p e

To obtain the Mixed Mode fracture envelope, from Equations (5) and (6), the following equations can be used: a KI - Crm f I"R"" KIC ~I fl (R,0) f

(13)

a

i .1 -- G m 2( ,13) KIC ~I

Fig. 12 -The normalized kink initiation stress (•m/O'i) versus notch orientation angle 13~ for HSCM.

(14)

fl(R,0 )

For the CND, a comparison of these criteria with the experimental results obtained in the present work is shown in Fig. 13. MHSC was used to predict the direction of crack propagation and the stability of crack growth. As seen in Fig. 13, the curve of IKIII/KIC v e r s u s IKII/KIC is plotted according to Maximum Hoop Stress Criterion. In compression - shear, there is a good fit between the curve plotted according to Maximum Hoop Stress Criterion and the test results, but in tension-shear, the theory underestimates the experimental results. Three theoretical criteria were used to predict the direction of crack propagation and the stability of crack growth. For CND specimens, a comparison of these criteria, the experimental results obtained in the present work, as well as the experimental results for sintered carbide, glass, and graphite (published by other researchers [13, 17, 25]) are shown in Fig. 13. Since CND specimens provide a wide range of KI/KI ratios, they are very useful in comparing design based predictive equations for MLxed-Mode fractures. In practice Krc KHc, and it is suggested that the fracture condition is put in the form of: (K r / K I c ) u + (K n/KIIc) u = 1

(15)

where u is a constant between 1.5 and 2 [25]. Fig. 14 compares the normalized Mode I and Mode II SIFs for mixed-mode fracture in CND specimens. It can be concluded that the mixed-mode fracture criteria that were formulated do not satisfactorily account for the CND test results. The experimental results obtained in this work show that Knc/Klc=l.22. On this basis, by taking u=l.9 it is possible

Fig. 13 - Mixed-mode fracture envelope for HSCMs.

Fig. 14 - Mixed-mode fracture envelope for sintered carbide, graphite and glass. to combine the equation Knc=l.22 Ktc and Equation 15 to propose the following fracture envelope KII"9 +0.67K~i 9 = K~c9

(16)

The curve given by Equation (l 6) is also plotted in Fig. 14, and it is seen that in tension-shear, there is a good fit between Equation (16) and the experimental results for glass and the other brittle materials.

H.N. Atahan et al. / Materials and Structures 38 (2005) 305-312

5. CONCLUSIONS The results obtained in this work can be summarized as follows: 1. The critical value of stress intensity factor for HSCM, Kic, can be determined under Mode I loading condition using disc specimens of different pre-existing crack lengths. This critical value can be considered as a material property for a HSCM, which is an ideally elastic and brittle material. 2. The crack initiation angles for HSCM subjected to Mixed Mode load lie somewhat below the theoretical curve according to the Maximum Hoop Stress Criterion. 3. The same testing geometry can be used under MixedMode loading conditions. By changing the notch orientation angle with respect to the loading direction, the mode of fracture can be varied from Mode I to Mixed Mode (tension-shear or compression-shear). The mixed mode fracture envelope of HSCM can be obtained for a very wide range of IKnI/K,ratios. 4. Based on the Maximum Hoop Stress Criterion, the normalized kink initiation stress is used to predict the experimental values for inclination angles lower than 50 ~. For higher inclinations, however, the experimental values lie below the theoretical curve. The normalized kink initiation stress passes through a minimum at an inclination angle of around 30 ~. In general, experimental results exhibit a plateau-like trend. 5. In pure Mode I loading condition, when Kic and the disc geometry are given, it is possible to obtain the graph of normalized critical load versus normalized crack length.

[7] [8] [9] [10] [11] [12] [13]

[14] [15]

[16] [17]

ACKNOWLEDGMENT This study is supported by T C M A (Turkish Cement Manufacturers' Association) grant, UNIPR: 98-5. The authors want to thank T C M A for the financial support. The authors also thank Sedat Erentug and Engin Coban for their help during the preparation of the specimens and the conducting of the mechanical tests.

[18]

[19] [20]

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