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Theoretical and experimental investigations on the mode II fracture toughness of brittle materials
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Theoretical and experimental investigations on the mode II fracture toughness of brittle materials Masoud Sistaninia, Mahjoubeh Sistaninia
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S0020-7403(15)00148-4 http://dx.doi.org/10.1016/j.ijmecsci.2015.04.003 MS2965
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International Journal of Mechanical Sciences
Received date: 4 September 2014 Revised date: 7 March 2015 Accepted date: 7 April 2015 Cite this article as: Masoud Sistaninia, Mahjoubeh Sistaninia, Theoretical and experimental investigations on the mode II fracture toughness of brittle materials, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j. ijmecsci.2015.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical and experimental investigations on the mode II fracture toughness of brittle materials Masoud Sistaniniaa,b,*, Mahjoubeh Sistaniniac a
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences Jahnstrasse 12, A-8700 Leoben, Austria b
Materials Center Leoben Forschung GmbH, Roseggerstrasse 12, A-8700 Leoben
c
Department of Materials Science and Engineering, Shahid Bahonar University of Kerman, P.O. Box No. 76135-133, Kerman, Iran
Abstract Recent theoretical and experimental investigations have shown that the mode II (shear mode) fracture resistance of brittle materials is strongly dependent on the loading conditions, the geometry and the size of components. For this reason, the shear mode fracture resistance of materials and components can be different from that measured from the laboratory fracture tests. To overcome this problem, this paper develops a new approach which can accurately estimate the size- and geometry-dependent shear mode fracture resistance of materials and components using the results obtained from the laboratory tests. Furthermore, a number of fracture tests are conducted in order to validate the predictions of the proposed approach and to experimentally investigate the specimen size effect on the mode II fracture toughness. A comparison between the theoretical predictions of the new approach and the experimental results shows that the proposed approach is able to estimate precisely the size- and geometry-dependent mode II fracture toughness of materials. The proposed analytical approach can be used for the design especially of fracture resistant materials and components.
Keywords: Crack; Mode II loading; Fracture toughness; Size and geometry effects; SCB specimen.
* Corresponding author: Tel.: +43 3842 45922 43; E-mail address: [email protected] (M. Sistaninia).
1
1. Introduction Fracture toughness and crack growth resistance are important material properties which describe the resistance of brittle and ductile materials against crack propagation [1-6]. In many practical applications, mode II (shear mode) fracture plays a key role in the failure process. In these practical applications, the determination of mode II fracture toughness (KIIc) is an indispensable issue for engineers and researchers. Whereas the mode II fracture toughness is a material property which depends on the geometry and the loading condition [7, 8], a standard method for determining mode II fracture toughness has not been yet established. However, many different test specimens have been proposed for determining the mode II fracture toughness of brittle materials. Some of the famous test configurations are the edge cracked semi-circular specimen subjected to three point bending [8-10], the centrally cracked Brazilian disk under diametral compression [8, 11-18], the single-edge crack specimen subjected to asymmetric four point bend loading [19, 20] and the compact shear-tension specimen [21, 22]. A review of literature shows that the edge cracked semi-circular specimen subjected to three point bending (SCB specimen) and the centrally cracked Brazilian disc (BD) specimen have been used most frequently in the mode II fracture analysis of rock types. For example, Khan and Al-Shayea [8], Aliha et al. [7, 11, 23], Al-Shayea [12], Awaji and Sato [14], Chang et al. [15], Krishnan et al. [16], Lanaro et al. [17], Chen et al. [24] and Ke et al. [25] have used the BD specimen to determine and study the mixed mode and mode II fracture toughness of various brittle materials. And also, Lim et al. [10], Khan and Al-Shayea [8], Ayatollahi et al. [7, 9] have applied the SCB specimen for the mode II fracture analysis. There are several advantages in using the SCB and BD specimens for the mode II fracture study of brittle materials, such as: simple geometry and loading configuration, easy test set-up procedure, application of compressive loads rather than tensile loads and the ability of introducing different mode mixities from pure mode I to pure mode II conveniently by changing the crack orientation relative to the direction of applied load. In addition to the mode II test specimens, there are three well-known fracture criteria proposed in the past for estimating the onset of fracture in brittle materials under mode II loading. These criteria are the Maximum Tangential Stress (MTS) criterion [26], the minimum Strain Energy Density (SED) criterion [27-29] and the maximum energy release rate or G criterion [30]. Among of these criteria, the Maximum Tangential Stress (MTS) criterion has 2
received more attentions by the researchers. Several researchers have shown that there is a large discrepancy between the results obtained experimentally from the BD and SCB specimens and the theoretical estimations of the conventional mode II fracture theories MTS, S and G criteria [9, 11, 23, 31-39]. MTS criterion only considers the effect of singular stress terms of William’s series expansion [40] for the elastic tangential stress near the crack tip and neglects the effect of nonsingular stress terms. As the researchers mentioned in the literatures [11, 32-35, 38, 39, 41], the non-singular stress terms (second and third stress terms) in the elastic tangential stress expansion have significant influences on the mode II fracture toughness and by taking into account these stress terms in addition to the singular stress terms, the estimates of MTS criterion can be improved significantly. For example, Ayatollahi and Aliha [34, 35] showed that the GMTS criterion, which takes into account the effect of T-stress (second stress term) in addition to the conventional singular stresses in the MTS criterion, can estimate the mixed mode and mode II fracture toughness in agreement with the experimental results. However, there is still a discrepancy between the GMTS estimates and the experimental results under mixed mode and pure mode II loading. The reason is that the third stress terms also have an influence on the fracture toughness. Recently, Ayatollahi and Sistaninia [38], Sistaninia et al. [42] and Saghafi et al. [39] took into account also the effect of third stress terms in the MTS criterion and proposed a new criterion (MMTS criterion) which could provide more accurate estimates for mixed mode and mode II fracture toughness in compared with GMTS criterion. Aliha et al. [43] have shown that the third order stress term has significant effects not only on the mixed mode and mode II, but also on the mode I fracture toughness. They have shown that the difference in fracture resistance of a rock obtained using the BD and SCB specimens can be related to the effect of the third stress term of the elastic stress field near the crack tip. Khan and Al-Shayea [8], Ayatollahi and Akbardoost [44] and the experimental results, which are presented in this paper, show that the mode II fracture toughness of brittle materials is dependent not only on the geometry and loading conditions but also on the specimen size. Khan and Al-Shayea [8] showed that by increasing the BD specimen diameter from 84 mm to 98 mm, 3
the mode II fracture toughness of a rock (Saudi Arabian limestone) increases by about 17%. The experimental results, which are presented later, also show that by increasing the size of SCB specimen, the mode II fracture toughness of a marble rock (Neiriz rock) increases substantially. Therefore, the mode II fracture toughness values measured from the laboratory specimens are not identical to the mode II fracture toughness of real size structures. Now the question arises, how we can estimate KIIc-values of real size structures or large scale materials from the fracture toughness values measured from the laboratory specimens. It is the aim of the current paper to present a new approach based on the MMTS criterion for predicting the size- and geometrydependent mode II fracture toughness of materials and components from their mode I fracture toughness. In the present paper, first the mode II fracture resistance of a marble rock is studied experimentally using the SCB specimens in four different sizes. Then, the new approach based on the MMTS criterion is introduced to take into account the influence of specimen size and the geometry conditions on KIIc. Finally, the theoretical predictions of the new approach are compared with the experimental results obtained in this paper and the experimental results reported in [8] and [44].
2. Elastic tangential stress around of the crack tip Williams [40] showed that the elastic stresses around the crack tip are written as an infinite series expansion. By considering the first three terms of mode I and mode II in the series expansion, the elastic tangential stress in the vicinity of the crack tip can be written as
σ θθ =
3 KΙ θ 1 3θ 3 K ΙΙ θ 3θ 2 cos + cos − sin + sin + T sin θ + 4 2π r 2 3 2 4 2π r 2 2 θ 1 15 0.5 5θ 15 5θ θ r A3 cos − cos + r 0.5 B3 sin − sin + O(r ), 4 2 5 2 4 2 2
(
)
(1)
where r and θ are the conventional crack tip co-ordinates, see Fig. 1. K Ι and K ΙΙ are the mode I and mode II stress intensity factors, respectively, and their corresponding terms are called the singular stress terms. In Eq. (1), T is a non-singular and constant stress term which is called the 4
T-stress. A3 and B3 are the third order crack parameters. Crack parameters K Ι , T , A3 (the mode I parameters) and K ΙΙ , B3 (the mode II parameters) are generally dependent on the geometry and loading conditions of the cracked specimen.
K Ι , K ΙΙ and T for each cracked configuration can be calculated readily by most commercial finite element codes. However, an independent numerical technique is required for determining the third order crack parameters A3 and B3. The FEOD method proposed by Ayatollahi and Nejati [45, 46] is a numerical technique for determining higher order crack parameters as well as K Ι , K ΙΙ and T. The FEOD method makes it possible to calculate not only the stress intensity factors but also the coefficients of higher order terms by using the displacement field obtained from finite element analysis. In this method, an over-determined set of simultaneous linear equations is obtained for a large number of nodes around the crack tip, and then by using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion can be calculated with a good accuracy for pure mode I, pure mode II and mixed mode I/II conditions. In this paper, the crack parameters of BD and SCB specimens are calculated using the FEOD method. The following section presents the normalized crack parameters of BD and SCB specimens under mode I and mode II loading, which are calculated using FEOD method.
3. BD and SCB specimens Fig. 2 shows the geometry and loading condition of the BD and SCB specimens used for fracture tests. The BD specimen is a circular disc of radius R with a central crack of length 2a and a thickness of t. The SCB specimen is a semi-circular disc of radius R with an edge crack of length a, a thickness of t and a loading support distance of 2S. Both specimens are subjected to a compressive load P. The crack makes an angle α with respect to the loading direction. By changing α, different combinations of mode I and mode II are achieved. When α is zero, the specimens are loaded in pure mode I and the mode II stress intensity factor is zero ( KΙΙ = 0 ). For specific inclination angles α ΙΙ , the mode I stress intensity factor is zero (KI=0) and the specimen 5
is subjected to pure mode II. The α ΙΙ -value for BD specimen, which depends on the crack length ratio a/R, can be found in the literature (i.e. [47]). Ayatollahi and Aliha [48] determined α ΙΙ for the SCB specimen using the finite element method and they showed that α ΙΙ depends on the ratio of a/R and S/R. Since the crack parameters for the BD and SCB specimens are dependent on the geometry and the magnitude of the load, the following equations are used to normalize them,
A1n =
Rt KΙ , P 2π a
(2)
B1n =
Rt K ΙΙ , P 2π a
(3)
Rt T, 4P
(4)
A3n =
Rt a A3 , P
(5)
B3n =
Rt a B3 , P
(6)
A2 n =
where A1n , B1n , A2 n , A3n and B3n are dimensionless geometry factors of the BD and SCB specimens. These dimensionless geometry factors are functions of α and a/R for the BD specimen and are functions of α, a/R and S/R for the SCB specimen. Ι Figs. 3a-3c and 4a-4c show the dimensionless geometry factors A1Ιn , A2Ι n and A3n of the BD Ι and SCB specimens, respectively, where A1Ιn , A2Ι n and A3n are the dimensionless geometry factors
of the specimens under pure mode I loading. As it can be seen in Figs. 3a-3c and 4a-4c, the dimensionless geometry factors of the BD specimen are presented for various crack length ratios and for the SCB specimen, they are presented for various ratios of a/R and S/R. Similarly, Figs. ΙΙ 5a-5d and 6a-6d show the dimensionless geometry factors B1ΙΙn , A2ΙΙn , B3ΙΙn and A3n for the BD (with
various ratios of a/R) and SCB (with various ratios of a/R and S/R) specimens, respectively, 6
ΙΙ where B1ΙΙn , A2ΙΙn , B3ΙΙn and A3n are the dimensionless geometry factors of the specimens under pure
mode II loading. It is worth noting that the superscripts I or II in the geometry factors refer to the mode of loading and the numbers 1, 2, and 3 in the subscripts refer to the corresponding term in the Williams series expansion. As mentioned before, the dimensionless geometry factors (in Figs. 3 to 6) are calculated by “finite element over-deterministic” or FEOD method [45]. It is noted that the dimensionless geometry factors of the BD specimen were calculated before by ΙΙ of the SCB specimen Ayatollahi and Sistaninia [38], but the geometry factors A3Ιn , A3ΙΙn and B3n
are calculated and presented for the first time in this paper. In the following section, the experimental tests are presented. First the mode II fracture tests, which are conducted on the SCB samples with four different radii, are presented and discussed. Then, the mode I fracture test, which is conducted on a different specimen (the BD specimen), is presented. The experimental results will be used later in order to check if the new approach, which will be proposed in section 5, can estimate the mode II fracture toughness of a material from its mode I fracture toughness which is obtained from a different test configuration with a different size.
4. Experimental procedure 4.1. Mode II fracture tests Using the SCB specimens, mode II fracture toughness experiments are performed on a kind of marble rock (Neiriz rock) excavated from Fars province mines in Iran. The structure of this marble rock is relatively homogenous and isotropic. In order to study the effect of specimen size on the mode II fracture toughness, the SCB samples with four different radii are manufactured, approximately R = 30, R = 50, R = 70 and R = 100 mm (see Fig. 7). In order to get sufficient accuracy, mode II fracture toughness for each size is measured four times. Thus, a total number of 16 SCB specimens are prepared for conducting the mode II fracture tests. The SCB specimens are manufactured with a crack length ratio a/R=0.5 and the loading span ratio of S/R=0.5. Table 1 shows the geometries of the SCB specimens. The crack generation in the specimens is performed as follows; firstly a thin notch of 0.8 mm width is generated by using a 7
high-precision CNC water jet cutting machine and then the notch tip is sharpened using a saw blade with the thickness of 0.2 mm. The crack is generated in the appropriate angle of
α ΙΙ = 40.5 (the pure mode II angle of SCB specimen with the ratios a/R=0.5 and S/R=0.5). The prepared samples are then tested using a universal tension/compression test machine with a load cell capacity of 15 kN. The tests are carried out under displacement control conditions with a constant rate of 0.05 mm/min. The specimens are tested using a three-point bend fixture with the loading span ratio S/R=0.5. Fig. 8a shows the loading setup for the SCB specimens. For each specimen the load versus displacement curve is recorded until the final fracture, the fractured samples are shown in Fig. 8b. Fig. 9 shows a sample load–displacement curve corresponding to one of the SCB specimens with R = 100 mm. The absolute values of load and displacement have been shown in this figure. As it can be seen, the specimen is loaded until a maximum load of 11.3 kN that is reached when the actuator of the test machine moves down to 0.62 mm. After the load reaches its maximum value ( Pcr ), the specimen is suddenly fractured in a brittle manner initiating from the crack tip, the fracture load (Pcr) of the specimen is determined as 11.3 kN. It was observed that all the samples showed a linear curve and fractured suddenly in a brittle manner like in Fig. 9. It is worth mentioning that for ideally brittle materials, the fracture occurs immediately after initiation of the crack growth. Indeed, for ideally brittle materials, catastrophic failure or unstable crack propagation takes place immediately once the stress intensity factor at the initial crack tip reaches the fracture initiation toughness (the fracture toughness at the onset of crack extension). This is the reason why the fracture initiation toughness of ideally brittle materials is nearly equal to their critical value of fracture toughness or KIIc (fracture toughness at the maximum load, beyond which the specimen fails) [1-3]. The mode II fracture resistance (KIIc) of the tested SCB specimens is determined from the following equation
K ΙΙc =
Pcr 2π a ΙΙ B1n , Rt
8
(7)
where Pcr is the fracture load and B1nΙΙ is the geometry factor of the SCB specimen which is presented in Fig. 6a. By replacing the geometry factor B1ΙΙn = 0.434 ( B1nΙΙ for the SCB specimen with a/R=0.5 and S/R=0.5 is 0.434) and Pcr (obtained from the fracture test) in Eq. (7), the mode II fracture toughness of Neiriz rock obtained using the SCB specimen is determined. Table 1 shows the mode II fracture toughness of Neiriz rock obtained from SCB specimens with four different sizes. As an average of four test results, the KIIc-values are measured to be 0.53, 0.70, 0.77 and 0.90 MPa√m from the samples with R = 30, 50, 70 and 100 mm, respectively. There is an increase of about 70% in the mode II fracture toughness when the radius is increased from 30 mm to 100 mm. The experimental results show that the test specimen size has a noticeable influence on the mode II fracture resistance of materials. 4.2. Mode I fracture and tensile strength tests
Mode I fracture toughness of Neiriz rock is measured by averaging the results obtained from the pure mode I tests conducted on the BD specimen. The BD specimens are manufactured with a crack length ratio a/R=0.5. Table 2 shows the geometries of the BD specimens conducted for mode I fracture tests. First, the crack is generated in the specimens as explained in the previous subsection. Then, the prepared samples are loaded using two flat loading fixtures when the direction of the applied load is along the crack line, the crack is simultaneously subjected to pure mode I (or opening) deformation. Fig. 10 shows a sample BD specimen inside the test machine before and after fracture test. The tests are carried out under displacement control conditions with a constant rate of 0.05 mm/min. After measuring the fracture load Pcr from the fracture tests, the mode I fracture resistance (KIc) of the tested BD specimens is determined from the following equation
K Ιc =
Pcr 2π a Ι A1n , Rt
(8)
where A1nΙ is the geometry factor of the BD specimen which is plotted versus a/R in Fig. 3a. By replacing the geometry factor A1Ιn = 0.313 ( A1nΙ for the BD specimen with a/R=0.5 is 0.313) and Pcr (obtained from the fracture tests) in Eq. (8), the mode I fracture toughness of Neiriz rock is 9
determined. As an average of four test results, the mode I fracture toughness KIC of Neiriz rock is determined as 1 MPa√m. The uncracked Brazilian disk subjected to compressive load [8] is used for obtaining the tensile strength of Neiriz rock. The uncracked Brazilian disk specimens are manufactured with a radius of 51 mm. Table 3 shows the geometries of the specimens conducted for obtaining the tensile strength. The specimens are loaded using two flat loading fixtures, which can be seen in Fig. 11a, under displacement control conditions. The load-displacement curve is demonstrated in Fig. 11b taken from the test record. Initially, the specimen has a non-linear behaviour caused probably by compression of existed porosity and micro-cracking of the specimen. Then follows a period of linear ascending, the load reaches its maximum, which corresponds to the initiation of a crack in the centre of the specimen. Then, the crack propagates unstably at this stage and the specimen fractures suddenly. After measuring the fracture load Pcr from the tests, the tensile strength σt can be determined as
σt =
Pcr . π Rt
(9)
The tensile strength (σt) of Neiriz rock is determined to be 5.70 MPa from an average of four uncracked Brazilian disk specimen tests. Table 3 summarizes the results of σt for Neiriz rock.
5. Brittle fracture criterion A new approach which is based on the MMTS criterion is proposed here in order to estimate the mode II fracture toughness (KIIc) of brittle materials for any configuration and size from their mode I fracture toughness data. It should be mentioned that the MMTS criterion is able to predict the fracture initiation toughness, not the critical value of fracture toughness (KIIc). However, as explained before, for an ideally brittle material, the fracture initiation toughness is equal to KIIc. The maximum tangential stress (MTS) criterion was first proposed by Erdogan and Sih [26] for brittle fracture in mixed mode I/II crack problems. According to MTS criterion, crack 10
growth initiates radially from the crack tip along the direction of maximum tangential stress θm. The crack growth takes place when the tangential stress σθθ along θm and at a critical distance rc from the crack tip attains a critical value σθθc. rc is the size of fracture process zone, and σθθc is the tensile strength of the material (σt), which is a material property. The classical MTS criterion only considers the effect of the singular stress terms and the non-singular stress terms are neglected in the MTS criterion. Ayatollahi and Sistaninia [38] and Saghafi et al. [39] showed that the first three terms of William’s series expansion for elastic stresses near the crack tip have to be taken into account in MTS criterion in order to provide more accurate estimates for mode II fracture toughness of brittle materials. By taking into account the first three terms of William’s series expansion for tangential stress, the tangential stress in pure mode II loading can be written as 15 0.5 ΙΙ θ 1 5θ r A3 cos − cos 4 2 5 2 3 K ΙΙ θ 3θ 15 0.5 ΙΙ θ 5θ − sin + sin + r B3 sin − sin . 4 2π r 2 2 4 2 2
σ θθΙΙ (r ,θ ) = T ΙΙ ( sin 2 θ ) +
(10)
K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ are the crack parameters for pure mode II loading. The angle of maximum
tangential stress θm is determined from
∂σ θθΙΙ θ 3θ 3 1 =− K ΙΙ 0.5cos m + 1.5 cos m + T ΙΙ ( sin 2θ m ) + 4 2π rc 2 2 ∂θ θ =θm
θ 5θ 15 rc A3ΙΙ −0.5sin m + 0.5sin m 4 2 2
(11)
θm 5θ m 15 ΙΙ + 4 rc B3 0.5cos 2 − 2.5 cos 2 = 0.
The fracture initiation angle θm for each specimen can be determined by solving Eq. (11). Since the crack parameters K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ are functions of geometry and loading condition of specimen, it is preferred to present Eq. (11) in terms of the normalized crack parameters. By replacing Eqs. (2)-(6) into Eq. (11), Eq. (11) can be rewritten as
11
−
θ 3θ 3 a ΙΙ B1n 0.5cos m + 1.5cos m + 4 A2ΙΙn ( sin 2θ m ) + 4 rc 2 2
(12)
5θ 15 rc ΙΙ 5θ θ θ 15 rc ΙΙ A3 n −0.5sin m + 0.5sin m + B3 n 0.5 cos m − 2.5cos m = 0. 4 a 2 2 4 a 2 2 By replacing the pure mode II geometry factors of each cracked body into Eq. (12), the fracture initiation angle ( θ m ) is estimated by the MMTS criterion in terms of a / rc . According to the MMTS criterion, brittle fracture takes place when the tangential stress
σθθ along θm and at a critical distance rc from the crack tip attains the critical value σθθc or σt. Thus, in a test configuration x which is under pure mode II loading, the mode II brittle fracture occurs when
σ θθΙΙ (rc ,θ m ) |x = σ t .
(13)
By replacing rc and θm into Eq. (10), for the test configuration x, one can write
θ mx 3θ mx θ mx 1 5θ mx 3 1 15 x ΙΙ , x 2 x ΙΙ , x − K ΙΙc sin + sin rc A3c cos − cos + Tc ( sin θ m ) + 4 2π rc 2 2 4 2 5 2 θ mx 5θ mx 15 ΙΙ , x + rc B3c sin − sin =σt, 4 2 2
(14)
where K ΙΙxc , TcΙΙ , x , B3ΙΙc, x , A3ΙΙc, x are the critical values of K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ corresponding to the fracture load for the test configuration x. It is noted that in Eq. (14), the superscript x refers to the test configuration x. Since σt is a material constant and is independent of the test configuration, mode mixity and the specimen size, an equation similar to Eq. (13) can be written for mode I brittle fracture of a test configuration y which is made of the same material as the test configuration x, but can have a different size. For mode I loading, KII and B3 are zero and the fracture initiation angle ( θ m ) is determined from Eq. (1) to be zero. Therefore, the mode I brittle fracture occurs in the test configuration y when
σ θθΙ ( rc , 0) | y = σ t →
1 K Ιyc + 3 rc A3Ιc, y = σ t , 2π rc
12
(15)
where K Ιyc , A3Ιc, y are the critical values of the mode I crack parameters K Ι and A3Ι corresponding to the fracture load for the test configuration y. Since the right side of Eqs. (14) and (15) is a constant material property, therefore, Eq. (15) can be replaced into Eq. (14) and the ratio of mode II fracture toughness of the test configuration x to mode I fracture toughness of the test configuration y, K ΙΙxc / K Ιyc , can be determined as
K ΙΙxc 1 A3Ιc, y = + 3 r c K Ιyc K Ιyc 2π rc
3 1 θ mx 3θ mx TcΙΙ , x 2 x − + sin sin + x ( sin θ m ) + 2 2 K ΙΙc 4 2π rc (16) ΙΙ , x x x ΙΙ , x x x 15 rc A3c cos θ m − 1 cos 5θ m + 15 rc B3c sin θ m − sin 5θ m 4 K ΙΙxc 2 5 2 4 K ΙΙxc 2 2
In Eq. (16), K Ιyc and A3Ιc, y are proportional to the fracture load, but the ratio of A3Ιc, y K Ιyc is independent of the fracture load. The ratio A3Ιc, y K Ιyc can be simplified as A3Ιc, y 1 A3Ιn, y = Ι, y K Ιyc 2π A1n
1 , a
(17)
where A1Ιn, y , A3Ιn, y are the geometry factors of the test configuration y. Similarly, K ΙΙxc , TcΙΙ , x , B3ΙΙc, x and A3ΙΙc, x are proportional to the fracture load, but the ratios TcΙΙ , x K ΙΙxc , B3ΙΙc K ΙΙxc , A3ΙΙc K ΙΙxc are
independent of the fracture load. These ratios can be rewritten as
TcΙΙ , x 4 A2ΙΙn, x 1 = ΙΙ , x K ΙΙxc 2π B1n a A3ΙΙc, x 1 A3ΙΙn, x 1 = , ΙΙ , x K ΙΙxc 2π B1n a
(18)
B3ΙΙc, x 1 B3ΙΙn, x 1 = ΙΙ , x K ΙΙxc 2π B1n a where B1ΙΙn, x , A2ΙΙn, x , B3ΙΙn, x and A3ΙΙn, x are the geometry factors of the test configuration x. By replacing Eqs. (17) and (18) into Eq. (16), the fracture toughness ratio K ΙΙxc / K Ιyc is rewritten as
13
x ΙΙc y Ιc
K K
r = 1 + 3 c a y
A A Ι, y 3n Ι, y 1n
3 θ mx 3θ mx rc A2ΙΙn, x + 4 sin 2 θ mx ) + − sin + sin ΙΙ , x ( 4 2 2 a B x 1n r AΙΙ , x x x ΙΙ , x 15 c 3ΙΙn, x cos θ m − 1 cos 5θ m + 15 rc B3ΙΙn, x 2 5 2 4 ax B1n 4 ax B1n
, (19) x x θm 5θ m sin − sin 2 2
where ay and ax are the crack lengths of the test configurations y and x, respectively. If the
(
)
geometry factors of the test configuration y A1Ιn, y , A3Ιn, y , the geometry factors of the test
(
)
configuration x B1ΙΙn, x , A2ΙΙn, x , A3ΙΙn, x , B3ΙΙn, x and the fracture initiation angle θ mx (which can be determined from Eq. (12)) are inserted in Eq. (19), the fracture toughness ratio K ΙΙxc / K Ιyc is determined in terms of a x / rc (or Rx / rc ) and a y / rc (or R y / rc ), the size of specimen and the crack length are related by the crack length ratio a/R. In order to use the proposed approach, rc must be known. A formulation based on the maximum principal stress theory has been suggested in [1, 49, 50] for evaluating rc . According to this model, rc can be calculated from
1 rc = 2π
2
K Ιc . σ t
(20)
Since the mode I fracture resistance K Ιc is dependent on the size and the geometry and loading conditions of the test specimen [43, 51], a shortcoming of Eq. (20) is that the effects of size, geometry and loading conditions of the test specimen are not taken into account. As it is shown in [43], when A3I is non-zero, K Ιc takes different values depending on the size and loading and geometry conditions of the specimen. Therefore, the effects of size, geometry and loading conditions of the test specimen on K Ιc , and the resulting rc , can be taken into account if one considers the effect of A3I for the tangential stress near the crack tip. For this reason, rc should be determined by solving the non-linear Eq. (15). When A3I = 0 , Eq. (15) becomes similar to Eq. (20). It is noted that Eq. (15) might have multiple solutions, but the lowest positive value is physically acceptable.
14
In summary, in order to estimate the mode II fracture toughness K ΙΙc of materials and structures with any configuration and size, first a tensile strength test specimen and a pure mode I fracture specimen (it can have any size or configuration) are manufactured and tested in the laboratory. Then, after measuring the fracture load Pcr of mode I test and the tensile strength σt (from the tensile test), K Ιc and A3cI are determined from
K Ιc = I 3c
A =
Pcr 2π a A1nI Rt , I Pcr A3n
(21)
Rt a
and then by inserting σt, K Ιc and A3cI in Eq. (15), rc can be determined by solving Eq. (15). After that, K ΙΙc of the material is readily estimated by Eq. (19). In the following section, a comparison between the experimental results and predictions of the proposed criterion demonstrates that the proposed criterion can estimate the mode II fracture resistance very well.
6. Results and discussion In [7, 8, 33, 39, 43, 44], it was shown that the mode II fracture toughness is a material property which depends also on the geometry of specimen and the type of loading. The experimental results in the current paper also show that the specimen size has a significant influence on the mode II fracture toughness. Therefore, the mode II fracture toughness of real size structures is not identical to that measured from the laboratory fracture tests. In the case of practical applications, it is essential to use an appropriate criterion for estimating the mode II fracture toughness of structures using the results obtained from the laboratory specimens. The new approach proposed in Section 5 is able to satisfy this object. In this section, the mode II fracture toughness results reported in [8, 44] and the experimental results presented in this paper are used to verify the estimates of the new proposed approach. Khan and Al-Shayea [8] used the SCB specimen with R = 49 mm and the ratios a/R = 0.3
and S/R = 0.8 to determine the mode I fracture toughness of Saudi Arabian limestone. As 15
reported in [8], the mode I fracture toughness measured by this test configuration is , R = 49 = 0.68 MPa m ). They also used the BD specimens with two different 0.68 MPa m ( K ΙSCB c
radii and crack lengths (a specimen with R = 49 mm and a/R = 0.3 and another specimen with R=42 mm and a/R = 0.4) for determining the mode II fracture toughness of Saudi Arabian
limestone. The average mode II fracture toughness obtained by the BD specimen with the radius R = 49 mm is noticeably higher than that of the BD specimen with the radius R = 42 mm. The
mode II fracture toughness value for the BD specimen with R=42 mm is K ΙΙBDc , R = 42 = 0.79 , R = 49 MPa√m, but the KIIc-value measured by the specimen with R=49 mm is K ΙΙBD = 0.92 MPa√m. c
, R =49 , R = 49 The fracture toughness ratios K ΙΙBDc ,R =42 / K ΙSCB and K ΙΙBDc , R =49 / K ΙSCB are 1.16 and 1.3, c c
respectively. If the SCB specimen with R = 42 mm, a/R = 0.3 and S/R=0.8 under mode I loading is considered as the test configuration y and the BD specimens with R = 42 mm, a/R = 0.4 and R = 49 mm, a/R = 0.3 under mode II loading are considered as test configurations x, the fracture , R = 42 , R = 49 , R = 49 , R = 49 toughness ratios K ΙΙBD / K ΙSCB and K ΙΙBD / K ΙSCB can be analytically determined by c c c c
Eq. (19), rc of Saudi Arabian limestone is 5.8 mm (determined by Eq. (15)). Table 4 compares the experimental results and the analytical predictions of the new approach for the fracture toughness ratios of Saudi Arabian limestone. It is seen that there is a good agreement between the experimental results and the predictions of the new approach. The average mode II fracture toughness of Neiriz rock obtained using SCB specimen (with a/R = 0.5 and S/R = 0.5) with four different sizes R = 30, 50, 70 and 100 mm are 0.53, 0.70, 0.77 and 0.90 MPa√m, respectively, (see Table 1). Using the BD specimen with R=50 mm and a/R = 0.5, the mode I fracture toughness of Neiriz rock is measured to be 1 MPa√m, see Table 2. According to the experimental results, the fracture toughness ratio K ΙΙc / K Ιc is 0.53, 0.70, 0.77 and 0.90 for four different specimen sizes R = 30, 50, 70 and 100 mm, respectively. If the BD specimen with a/R = 0.5 and R = 50 mm under mode I loading is considered as the test configuration y and the SCB specimens (with a/R = S/R= 0.5) with the radii R = 30, 50, 70 and 100 mm under mode II loading are considered as the test configurations x, the fracture toughness ratio K ΙΙxc / K Ιyc for Neiriz rock is analytically estimated from Eq. (19) by inserting the dimensionless crack parameters (extracted from Figs. 3 to 6), the crack initiation angle θm 16
(obtained from Eq. (12)) and the critical distance rc . rc of Neiriz rock is 6.6 mm, which is obtained from Eq. (15). Fig. 12 compares the fracture toughness ratio K ΙΙxc / K Ιyc estimated by the new approach (Eq. (19)) to those obtained by the experimental tests, K ΙΙxc / K Ιyc is plotted versus the radius of the mode II fracture test (Rx). As it can be seen, there is a good coincidence between the analytical predictions of the new approach and the experimental results; the maximum discrepancy is 7%. Our experimental results show that the new proposed approach can accurately estimate the mode II fracture resistance of a material from its mode I fracture toughness which is obtained from a different test configuration with a different size (smaller or larger). As it can be seen in Fig. 12, the mode II fracture toughness K ΙΙxc ( or K ΙΙxc / K Ιyc ) strongly increases with increasing the specimen size until it approaches a saturation value when the specimen size is very large. Indeed, for very large specimens, the mode II fracture resistance K ΙΙc becomes independent of specimen size and it can be considered as the inherent mode II fracture resistance of the material. The size-independent value of K ΙΙxc / K Ιyc of Neiriz rock is 1.2, see Fig. 12. Since K Ιyc = 1 MPa m , the size-independent mode II fracture resistance of Neiriz rock is determined to be 1.2 MPa√m, which can be reached when Rx ≥ 200 mm. It should be mentioned that if both the mode I and mode II fracture tests are infinitely large, the size-independent value of K ΙΙxc / K Ιyc becomes 0.866 which is the same value suggested by the conventional MTS criterion. Recently, Ayatollahi and Akbardoost [44] proposed a new criterion in order to predict the size-dependent mode II fracture toughness of rock materials. Although their proposed criterion can predict the mode II fracture toughness in agreement with the experimental results, the criterion is complicated compared to the approach proposed in this paper. The reason is that their proposed criterion needs a set of mode I experimental tests for different sizes in order to predict the mode II fracture resistance of a given material, while the new approach proposed in this paper needs only one mode I experiment. They also conducted experimental tests on the SCB samples of different radii made of a marble (Ghorveh marble) in order to experimentally investigate the influence of the specimen size on K ΙΙc . The mode II fracture tests were conducted 17
on the SCB samples with S/R = a/R = 0.5 and four different radii R = 25, 50, 95 and 190 mm. As reported, the K ΙΙc values were measured to be 0.55, 0.71, 0.82 and 0.91 MPa√m from the samples with R = 25, 50, 95 and 190 mm, respectively. The K Ιc of Ghorveh marble was measured to be 1.32 MPa√m from the SCB sample with S/R = a/R = 0.5 and R = 50 mm. The tensile strength σt of Ghorveh marble was measured as 5.37 MPa. The critical distance rc can be determined as 4.49 mm by Eq. (15). Fig. 13 compares the fracture toughness ratio K ΙΙxc / K Ιyc of Ghorveh marble estimated by the new approach (Eq. (19)) to those obtained by the experimental tests. It is seen that the proposed approach predicts the size dependent fracture resistance of the SCB specimens under mode II loading very well even for large specimens. Both analytical and experimental results show that the mode II fracture toughness K ΙΙxc reaches a constant value when the specimen size is large, Rx ≥ 200 mm. For convenient using of the new approach, Fig. 14 is prepared which shows the fracture toughness ratio K ΙΙxc / K Ιyc versus the normalized crack length ax / rc (ax is the crack length of the mode II fracture test). Fig. 14a displays the K ΙΙxc / K Ιyc -values plotted against ax / rc for different values of the normalized crack length a y / rc (ay is the crack length of the mode I fracture test), where the mode II fracture test is the SCB specimen with S/R = a/R = 0.5 and the mode I fracture test is the BD specimen with a/R = 0.5. As discussed before, the mode II fracture toughness K ΙΙxc strongly increases with the specimen size until it approaches a saturation value when the specimen size is very large. When the dimensionless parameter ax / rc is kept constant, K ΙΙxc / K Ιyc decreases with increasing a y / rc , because K Ιyc increases with increasing a y or the mode I specimen size. In Fig. 14b, the K ΙΙxc / K Ιyc -values are plotted against ax / rc for four different values of S/R, where a y / rc = 0.5 and the mode II fracture test is the SCB specimen with a/R = 0.5 and the mode I fracture test is the BD specimen with a/R = 0.5. This figure shows the influence of the support condition on the mode II fracture resistance of materials. The mode II fracture resistance increases with decreasing the support distance S. However, when the size of specimen becomes large (approximately when ax / rc > 12 ), the mode II fracture resistance K ΙΙxc 18
strongly decreases with decreasing the support distance S. Furthermore, the influence of the crack length ratio a/R on the mode II fracture resistance of materials can be seen in Fig. 14c. Fig. 14c shows the K ΙΙxc / K Ιyc -values plotted against ax / rc for four different values of a/R, where the mode II fracture test is the SCB specimen with S/R = 0.5 and the mode I fracture test is the same as before (the BD specimen with a/R = 0.5) and a y / rc = 0.5 . As one can see, when ax / rc is kept constant, the mode II fracture resistance increases with increasing a/R, but when ax / rc > 15 , the mode II fracture resistance decreases with increasing a/R. As shown in this section, the new approach can precisely predict the size- and geometrydependent mode II fracture resistance of materials from their mode I fracture test. The theoretical predictions of the new approach were compared to the experimental results of three different rock materials and there was a good agreement between the theoretical predictions and the experimental results at all three cases. Finally, it should be remarked that since the geometry and loading conditions influence the mode II fracture toughness of materials, fracture resistant materials and structures can be designed by this fact. In order to work out design rules of more fracture resistant materials, it is necessary to quantify the influence of the geometry and loading conditions on the mode II fracture toughness. This will be investigated extensively in a separate paper.
7. Conclusion •
The mode II fracture toughness of a marble rock (Neiriz rock) is investigated experimentally by means of the SCB specimens in four different sizes.
•
The experimental results show that the specimen size has a substantial influence on the mode II fracture toughness of Neiriz rock. The mode II fracture toughness of Neiriz rock obtained using the SCB specimen increases with the size of specimen.
•
A new approach is proposed to estimate the size- and geometry-dependent mode II fracture toughness of materials and components from their mode I fracture data measured from the laboratory fracture tests. 19
•
Comparison between the theoretical predictions of the new approach and the experimental results of three different rock materials showed that the proposed approach is able to estimate precisely the size- and geometry-dependent mode II fracture toughness of brittle materials.
References [1] Anderson TL. Fracture mechanics: fundamentals and applications. 3rd ed. Boca Raton: Taylor & Francis; 2005. [2] Gao DY, Ogden RW. Advances in Mechanics and Mathematics: Springer; 2003. [3] Kolednik O, Predan J, Fischer FD, Fratzl P. Improvements of strength and fracture resistance by spatial material property variations. Acta Materialia. 2014;68:279-94. [4] Kolednik O, Predan J, Gubeljak N, Fischer DF. Modeling fatigue crack growth in a bimaterial specimen with the configurational forces concept. Materials Science and Engineering A. 2009;519:172-83. [5] Sistaninia M, Kolednik O. Effect of a single soft interlayer on the crack driving force. Engineering Fracture Mechanics. 2014;130:21–41. [6] Xia Z, Curtin WA, Sheldon BW. A new method to evaluate the fracture toughness of thin films. Acta Materialia. 2004;52:3507-17. [7] Aliha MRM, Ayatollah MR, Smith DJ, Pavier MJ. Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Engineering Fracture Mechanics. 2010;77:2200-12. [8] Khan K, Al-Shayea NA. Effect of specimen geometry and testing method on mixed Mode III fracture toughness of a limestone rock from Saudi Arabia. Rock Mechanics and Rock Engineering. 2000;33:179-206. [9] Ayatollahi MR, Aliha MRM, Hassani MM. Mixed mode brittle fracture in PMMA - An experimental study using SCB specimens. Materials Science and Engineering A. 2006;417:34856. [10] Lim IL, Johnston IW, Choi SK, Boland JN. Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2-mixed-mode. International Journal of Rock Mechanics and Mining Sciences and. 1994;31:199-212. [11] Aliha MRM, Ashtari R, Ayatollahi MR. Mode I and mode II fracture toughness testing for a coarse grain marble. Applied Mechanics and Materials. 2006;5-6:181-8. [12] Al-Shayea N. Comparing reservoir and outcrop specimens for mixed mode I-II fracture toughness of a limestone rock formation at various conditions. Rock Mechanics and Rock Engineering. 2002;35:271-97. 20
[13] Atkinson C, Smelser RE, Sanchez J. Combined mode fracture via the cracked Brazilian disk test. International Journal of Fracture. 1982;18:279-91. [14] Awaji H, Sato S. Combined mode fracture toughness measurement by the disk test. Journal of Engineering Materials and Technology, Transactions of the ASME. 1978;100:175-82. [15] Chang SH, Lee CI, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Engineering Geology. 2002;66:79-97. [16] Krishnan GR, Zhao XL, Zaman M, Roegiers JC. Fracture toughness of a soft sandstone. International Journal of Rock Mechanics and Mining Sciences. 1998;35:695-710. [17] Lanaro F, Sato T, Stephansson O. Microcrack modelling of Brazilian tensile tests with the boundary element method. International Journal of Rock Mechanics and Mining Sciences. 2009;46:450-61. [18] Shetty DK, Rosenfield AR, Duckworth WH. Mixed-mode fracture in biaxial stress state: Application of the diametral-compression (Brazilian disk) test. Engineering Fracture Mechanics. 1987;26:825-40. [19] Margevicius RW, Riedle J, Gumbsch P. Fracture toughness of polycrystalline tungsten under mode I and mixed mode I/II loading. Materials Science and Engineering A. 1999;270:197209. [20] Suresh S, Shih CF, Morrone A, O'Dowd NP. Mixed-mode fracture toughness of ceramic materials. Journal of the American Ceramic Society. 1990;73:1257-67. [21] Richard HA, Benitz K. A loading device for the creation of mixed mode in fracture mechanics. International Journal of Fracture. 1983;22:R55-R8. [22] Zipf Jr RK, Bieniawski ZT. Mixed mode testing for fracture toughness of coal based on critical-energy-density. 1986. p. 16-23. [23] Aliha MRM, Ayatollahi MR. On mixed-mode I/II crack growth in dental resin materials. Scripta Materialia. 2008;59:258-61. [24] Chen CS, Pan E, Amadei B. Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method. International Journal of Rock Mechanics and Mining Sciences. 1998;35:195-218. [25] Ke CC, Chen CS, Tu CH. Determination of fracture toughness of anisotropic rocks by boundary element method. Rock Mechanics and Rock Engineering. 2008;41:509-38. [26] Erdogan F, Sih GC. On the Crack Extension in Plates under Plane Loading and Transverse Shear. J Basic Eng Trans ASME. 1963;85:519–25. [27] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture. 1974;10:305-21. [28] Berto F, Ayatollahi MR. Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy. Materials & Design. 2011;32:2858-69.
21
[29] Berto F, Barati E. Fracture assessment of U-notches under three point bending by means of local energy density. Materials & Design. 2011;32:822-30. [30] Hussain MA, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II. American Society for Testing and Materials. Philadelphia1974. p. 2-28. [31] Aliha MRM, Ayatollahi MR. Brittle fracture evaluation of a fine grain cement mortar in combined tensile-shear deformation. Fatigue and Fracture of Engineering Materials and Structures. 2009;32:987-94. [32] Ayatollahi MR, Aliha MRM. Cracked Brazilian disc specimen subjected to mode II deformation. Engineering Fracture Mechanics. 2005;72:493-503. [33] Ayatollahi MR, Aliha MRM. On determination of mode II fracture toughness using semicircular bend specimen. International Journal of Solids and Structures. 2006;43:5217-27. [34] Ayatollahi MR, Aliha MRM. Fracture toughness study for a brittle rock subjected to mixed mode I/II loading. International Journal of Rock Mechanics and Mining Sciences. 2007;44:61724. [35] Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials. Engineering Fracture Mechanics. 2008;75:463141. [36] Ayatollahi MR, Aliha MRM. Mixed mode fracture analysis of polycrystalline graphite - A modified MTS criterion. Carbon. 2008;46:1302-8. [37] Ayatollahi MR, Aliha MRM. Mixed mode fracture in soda lime glass analyzed by using the generalized MTS criterion. International Journal of Solids and Structures. 2009;46:311-21. [38] Ayatollahi MR, Sistaninia M. Mode II fracture study of rocks using Brazilian disk specimens. International Journal of Rock Mechanics and Mining Sciences. 2011;48:819-26. [39] Saghafi H, Ayatollahi MR, Sistaninia M. A modified MTS criterion (MMTS) for mixedmode fracture toughness assessment of brittle materials. Materials Science and Engineering A. 2010;527:5624-30. [40] Williams ML. On the Stress Distribution at the Base of a Stationary Crack. Journal of Applied Mechanics. 1957;24:109-14. [41] Mirsayar MM. On fracture of kinked interface cracks – The role of T-stress. Materials & Design. 2014;61:117-23. [42] Sistaninia M, Ayatollahi MR, Sistaninia M. On Fracture Analysis of Cracked Graphite Components under Mixed Mode Loading. Mechanics of Advanced Materials and Structures. 2014;21:781-91. [43] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. International Journal of Rock Mechanics and Mining Sciences. 2012;51:128-35. [44] Ayatollahi MR, Akbardoost J. Size effects in mode II brittle fracture of rocks. Engineering Fracture Mechanics. 2013;112-113:165-80. 22
[45] Ayatollahi MR, Nejati M. An over-deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis. Fatigue Fract Eng Mater Struct. 2011;34:159-76. [46] Ayatollahi MR, Nejati M. Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method. International Journal of Mechanical Sciences. 2011;53:164-77. [47] Dong SM, Wang Y, Xia YM. Stress intensity factors for central cracked circular disk subjected to compression. Engineering Fracture Mechanics. 2004;71:1135-48. [48] Ayatollahi MR, Aliha MRM. Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Computational Materials Science. 2007;38:660-70. [49] Schmidt RA. A microcrack model and its significance to hydraulic fracturing and fracture toughness testing. Proceedings of the 21st US rock mechanics symposium; 1980, p. 581-90. [50] Ayatollahi MR, Torabi AR. Brittle fracture in rounded-tip V-shaped notches. Materials & Design. 2010;31:60-7. [51] Ayatollahi MR, Akbardoost J. Size and geometry effects on rock fracture toughness: Mode i fracture. Rock Mechanics and Rock Engineering. 2014;47:677-87.
23
Table 1 Summery of mode II fracture tests conducted on SCB specimens manufactured from Neiriz rock. Sample no.
R (mm)
a/ R
S/R
t (mm)
Pcr (kN)
1 2 3 4
31.2 29.6 29.3 30.7
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
20.2 20.1 19.8 18.6
2.11 2.64 2.30 2.48
KIIc (MPa√m) 0.45 0.58 0.52 0.58
5.27 5.82 5.68 4.73
0.53 0.68 0.75 0.72 0.65
8.20 7.47 7.06 7.04
0.70 0.72 0.78 0.78 0.8
11.3 10.6 10.33 12.43
0.77 1.00 0.78 0.85 0.94
Average 1 2 3 4 Average 1 2 3 4 Average 1 2 3 4
49.6 50.4 50.2 49.4
71.5 70.0 70.0 70.0
97.0 98.7 98.5 101.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
26.7 26.5 27.0 25.0
0.5 0.5 0.5 0.5
32.6 28.0 25.8 25.8
0.5 0.5 0.5 0.5
28.0 33.7 30.0 32.1
Average
0.90
24
Table 2 Summary of mode I fracture tests conducted on BD specimens manufactured from Neiriz rock. Sample no.
R (mm)
a/ R
t (mm)
Pcr (kN)
1 2 3 4 Average
52.6 51.3 51.2 51.2
0.5 0.5 0.5 0.5
28.0 24.0 26.0 26.0
12.81 10.46 10.64 8.84
KIc (MPa√m) 1.11 1.07 1.00 0.83 1.00
Table 3 Summary of tensile strength tests conducted on uncracked BD specimens manufactured from Neiriz rock. Sample no. 1 2 3 4 Average
R (mm) 51 51 51 51
t (mm) 23.0 24.0 23.0 25.6
σt (MPa) 5.65 6.58 5.98 4.64 5.70
Pcr (kN) 20.82 25.31 22.06 19.06
Table 4 The fracture toughness ratio K ΙΙc / K Ιc predicted for Saudi Arabian limestone using the new proposed approach. Rock Material
Saudi Arabian limestone
rc (mm) 5.8
, R = 49 K ΙΙBDc , R=42 / K ΙSCB c
, R = 49 K ΙΙBDc , R =49 / K ΙSCB c
Experiment
Analytical prediction
Experiment
Analytical prediction
1.16
1.17
1.3
1.24
25
co-ordinate. Fig. 1. The elastic stresses around of the crack tip in polar co
26
Fig. 2. Geometry and loading conditions of (a) Centrally cracked Brazilian disc specimen (BD specimen) (b) Edge cracked semi semi-circular circular specimen subjected to three point bending (SCB specimen).
27
Fig. 3. The dimensionless geometry factors calculated for various a/R in the BD specimen under Ι Ι pure mode I loading. (a) A1nΙ , (b) A2n , (c) A3n .
28
Fig. 4. The dimensionless geometry factors calculated for various a/R and S/R in the SCB Ι Ι specimen under pure mode I loading loading. (a) A1nΙ , (b) A2n , (c) A3n .
29
Fig. 5. The dimensionless geometry factors calculated for various a/R in the BD specimen under ΙΙ ΙΙ ΙΙ pure mode II loading. (a) B1nΙΙ , (b) A2n , (c) B3n , (d) A3n .
30
Fig. 6. The dimensionless geometry factors calculated for various a/R and S/R in the SCB ΙΙ ΙΙ ΙΙ specimen under pure mode III loading loading. (a) B1nΙΙ , (b) A2n , (c) B3n , (d) A3n .
31
Fig. 7. The SCB specimens manufactured in four different sizes for studying the size effect on the mode II fracture toughness.
32
Fig. 8. (a) Loading set-up up utilized for conducting mode II fracture tests.. (b) Fracture samples.
33
Fig. 9. Load–displacement displacement curve corresponding to one of the SCB specimens with R = 100 mm. The curve indicates the brittle fracture behaviour of the tested rock material.
34
Fig. 10. Loading set-up up utilized for conducting mode I fracture tests.. (a) Sample before test. (b) Sample after test.
35
Fig. 11. (a) Loading set-up up utilized for conducting the tensile strength tests.. (b) Load– L displacement curve corresponding to one of the tensile strength tests. 36
Fig. 12. Comparison between the experimental results of Neiriz rock and the predictions of the new approach for the fracture toughness ratio.
37
Fig. 13. Comparison between the experimental results of Ghorveh marble and the predictions of the new approach for the fracture racture toughness ratio.
38
Fig. 14. Analytical predictions of K ΙΙxc / K Ιyc plotted against ax / rc for different values of (a) a y / rc , (b) S/R, (c) a/R.
39
Highlights:
• Specimen size effect on KIIc of brittle materials is investigated experimentally • A new analytical approach is proposed to estimate size- and geometry-dependent KIIc •
Good estimates can be provided for size- and geometry-dependent KIIc
40
Theoretical and experimental investigations on the mode II fracture toughness of brittle materials Masoud Sistaninia, Mahjoubeh Sistaninia
www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(15)00148-4 http://dx.doi.org/10.1016/j.ijmecsci.2015.04.003 MS2965
To appear in:
International Journal of Mechanical Sciences
Received date: 4 September 2014 Revised date: 7 March 2015 Accepted date: 7 April 2015 Cite this article as: Masoud Sistaninia, Mahjoubeh Sistaninia, Theoretical and experimental investigations on the mode II fracture toughness of brittle materials, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j. ijmecsci.2015.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical and experimental investigations on the mode II fracture toughness of brittle materials Masoud Sistaniniaa,b,*, Mahjoubeh Sistaniniac a
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences Jahnstrasse 12, A-8700 Leoben, Austria b
Materials Center Leoben Forschung GmbH, Roseggerstrasse 12, A-8700 Leoben
c
Department of Materials Science and Engineering, Shahid Bahonar University of Kerman, P.O. Box No. 76135-133, Kerman, Iran
Abstract Recent theoretical and experimental investigations have shown that the mode II (shear mode) fracture resistance of brittle materials is strongly dependent on the loading conditions, the geometry and the size of components. For this reason, the shear mode fracture resistance of materials and components can be different from that measured from the laboratory fracture tests. To overcome this problem, this paper develops a new approach which can accurately estimate the size- and geometry-dependent shear mode fracture resistance of materials and components using the results obtained from the laboratory tests. Furthermore, a number of fracture tests are conducted in order to validate the predictions of the proposed approach and to experimentally investigate the specimen size effect on the mode II fracture toughness. A comparison between the theoretical predictions of the new approach and the experimental results shows that the proposed approach is able to estimate precisely the size- and geometry-dependent mode II fracture toughness of materials. The proposed analytical approach can be used for the design especially of fracture resistant materials and components.
Keywords: Crack; Mode II loading; Fracture toughness; Size and geometry effects; SCB specimen.
* Corresponding author: Tel.: +43 3842 45922 43; E-mail address: [email protected] (M. Sistaninia).
1
1. Introduction Fracture toughness and crack growth resistance are important material properties which describe the resistance of brittle and ductile materials against crack propagation [1-6]. In many practical applications, mode II (shear mode) fracture plays a key role in the failure process. In these practical applications, the determination of mode II fracture toughness (KIIc) is an indispensable issue for engineers and researchers. Whereas the mode II fracture toughness is a material property which depends on the geometry and the loading condition [7, 8], a standard method for determining mode II fracture toughness has not been yet established. However, many different test specimens have been proposed for determining the mode II fracture toughness of brittle materials. Some of the famous test configurations are the edge cracked semi-circular specimen subjected to three point bending [8-10], the centrally cracked Brazilian disk under diametral compression [8, 11-18], the single-edge crack specimen subjected to asymmetric four point bend loading [19, 20] and the compact shear-tension specimen [21, 22]. A review of literature shows that the edge cracked semi-circular specimen subjected to three point bending (SCB specimen) and the centrally cracked Brazilian disc (BD) specimen have been used most frequently in the mode II fracture analysis of rock types. For example, Khan and Al-Shayea [8], Aliha et al. [7, 11, 23], Al-Shayea [12], Awaji and Sato [14], Chang et al. [15], Krishnan et al. [16], Lanaro et al. [17], Chen et al. [24] and Ke et al. [25] have used the BD specimen to determine and study the mixed mode and mode II fracture toughness of various brittle materials. And also, Lim et al. [10], Khan and Al-Shayea [8], Ayatollahi et al. [7, 9] have applied the SCB specimen for the mode II fracture analysis. There are several advantages in using the SCB and BD specimens for the mode II fracture study of brittle materials, such as: simple geometry and loading configuration, easy test set-up procedure, application of compressive loads rather than tensile loads and the ability of introducing different mode mixities from pure mode I to pure mode II conveniently by changing the crack orientation relative to the direction of applied load. In addition to the mode II test specimens, there are three well-known fracture criteria proposed in the past for estimating the onset of fracture in brittle materials under mode II loading. These criteria are the Maximum Tangential Stress (MTS) criterion [26], the minimum Strain Energy Density (SED) criterion [27-29] and the maximum energy release rate or G criterion [30]. Among of these criteria, the Maximum Tangential Stress (MTS) criterion has 2
received more attentions by the researchers. Several researchers have shown that there is a large discrepancy between the results obtained experimentally from the BD and SCB specimens and the theoretical estimations of the conventional mode II fracture theories MTS, S and G criteria [9, 11, 23, 31-39]. MTS criterion only considers the effect of singular stress terms of William’s series expansion [40] for the elastic tangential stress near the crack tip and neglects the effect of nonsingular stress terms. As the researchers mentioned in the literatures [11, 32-35, 38, 39, 41], the non-singular stress terms (second and third stress terms) in the elastic tangential stress expansion have significant influences on the mode II fracture toughness and by taking into account these stress terms in addition to the singular stress terms, the estimates of MTS criterion can be improved significantly. For example, Ayatollahi and Aliha [34, 35] showed that the GMTS criterion, which takes into account the effect of T-stress (second stress term) in addition to the conventional singular stresses in the MTS criterion, can estimate the mixed mode and mode II fracture toughness in agreement with the experimental results. However, there is still a discrepancy between the GMTS estimates and the experimental results under mixed mode and pure mode II loading. The reason is that the third stress terms also have an influence on the fracture toughness. Recently, Ayatollahi and Sistaninia [38], Sistaninia et al. [42] and Saghafi et al. [39] took into account also the effect of third stress terms in the MTS criterion and proposed a new criterion (MMTS criterion) which could provide more accurate estimates for mixed mode and mode II fracture toughness in compared with GMTS criterion. Aliha et al. [43] have shown that the third order stress term has significant effects not only on the mixed mode and mode II, but also on the mode I fracture toughness. They have shown that the difference in fracture resistance of a rock obtained using the BD and SCB specimens can be related to the effect of the third stress term of the elastic stress field near the crack tip. Khan and Al-Shayea [8], Ayatollahi and Akbardoost [44] and the experimental results, which are presented in this paper, show that the mode II fracture toughness of brittle materials is dependent not only on the geometry and loading conditions but also on the specimen size. Khan and Al-Shayea [8] showed that by increasing the BD specimen diameter from 84 mm to 98 mm, 3
the mode II fracture toughness of a rock (Saudi Arabian limestone) increases by about 17%. The experimental results, which are presented later, also show that by increasing the size of SCB specimen, the mode II fracture toughness of a marble rock (Neiriz rock) increases substantially. Therefore, the mode II fracture toughness values measured from the laboratory specimens are not identical to the mode II fracture toughness of real size structures. Now the question arises, how we can estimate KIIc-values of real size structures or large scale materials from the fracture toughness values measured from the laboratory specimens. It is the aim of the current paper to present a new approach based on the MMTS criterion for predicting the size- and geometrydependent mode II fracture toughness of materials and components from their mode I fracture toughness. In the present paper, first the mode II fracture resistance of a marble rock is studied experimentally using the SCB specimens in four different sizes. Then, the new approach based on the MMTS criterion is introduced to take into account the influence of specimen size and the geometry conditions on KIIc. Finally, the theoretical predictions of the new approach are compared with the experimental results obtained in this paper and the experimental results reported in [8] and [44].
2. Elastic tangential stress around of the crack tip Williams [40] showed that the elastic stresses around the crack tip are written as an infinite series expansion. By considering the first three terms of mode I and mode II in the series expansion, the elastic tangential stress in the vicinity of the crack tip can be written as
σ θθ =
3 KΙ θ 1 3θ 3 K ΙΙ θ 3θ 2 cos + cos − sin + sin + T sin θ + 4 2π r 2 3 2 4 2π r 2 2 θ 1 15 0.5 5θ 15 5θ θ r A3 cos − cos + r 0.5 B3 sin − sin + O(r ), 4 2 5 2 4 2 2
(
)
(1)
where r and θ are the conventional crack tip co-ordinates, see Fig. 1. K Ι and K ΙΙ are the mode I and mode II stress intensity factors, respectively, and their corresponding terms are called the singular stress terms. In Eq. (1), T is a non-singular and constant stress term which is called the 4
T-stress. A3 and B3 are the third order crack parameters. Crack parameters K Ι , T , A3 (the mode I parameters) and K ΙΙ , B3 (the mode II parameters) are generally dependent on the geometry and loading conditions of the cracked specimen.
K Ι , K ΙΙ and T for each cracked configuration can be calculated readily by most commercial finite element codes. However, an independent numerical technique is required for determining the third order crack parameters A3 and B3. The FEOD method proposed by Ayatollahi and Nejati [45, 46] is a numerical technique for determining higher order crack parameters as well as K Ι , K ΙΙ and T. The FEOD method makes it possible to calculate not only the stress intensity factors but also the coefficients of higher order terms by using the displacement field obtained from finite element analysis. In this method, an over-determined set of simultaneous linear equations is obtained for a large number of nodes around the crack tip, and then by using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion can be calculated with a good accuracy for pure mode I, pure mode II and mixed mode I/II conditions. In this paper, the crack parameters of BD and SCB specimens are calculated using the FEOD method. The following section presents the normalized crack parameters of BD and SCB specimens under mode I and mode II loading, which are calculated using FEOD method.
3. BD and SCB specimens Fig. 2 shows the geometry and loading condition of the BD and SCB specimens used for fracture tests. The BD specimen is a circular disc of radius R with a central crack of length 2a and a thickness of t. The SCB specimen is a semi-circular disc of radius R with an edge crack of length a, a thickness of t and a loading support distance of 2S. Both specimens are subjected to a compressive load P. The crack makes an angle α with respect to the loading direction. By changing α, different combinations of mode I and mode II are achieved. When α is zero, the specimens are loaded in pure mode I and the mode II stress intensity factor is zero ( KΙΙ = 0 ). For specific inclination angles α ΙΙ , the mode I stress intensity factor is zero (KI=0) and the specimen 5
is subjected to pure mode II. The α ΙΙ -value for BD specimen, which depends on the crack length ratio a/R, can be found in the literature (i.e. [47]). Ayatollahi and Aliha [48] determined α ΙΙ for the SCB specimen using the finite element method and they showed that α ΙΙ depends on the ratio of a/R and S/R. Since the crack parameters for the BD and SCB specimens are dependent on the geometry and the magnitude of the load, the following equations are used to normalize them,
A1n =
Rt KΙ , P 2π a
(2)
B1n =
Rt K ΙΙ , P 2π a
(3)
Rt T, 4P
(4)
A3n =
Rt a A3 , P
(5)
B3n =
Rt a B3 , P
(6)
A2 n =
where A1n , B1n , A2 n , A3n and B3n are dimensionless geometry factors of the BD and SCB specimens. These dimensionless geometry factors are functions of α and a/R for the BD specimen and are functions of α, a/R and S/R for the SCB specimen. Ι Figs. 3a-3c and 4a-4c show the dimensionless geometry factors A1Ιn , A2Ι n and A3n of the BD Ι and SCB specimens, respectively, where A1Ιn , A2Ι n and A3n are the dimensionless geometry factors
of the specimens under pure mode I loading. As it can be seen in Figs. 3a-3c and 4a-4c, the dimensionless geometry factors of the BD specimen are presented for various crack length ratios and for the SCB specimen, they are presented for various ratios of a/R and S/R. Similarly, Figs. ΙΙ 5a-5d and 6a-6d show the dimensionless geometry factors B1ΙΙn , A2ΙΙn , B3ΙΙn and A3n for the BD (with
various ratios of a/R) and SCB (with various ratios of a/R and S/R) specimens, respectively, 6
ΙΙ where B1ΙΙn , A2ΙΙn , B3ΙΙn and A3n are the dimensionless geometry factors of the specimens under pure
mode II loading. It is worth noting that the superscripts I or II in the geometry factors refer to the mode of loading and the numbers 1, 2, and 3 in the subscripts refer to the corresponding term in the Williams series expansion. As mentioned before, the dimensionless geometry factors (in Figs. 3 to 6) are calculated by “finite element over-deterministic” or FEOD method [45]. It is noted that the dimensionless geometry factors of the BD specimen were calculated before by ΙΙ of the SCB specimen Ayatollahi and Sistaninia [38], but the geometry factors A3Ιn , A3ΙΙn and B3n
are calculated and presented for the first time in this paper. In the following section, the experimental tests are presented. First the mode II fracture tests, which are conducted on the SCB samples with four different radii, are presented and discussed. Then, the mode I fracture test, which is conducted on a different specimen (the BD specimen), is presented. The experimental results will be used later in order to check if the new approach, which will be proposed in section 5, can estimate the mode II fracture toughness of a material from its mode I fracture toughness which is obtained from a different test configuration with a different size.
4. Experimental procedure 4.1. Mode II fracture tests Using the SCB specimens, mode II fracture toughness experiments are performed on a kind of marble rock (Neiriz rock) excavated from Fars province mines in Iran. The structure of this marble rock is relatively homogenous and isotropic. In order to study the effect of specimen size on the mode II fracture toughness, the SCB samples with four different radii are manufactured, approximately R = 30, R = 50, R = 70 and R = 100 mm (see Fig. 7). In order to get sufficient accuracy, mode II fracture toughness for each size is measured four times. Thus, a total number of 16 SCB specimens are prepared for conducting the mode II fracture tests. The SCB specimens are manufactured with a crack length ratio a/R=0.5 and the loading span ratio of S/R=0.5. Table 1 shows the geometries of the SCB specimens. The crack generation in the specimens is performed as follows; firstly a thin notch of 0.8 mm width is generated by using a 7
high-precision CNC water jet cutting machine and then the notch tip is sharpened using a saw blade with the thickness of 0.2 mm. The crack is generated in the appropriate angle of
α ΙΙ = 40.5 (the pure mode II angle of SCB specimen with the ratios a/R=0.5 and S/R=0.5). The prepared samples are then tested using a universal tension/compression test machine with a load cell capacity of 15 kN. The tests are carried out under displacement control conditions with a constant rate of 0.05 mm/min. The specimens are tested using a three-point bend fixture with the loading span ratio S/R=0.5. Fig. 8a shows the loading setup for the SCB specimens. For each specimen the load versus displacement curve is recorded until the final fracture, the fractured samples are shown in Fig. 8b. Fig. 9 shows a sample load–displacement curve corresponding to one of the SCB specimens with R = 100 mm. The absolute values of load and displacement have been shown in this figure. As it can be seen, the specimen is loaded until a maximum load of 11.3 kN that is reached when the actuator of the test machine moves down to 0.62 mm. After the load reaches its maximum value ( Pcr ), the specimen is suddenly fractured in a brittle manner initiating from the crack tip, the fracture load (Pcr) of the specimen is determined as 11.3 kN. It was observed that all the samples showed a linear curve and fractured suddenly in a brittle manner like in Fig. 9. It is worth mentioning that for ideally brittle materials, the fracture occurs immediately after initiation of the crack growth. Indeed, for ideally brittle materials, catastrophic failure or unstable crack propagation takes place immediately once the stress intensity factor at the initial crack tip reaches the fracture initiation toughness (the fracture toughness at the onset of crack extension). This is the reason why the fracture initiation toughness of ideally brittle materials is nearly equal to their critical value of fracture toughness or KIIc (fracture toughness at the maximum load, beyond which the specimen fails) [1-3]. The mode II fracture resistance (KIIc) of the tested SCB specimens is determined from the following equation
K ΙΙc =
Pcr 2π a ΙΙ B1n , Rt
8
(7)
where Pcr is the fracture load and B1nΙΙ is the geometry factor of the SCB specimen which is presented in Fig. 6a. By replacing the geometry factor B1ΙΙn = 0.434 ( B1nΙΙ for the SCB specimen with a/R=0.5 and S/R=0.5 is 0.434) and Pcr (obtained from the fracture test) in Eq. (7), the mode II fracture toughness of Neiriz rock obtained using the SCB specimen is determined. Table 1 shows the mode II fracture toughness of Neiriz rock obtained from SCB specimens with four different sizes. As an average of four test results, the KIIc-values are measured to be 0.53, 0.70, 0.77 and 0.90 MPa√m from the samples with R = 30, 50, 70 and 100 mm, respectively. There is an increase of about 70% in the mode II fracture toughness when the radius is increased from 30 mm to 100 mm. The experimental results show that the test specimen size has a noticeable influence on the mode II fracture resistance of materials. 4.2. Mode I fracture and tensile strength tests
Mode I fracture toughness of Neiriz rock is measured by averaging the results obtained from the pure mode I tests conducted on the BD specimen. The BD specimens are manufactured with a crack length ratio a/R=0.5. Table 2 shows the geometries of the BD specimens conducted for mode I fracture tests. First, the crack is generated in the specimens as explained in the previous subsection. Then, the prepared samples are loaded using two flat loading fixtures when the direction of the applied load is along the crack line, the crack is simultaneously subjected to pure mode I (or opening) deformation. Fig. 10 shows a sample BD specimen inside the test machine before and after fracture test. The tests are carried out under displacement control conditions with a constant rate of 0.05 mm/min. After measuring the fracture load Pcr from the fracture tests, the mode I fracture resistance (KIc) of the tested BD specimens is determined from the following equation
K Ιc =
Pcr 2π a Ι A1n , Rt
(8)
where A1nΙ is the geometry factor of the BD specimen which is plotted versus a/R in Fig. 3a. By replacing the geometry factor A1Ιn = 0.313 ( A1nΙ for the BD specimen with a/R=0.5 is 0.313) and Pcr (obtained from the fracture tests) in Eq. (8), the mode I fracture toughness of Neiriz rock is 9
determined. As an average of four test results, the mode I fracture toughness KIC of Neiriz rock is determined as 1 MPa√m. The uncracked Brazilian disk subjected to compressive load [8] is used for obtaining the tensile strength of Neiriz rock. The uncracked Brazilian disk specimens are manufactured with a radius of 51 mm. Table 3 shows the geometries of the specimens conducted for obtaining the tensile strength. The specimens are loaded using two flat loading fixtures, which can be seen in Fig. 11a, under displacement control conditions. The load-displacement curve is demonstrated in Fig. 11b taken from the test record. Initially, the specimen has a non-linear behaviour caused probably by compression of existed porosity and micro-cracking of the specimen. Then follows a period of linear ascending, the load reaches its maximum, which corresponds to the initiation of a crack in the centre of the specimen. Then, the crack propagates unstably at this stage and the specimen fractures suddenly. After measuring the fracture load Pcr from the tests, the tensile strength σt can be determined as
σt =
Pcr . π Rt
(9)
The tensile strength (σt) of Neiriz rock is determined to be 5.70 MPa from an average of four uncracked Brazilian disk specimen tests. Table 3 summarizes the results of σt for Neiriz rock.
5. Brittle fracture criterion A new approach which is based on the MMTS criterion is proposed here in order to estimate the mode II fracture toughness (KIIc) of brittle materials for any configuration and size from their mode I fracture toughness data. It should be mentioned that the MMTS criterion is able to predict the fracture initiation toughness, not the critical value of fracture toughness (KIIc). However, as explained before, for an ideally brittle material, the fracture initiation toughness is equal to KIIc. The maximum tangential stress (MTS) criterion was first proposed by Erdogan and Sih [26] for brittle fracture in mixed mode I/II crack problems. According to MTS criterion, crack 10
growth initiates radially from the crack tip along the direction of maximum tangential stress θm. The crack growth takes place when the tangential stress σθθ along θm and at a critical distance rc from the crack tip attains a critical value σθθc. rc is the size of fracture process zone, and σθθc is the tensile strength of the material (σt), which is a material property. The classical MTS criterion only considers the effect of the singular stress terms and the non-singular stress terms are neglected in the MTS criterion. Ayatollahi and Sistaninia [38] and Saghafi et al. [39] showed that the first three terms of William’s series expansion for elastic stresses near the crack tip have to be taken into account in MTS criterion in order to provide more accurate estimates for mode II fracture toughness of brittle materials. By taking into account the first three terms of William’s series expansion for tangential stress, the tangential stress in pure mode II loading can be written as 15 0.5 ΙΙ θ 1 5θ r A3 cos − cos 4 2 5 2 3 K ΙΙ θ 3θ 15 0.5 ΙΙ θ 5θ − sin + sin + r B3 sin − sin . 4 2π r 2 2 4 2 2
σ θθΙΙ (r ,θ ) = T ΙΙ ( sin 2 θ ) +
(10)
K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ are the crack parameters for pure mode II loading. The angle of maximum
tangential stress θm is determined from
∂σ θθΙΙ θ 3θ 3 1 =− K ΙΙ 0.5cos m + 1.5 cos m + T ΙΙ ( sin 2θ m ) + 4 2π rc 2 2 ∂θ θ =θm
θ 5θ 15 rc A3ΙΙ −0.5sin m + 0.5sin m 4 2 2
(11)
θm 5θ m 15 ΙΙ + 4 rc B3 0.5cos 2 − 2.5 cos 2 = 0.
The fracture initiation angle θm for each specimen can be determined by solving Eq. (11). Since the crack parameters K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ are functions of geometry and loading condition of specimen, it is preferred to present Eq. (11) in terms of the normalized crack parameters. By replacing Eqs. (2)-(6) into Eq. (11), Eq. (11) can be rewritten as
11
−
θ 3θ 3 a ΙΙ B1n 0.5cos m + 1.5cos m + 4 A2ΙΙn ( sin 2θ m ) + 4 rc 2 2
(12)
5θ 15 rc ΙΙ 5θ θ θ 15 rc ΙΙ A3 n −0.5sin m + 0.5sin m + B3 n 0.5 cos m − 2.5cos m = 0. 4 a 2 2 4 a 2 2 By replacing the pure mode II geometry factors of each cracked body into Eq. (12), the fracture initiation angle ( θ m ) is estimated by the MMTS criterion in terms of a / rc . According to the MMTS criterion, brittle fracture takes place when the tangential stress
σθθ along θm and at a critical distance rc from the crack tip attains the critical value σθθc or σt. Thus, in a test configuration x which is under pure mode II loading, the mode II brittle fracture occurs when
σ θθΙΙ (rc ,θ m ) |x = σ t .
(13)
By replacing rc and θm into Eq. (10), for the test configuration x, one can write
θ mx 3θ mx θ mx 1 5θ mx 3 1 15 x ΙΙ , x 2 x ΙΙ , x − K ΙΙc sin + sin rc A3c cos − cos + Tc ( sin θ m ) + 4 2π rc 2 2 4 2 5 2 θ mx 5θ mx 15 ΙΙ , x + rc B3c sin − sin =σt, 4 2 2
(14)
where K ΙΙxc , TcΙΙ , x , B3ΙΙc, x , A3ΙΙc, x are the critical values of K ΙΙ , T ΙΙ , B3ΙΙ , A3ΙΙ corresponding to the fracture load for the test configuration x. It is noted that in Eq. (14), the superscript x refers to the test configuration x. Since σt is a material constant and is independent of the test configuration, mode mixity and the specimen size, an equation similar to Eq. (13) can be written for mode I brittle fracture of a test configuration y which is made of the same material as the test configuration x, but can have a different size. For mode I loading, KII and B3 are zero and the fracture initiation angle ( θ m ) is determined from Eq. (1) to be zero. Therefore, the mode I brittle fracture occurs in the test configuration y when
σ θθΙ ( rc , 0) | y = σ t →
1 K Ιyc + 3 rc A3Ιc, y = σ t , 2π rc
12
(15)
where K Ιyc , A3Ιc, y are the critical values of the mode I crack parameters K Ι and A3Ι corresponding to the fracture load for the test configuration y. Since the right side of Eqs. (14) and (15) is a constant material property, therefore, Eq. (15) can be replaced into Eq. (14) and the ratio of mode II fracture toughness of the test configuration x to mode I fracture toughness of the test configuration y, K ΙΙxc / K Ιyc , can be determined as
K ΙΙxc 1 A3Ιc, y = + 3 r c K Ιyc K Ιyc 2π rc
3 1 θ mx 3θ mx TcΙΙ , x 2 x − + sin sin + x ( sin θ m ) + 2 2 K ΙΙc 4 2π rc (16) ΙΙ , x x x ΙΙ , x x x 15 rc A3c cos θ m − 1 cos 5θ m + 15 rc B3c sin θ m − sin 5θ m 4 K ΙΙxc 2 5 2 4 K ΙΙxc 2 2
In Eq. (16), K Ιyc and A3Ιc, y are proportional to the fracture load, but the ratio of A3Ιc, y K Ιyc is independent of the fracture load. The ratio A3Ιc, y K Ιyc can be simplified as A3Ιc, y 1 A3Ιn, y = Ι, y K Ιyc 2π A1n
1 , a
(17)
where A1Ιn, y , A3Ιn, y are the geometry factors of the test configuration y. Similarly, K ΙΙxc , TcΙΙ , x , B3ΙΙc, x and A3ΙΙc, x are proportional to the fracture load, but the ratios TcΙΙ , x K ΙΙxc , B3ΙΙc K ΙΙxc , A3ΙΙc K ΙΙxc are
independent of the fracture load. These ratios can be rewritten as
TcΙΙ , x 4 A2ΙΙn, x 1 = ΙΙ , x K ΙΙxc 2π B1n a A3ΙΙc, x 1 A3ΙΙn, x 1 = , ΙΙ , x K ΙΙxc 2π B1n a
(18)
B3ΙΙc, x 1 B3ΙΙn, x 1 = ΙΙ , x K ΙΙxc 2π B1n a where B1ΙΙn, x , A2ΙΙn, x , B3ΙΙn, x and A3ΙΙn, x are the geometry factors of the test configuration x. By replacing Eqs. (17) and (18) into Eq. (16), the fracture toughness ratio K ΙΙxc / K Ιyc is rewritten as
13
x ΙΙc y Ιc
K K
r = 1 + 3 c a y
A A Ι, y 3n Ι, y 1n
3 θ mx 3θ mx rc A2ΙΙn, x + 4 sin 2 θ mx ) + − sin + sin ΙΙ , x ( 4 2 2 a B x 1n r AΙΙ , x x x ΙΙ , x 15 c 3ΙΙn, x cos θ m − 1 cos 5θ m + 15 rc B3ΙΙn, x 2 5 2 4 ax B1n 4 ax B1n
, (19) x x θm 5θ m sin − sin 2 2
where ay and ax are the crack lengths of the test configurations y and x, respectively. If the
(
)
geometry factors of the test configuration y A1Ιn, y , A3Ιn, y , the geometry factors of the test
(
)
configuration x B1ΙΙn, x , A2ΙΙn, x , A3ΙΙn, x , B3ΙΙn, x and the fracture initiation angle θ mx (which can be determined from Eq. (12)) are inserted in Eq. (19), the fracture toughness ratio K ΙΙxc / K Ιyc is determined in terms of a x / rc (or Rx / rc ) and a y / rc (or R y / rc ), the size of specimen and the crack length are related by the crack length ratio a/R. In order to use the proposed approach, rc must be known. A formulation based on the maximum principal stress theory has been suggested in [1, 49, 50] for evaluating rc . According to this model, rc can be calculated from
1 rc = 2π
2
K Ιc . σ t
(20)
Since the mode I fracture resistance K Ιc is dependent on the size and the geometry and loading conditions of the test specimen [43, 51], a shortcoming of Eq. (20) is that the effects of size, geometry and loading conditions of the test specimen are not taken into account. As it is shown in [43], when A3I is non-zero, K Ιc takes different values depending on the size and loading and geometry conditions of the specimen. Therefore, the effects of size, geometry and loading conditions of the test specimen on K Ιc , and the resulting rc , can be taken into account if one considers the effect of A3I for the tangential stress near the crack tip. For this reason, rc should be determined by solving the non-linear Eq. (15). When A3I = 0 , Eq. (15) becomes similar to Eq. (20). It is noted that Eq. (15) might have multiple solutions, but the lowest positive value is physically acceptable.
14
In summary, in order to estimate the mode II fracture toughness K ΙΙc of materials and structures with any configuration and size, first a tensile strength test specimen and a pure mode I fracture specimen (it can have any size or configuration) are manufactured and tested in the laboratory. Then, after measuring the fracture load Pcr of mode I test and the tensile strength σt (from the tensile test), K Ιc and A3cI are determined from
K Ιc = I 3c
A =
Pcr 2π a A1nI Rt , I Pcr A3n
(21)
Rt a
and then by inserting σt, K Ιc and A3cI in Eq. (15), rc can be determined by solving Eq. (15). After that, K ΙΙc of the material is readily estimated by Eq. (19). In the following section, a comparison between the experimental results and predictions of the proposed criterion demonstrates that the proposed criterion can estimate the mode II fracture resistance very well.
6. Results and discussion In [7, 8, 33, 39, 43, 44], it was shown that the mode II fracture toughness is a material property which depends also on the geometry of specimen and the type of loading. The experimental results in the current paper also show that the specimen size has a significant influence on the mode II fracture toughness. Therefore, the mode II fracture toughness of real size structures is not identical to that measured from the laboratory fracture tests. In the case of practical applications, it is essential to use an appropriate criterion for estimating the mode II fracture toughness of structures using the results obtained from the laboratory specimens. The new approach proposed in Section 5 is able to satisfy this object. In this section, the mode II fracture toughness results reported in [8, 44] and the experimental results presented in this paper are used to verify the estimates of the new proposed approach. Khan and Al-Shayea [8] used the SCB specimen with R = 49 mm and the ratios a/R = 0.3
and S/R = 0.8 to determine the mode I fracture toughness of Saudi Arabian limestone. As 15
reported in [8], the mode I fracture toughness measured by this test configuration is , R = 49 = 0.68 MPa m ). They also used the BD specimens with two different 0.68 MPa m ( K ΙSCB c
radii and crack lengths (a specimen with R = 49 mm and a/R = 0.3 and another specimen with R=42 mm and a/R = 0.4) for determining the mode II fracture toughness of Saudi Arabian
limestone. The average mode II fracture toughness obtained by the BD specimen with the radius R = 49 mm is noticeably higher than that of the BD specimen with the radius R = 42 mm. The
mode II fracture toughness value for the BD specimen with R=42 mm is K ΙΙBDc , R = 42 = 0.79 , R = 49 MPa√m, but the KIIc-value measured by the specimen with R=49 mm is K ΙΙBD = 0.92 MPa√m. c
, R =49 , R = 49 The fracture toughness ratios K ΙΙBDc ,R =42 / K ΙSCB and K ΙΙBDc , R =49 / K ΙSCB are 1.16 and 1.3, c c
respectively. If the SCB specimen with R = 42 mm, a/R = 0.3 and S/R=0.8 under mode I loading is considered as the test configuration y and the BD specimens with R = 42 mm, a/R = 0.4 and R = 49 mm, a/R = 0.3 under mode II loading are considered as test configurations x, the fracture , R = 42 , R = 49 , R = 49 , R = 49 toughness ratios K ΙΙBD / K ΙSCB and K ΙΙBD / K ΙSCB can be analytically determined by c c c c
Eq. (19), rc of Saudi Arabian limestone is 5.8 mm (determined by Eq. (15)). Table 4 compares the experimental results and the analytical predictions of the new approach for the fracture toughness ratios of Saudi Arabian limestone. It is seen that there is a good agreement between the experimental results and the predictions of the new approach. The average mode II fracture toughness of Neiriz rock obtained using SCB specimen (with a/R = 0.5 and S/R = 0.5) with four different sizes R = 30, 50, 70 and 100 mm are 0.53, 0.70, 0.77 and 0.90 MPa√m, respectively, (see Table 1). Using the BD specimen with R=50 mm and a/R = 0.5, the mode I fracture toughness of Neiriz rock is measured to be 1 MPa√m, see Table 2. According to the experimental results, the fracture toughness ratio K ΙΙc / K Ιc is 0.53, 0.70, 0.77 and 0.90 for four different specimen sizes R = 30, 50, 70 and 100 mm, respectively. If the BD specimen with a/R = 0.5 and R = 50 mm under mode I loading is considered as the test configuration y and the SCB specimens (with a/R = S/R= 0.5) with the radii R = 30, 50, 70 and 100 mm under mode II loading are considered as the test configurations x, the fracture toughness ratio K ΙΙxc / K Ιyc for Neiriz rock is analytically estimated from Eq. (19) by inserting the dimensionless crack parameters (extracted from Figs. 3 to 6), the crack initiation angle θm 16
(obtained from Eq. (12)) and the critical distance rc . rc of Neiriz rock is 6.6 mm, which is obtained from Eq. (15). Fig. 12 compares the fracture toughness ratio K ΙΙxc / K Ιyc estimated by the new approach (Eq. (19)) to those obtained by the experimental tests, K ΙΙxc / K Ιyc is plotted versus the radius of the mode II fracture test (Rx). As it can be seen, there is a good coincidence between the analytical predictions of the new approach and the experimental results; the maximum discrepancy is 7%. Our experimental results show that the new proposed approach can accurately estimate the mode II fracture resistance of a material from its mode I fracture toughness which is obtained from a different test configuration with a different size (smaller or larger). As it can be seen in Fig. 12, the mode II fracture toughness K ΙΙxc ( or K ΙΙxc / K Ιyc ) strongly increases with increasing the specimen size until it approaches a saturation value when the specimen size is very large. Indeed, for very large specimens, the mode II fracture resistance K ΙΙc becomes independent of specimen size and it can be considered as the inherent mode II fracture resistance of the material. The size-independent value of K ΙΙxc / K Ιyc of Neiriz rock is 1.2, see Fig. 12. Since K Ιyc = 1 MPa m , the size-independent mode II fracture resistance of Neiriz rock is determined to be 1.2 MPa√m, which can be reached when Rx ≥ 200 mm. It should be mentioned that if both the mode I and mode II fracture tests are infinitely large, the size-independent value of K ΙΙxc / K Ιyc becomes 0.866 which is the same value suggested by the conventional MTS criterion. Recently, Ayatollahi and Akbardoost [44] proposed a new criterion in order to predict the size-dependent mode II fracture toughness of rock materials. Although their proposed criterion can predict the mode II fracture toughness in agreement with the experimental results, the criterion is complicated compared to the approach proposed in this paper. The reason is that their proposed criterion needs a set of mode I experimental tests for different sizes in order to predict the mode II fracture resistance of a given material, while the new approach proposed in this paper needs only one mode I experiment. They also conducted experimental tests on the SCB samples of different radii made of a marble (Ghorveh marble) in order to experimentally investigate the influence of the specimen size on K ΙΙc . The mode II fracture tests were conducted 17
on the SCB samples with S/R = a/R = 0.5 and four different radii R = 25, 50, 95 and 190 mm. As reported, the K ΙΙc values were measured to be 0.55, 0.71, 0.82 and 0.91 MPa√m from the samples with R = 25, 50, 95 and 190 mm, respectively. The K Ιc of Ghorveh marble was measured to be 1.32 MPa√m from the SCB sample with S/R = a/R = 0.5 and R = 50 mm. The tensile strength σt of Ghorveh marble was measured as 5.37 MPa. The critical distance rc can be determined as 4.49 mm by Eq. (15). Fig. 13 compares the fracture toughness ratio K ΙΙxc / K Ιyc of Ghorveh marble estimated by the new approach (Eq. (19)) to those obtained by the experimental tests. It is seen that the proposed approach predicts the size dependent fracture resistance of the SCB specimens under mode II loading very well even for large specimens. Both analytical and experimental results show that the mode II fracture toughness K ΙΙxc reaches a constant value when the specimen size is large, Rx ≥ 200 mm. For convenient using of the new approach, Fig. 14 is prepared which shows the fracture toughness ratio K ΙΙxc / K Ιyc versus the normalized crack length ax / rc (ax is the crack length of the mode II fracture test). Fig. 14a displays the K ΙΙxc / K Ιyc -values plotted against ax / rc for different values of the normalized crack length a y / rc (ay is the crack length of the mode I fracture test), where the mode II fracture test is the SCB specimen with S/R = a/R = 0.5 and the mode I fracture test is the BD specimen with a/R = 0.5. As discussed before, the mode II fracture toughness K ΙΙxc strongly increases with the specimen size until it approaches a saturation value when the specimen size is very large. When the dimensionless parameter ax / rc is kept constant, K ΙΙxc / K Ιyc decreases with increasing a y / rc , because K Ιyc increases with increasing a y or the mode I specimen size. In Fig. 14b, the K ΙΙxc / K Ιyc -values are plotted against ax / rc for four different values of S/R, where a y / rc = 0.5 and the mode II fracture test is the SCB specimen with a/R = 0.5 and the mode I fracture test is the BD specimen with a/R = 0.5. This figure shows the influence of the support condition on the mode II fracture resistance of materials. The mode II fracture resistance increases with decreasing the support distance S. However, when the size of specimen becomes large (approximately when ax / rc > 12 ), the mode II fracture resistance K ΙΙxc 18
strongly decreases with decreasing the support distance S. Furthermore, the influence of the crack length ratio a/R on the mode II fracture resistance of materials can be seen in Fig. 14c. Fig. 14c shows the K ΙΙxc / K Ιyc -values plotted against ax / rc for four different values of a/R, where the mode II fracture test is the SCB specimen with S/R = 0.5 and the mode I fracture test is the same as before (the BD specimen with a/R = 0.5) and a y / rc = 0.5 . As one can see, when ax / rc is kept constant, the mode II fracture resistance increases with increasing a/R, but when ax / rc > 15 , the mode II fracture resistance decreases with increasing a/R. As shown in this section, the new approach can precisely predict the size- and geometrydependent mode II fracture resistance of materials from their mode I fracture test. The theoretical predictions of the new approach were compared to the experimental results of three different rock materials and there was a good agreement between the theoretical predictions and the experimental results at all three cases. Finally, it should be remarked that since the geometry and loading conditions influence the mode II fracture toughness of materials, fracture resistant materials and structures can be designed by this fact. In order to work out design rules of more fracture resistant materials, it is necessary to quantify the influence of the geometry and loading conditions on the mode II fracture toughness. This will be investigated extensively in a separate paper.
7. Conclusion •
The mode II fracture toughness of a marble rock (Neiriz rock) is investigated experimentally by means of the SCB specimens in four different sizes.
•
The experimental results show that the specimen size has a substantial influence on the mode II fracture toughness of Neiriz rock. The mode II fracture toughness of Neiriz rock obtained using the SCB specimen increases with the size of specimen.
•
A new approach is proposed to estimate the size- and geometry-dependent mode II fracture toughness of materials and components from their mode I fracture data measured from the laboratory fracture tests. 19
•
Comparison between the theoretical predictions of the new approach and the experimental results of three different rock materials showed that the proposed approach is able to estimate precisely the size- and geometry-dependent mode II fracture toughness of brittle materials.
References [1] Anderson TL. Fracture mechanics: fundamentals and applications. 3rd ed. Boca Raton: Taylor & Francis; 2005. [2] Gao DY, Ogden RW. Advances in Mechanics and Mathematics: Springer; 2003. [3] Kolednik O, Predan J, Fischer FD, Fratzl P. Improvements of strength and fracture resistance by spatial material property variations. Acta Materialia. 2014;68:279-94. [4] Kolednik O, Predan J, Gubeljak N, Fischer DF. Modeling fatigue crack growth in a bimaterial specimen with the configurational forces concept. Materials Science and Engineering A. 2009;519:172-83. [5] Sistaninia M, Kolednik O. Effect of a single soft interlayer on the crack driving force. Engineering Fracture Mechanics. 2014;130:21–41. [6] Xia Z, Curtin WA, Sheldon BW. A new method to evaluate the fracture toughness of thin films. Acta Materialia. 2004;52:3507-17. [7] Aliha MRM, Ayatollah MR, Smith DJ, Pavier MJ. Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Engineering Fracture Mechanics. 2010;77:2200-12. [8] Khan K, Al-Shayea NA. Effect of specimen geometry and testing method on mixed Mode III fracture toughness of a limestone rock from Saudi Arabia. Rock Mechanics and Rock Engineering. 2000;33:179-206. [9] Ayatollahi MR, Aliha MRM, Hassani MM. Mixed mode brittle fracture in PMMA - An experimental study using SCB specimens. Materials Science and Engineering A. 2006;417:34856. [10] Lim IL, Johnston IW, Choi SK, Boland JN. Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2-mixed-mode. International Journal of Rock Mechanics and Mining Sciences and. 1994;31:199-212. [11] Aliha MRM, Ashtari R, Ayatollahi MR. Mode I and mode II fracture toughness testing for a coarse grain marble. Applied Mechanics and Materials. 2006;5-6:181-8. [12] Al-Shayea N. Comparing reservoir and outcrop specimens for mixed mode I-II fracture toughness of a limestone rock formation at various conditions. Rock Mechanics and Rock Engineering. 2002;35:271-97. 20
[13] Atkinson C, Smelser RE, Sanchez J. Combined mode fracture via the cracked Brazilian disk test. International Journal of Fracture. 1982;18:279-91. [14] Awaji H, Sato S. Combined mode fracture toughness measurement by the disk test. Journal of Engineering Materials and Technology, Transactions of the ASME. 1978;100:175-82. [15] Chang SH, Lee CI, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Engineering Geology. 2002;66:79-97. [16] Krishnan GR, Zhao XL, Zaman M, Roegiers JC. Fracture toughness of a soft sandstone. International Journal of Rock Mechanics and Mining Sciences. 1998;35:695-710. [17] Lanaro F, Sato T, Stephansson O. Microcrack modelling of Brazilian tensile tests with the boundary element method. International Journal of Rock Mechanics and Mining Sciences. 2009;46:450-61. [18] Shetty DK, Rosenfield AR, Duckworth WH. Mixed-mode fracture in biaxial stress state: Application of the diametral-compression (Brazilian disk) test. Engineering Fracture Mechanics. 1987;26:825-40. [19] Margevicius RW, Riedle J, Gumbsch P. Fracture toughness of polycrystalline tungsten under mode I and mixed mode I/II loading. Materials Science and Engineering A. 1999;270:197209. [20] Suresh S, Shih CF, Morrone A, O'Dowd NP. Mixed-mode fracture toughness of ceramic materials. Journal of the American Ceramic Society. 1990;73:1257-67. [21] Richard HA, Benitz K. A loading device for the creation of mixed mode in fracture mechanics. International Journal of Fracture. 1983;22:R55-R8. [22] Zipf Jr RK, Bieniawski ZT. Mixed mode testing for fracture toughness of coal based on critical-energy-density. 1986. p. 16-23. [23] Aliha MRM, Ayatollahi MR. On mixed-mode I/II crack growth in dental resin materials. Scripta Materialia. 2008;59:258-61. [24] Chen CS, Pan E, Amadei B. Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method. International Journal of Rock Mechanics and Mining Sciences. 1998;35:195-218. [25] Ke CC, Chen CS, Tu CH. Determination of fracture toughness of anisotropic rocks by boundary element method. Rock Mechanics and Rock Engineering. 2008;41:509-38. [26] Erdogan F, Sih GC. On the Crack Extension in Plates under Plane Loading and Transverse Shear. J Basic Eng Trans ASME. 1963;85:519–25. [27] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture. 1974;10:305-21. [28] Berto F, Ayatollahi MR. Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy. Materials & Design. 2011;32:2858-69.
21
[29] Berto F, Barati E. Fracture assessment of U-notches under three point bending by means of local energy density. Materials & Design. 2011;32:822-30. [30] Hussain MA, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II. American Society for Testing and Materials. Philadelphia1974. p. 2-28. [31] Aliha MRM, Ayatollahi MR. Brittle fracture evaluation of a fine grain cement mortar in combined tensile-shear deformation. Fatigue and Fracture of Engineering Materials and Structures. 2009;32:987-94. [32] Ayatollahi MR, Aliha MRM. Cracked Brazilian disc specimen subjected to mode II deformation. Engineering Fracture Mechanics. 2005;72:493-503. [33] Ayatollahi MR, Aliha MRM. On determination of mode II fracture toughness using semicircular bend specimen. International Journal of Solids and Structures. 2006;43:5217-27. [34] Ayatollahi MR, Aliha MRM. Fracture toughness study for a brittle rock subjected to mixed mode I/II loading. International Journal of Rock Mechanics and Mining Sciences. 2007;44:61724. [35] Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials. Engineering Fracture Mechanics. 2008;75:463141. [36] Ayatollahi MR, Aliha MRM. Mixed mode fracture analysis of polycrystalline graphite - A modified MTS criterion. Carbon. 2008;46:1302-8. [37] Ayatollahi MR, Aliha MRM. Mixed mode fracture in soda lime glass analyzed by using the generalized MTS criterion. International Journal of Solids and Structures. 2009;46:311-21. [38] Ayatollahi MR, Sistaninia M. Mode II fracture study of rocks using Brazilian disk specimens. International Journal of Rock Mechanics and Mining Sciences. 2011;48:819-26. [39] Saghafi H, Ayatollahi MR, Sistaninia M. A modified MTS criterion (MMTS) for mixedmode fracture toughness assessment of brittle materials. Materials Science and Engineering A. 2010;527:5624-30. [40] Williams ML. On the Stress Distribution at the Base of a Stationary Crack. Journal of Applied Mechanics. 1957;24:109-14. [41] Mirsayar MM. On fracture of kinked interface cracks – The role of T-stress. Materials & Design. 2014;61:117-23. [42] Sistaninia M, Ayatollahi MR, Sistaninia M. On Fracture Analysis of Cracked Graphite Components under Mixed Mode Loading. Mechanics of Advanced Materials and Structures. 2014;21:781-91. [43] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. International Journal of Rock Mechanics and Mining Sciences. 2012;51:128-35. [44] Ayatollahi MR, Akbardoost J. Size effects in mode II brittle fracture of rocks. Engineering Fracture Mechanics. 2013;112-113:165-80. 22
[45] Ayatollahi MR, Nejati M. An over-deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis. Fatigue Fract Eng Mater Struct. 2011;34:159-76. [46] Ayatollahi MR, Nejati M. Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method. International Journal of Mechanical Sciences. 2011;53:164-77. [47] Dong SM, Wang Y, Xia YM. Stress intensity factors for central cracked circular disk subjected to compression. Engineering Fracture Mechanics. 2004;71:1135-48. [48] Ayatollahi MR, Aliha MRM. Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Computational Materials Science. 2007;38:660-70. [49] Schmidt RA. A microcrack model and its significance to hydraulic fracturing and fracture toughness testing. Proceedings of the 21st US rock mechanics symposium; 1980, p. 581-90. [50] Ayatollahi MR, Torabi AR. Brittle fracture in rounded-tip V-shaped notches. Materials & Design. 2010;31:60-7. [51] Ayatollahi MR, Akbardoost J. Size and geometry effects on rock fracture toughness: Mode i fracture. Rock Mechanics and Rock Engineering. 2014;47:677-87.
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Table 1 Summery of mode II fracture tests conducted on SCB specimens manufactured from Neiriz rock. Sample no.
R (mm)
a/ R
S/R
t (mm)
Pcr (kN)
1 2 3 4
31.2 29.6 29.3 30.7
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
20.2 20.1 19.8 18.6
2.11 2.64 2.30 2.48
KIIc (MPa√m) 0.45 0.58 0.52 0.58
5.27 5.82 5.68 4.73
0.53 0.68 0.75 0.72 0.65
8.20 7.47 7.06 7.04
0.70 0.72 0.78 0.78 0.8
11.3 10.6 10.33 12.43
0.77 1.00 0.78 0.85 0.94
Average 1 2 3 4 Average 1 2 3 4 Average 1 2 3 4
49.6 50.4 50.2 49.4
71.5 70.0 70.0 70.0
97.0 98.7 98.5 101.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
26.7 26.5 27.0 25.0
0.5 0.5 0.5 0.5
32.6 28.0 25.8 25.8
0.5 0.5 0.5 0.5
28.0 33.7 30.0 32.1
Average
0.90
24
Table 2 Summary of mode I fracture tests conducted on BD specimens manufactured from Neiriz rock. Sample no.
R (mm)
a/ R
t (mm)
Pcr (kN)
1 2 3 4 Average
52.6 51.3 51.2 51.2
0.5 0.5 0.5 0.5
28.0 24.0 26.0 26.0
12.81 10.46 10.64 8.84
KIc (MPa√m) 1.11 1.07 1.00 0.83 1.00
Table 3 Summary of tensile strength tests conducted on uncracked BD specimens manufactured from Neiriz rock. Sample no. 1 2 3 4 Average
R (mm) 51 51 51 51
t (mm) 23.0 24.0 23.0 25.6
σt (MPa) 5.65 6.58 5.98 4.64 5.70
Pcr (kN) 20.82 25.31 22.06 19.06
Table 4 The fracture toughness ratio K ΙΙc / K Ιc predicted for Saudi Arabian limestone using the new proposed approach. Rock Material
Saudi Arabian limestone
rc (mm) 5.8
, R = 49 K ΙΙBDc , R=42 / K ΙSCB c
, R = 49 K ΙΙBDc , R =49 / K ΙSCB c
Experiment
Analytical prediction
Experiment
Analytical prediction
1.16
1.17
1.3
1.24
25
co-ordinate. Fig. 1. The elastic stresses around of the crack tip in polar co
26
Fig. 2. Geometry and loading conditions of (a) Centrally cracked Brazilian disc specimen (BD specimen) (b) Edge cracked semi semi-circular circular specimen subjected to three point bending (SCB specimen).
27
Fig. 3. The dimensionless geometry factors calculated for various a/R in the BD specimen under Ι Ι pure mode I loading. (a) A1nΙ , (b) A2n , (c) A3n .
28
Fig. 4. The dimensionless geometry factors calculated for various a/R and S/R in the SCB Ι Ι specimen under pure mode I loading loading. (a) A1nΙ , (b) A2n , (c) A3n .
29
Fig. 5. The dimensionless geometry factors calculated for various a/R in the BD specimen under ΙΙ ΙΙ ΙΙ pure mode II loading. (a) B1nΙΙ , (b) A2n , (c) B3n , (d) A3n .
30
Fig. 6. The dimensionless geometry factors calculated for various a/R and S/R in the SCB ΙΙ ΙΙ ΙΙ specimen under pure mode III loading loading. (a) B1nΙΙ , (b) A2n , (c) B3n , (d) A3n .
31
Fig. 7. The SCB specimens manufactured in four different sizes for studying the size effect on the mode II fracture toughness.
32
Fig. 8. (a) Loading set-up up utilized for conducting mode II fracture tests.. (b) Fracture samples.
33
Fig. 9. Load–displacement displacement curve corresponding to one of the SCB specimens with R = 100 mm. The curve indicates the brittle fracture behaviour of the tested rock material.
34
Fig. 10. Loading set-up up utilized for conducting mode I fracture tests.. (a) Sample before test. (b) Sample after test.
35
Fig. 11. (a) Loading set-up up utilized for conducting the tensile strength tests.. (b) Load– L displacement curve corresponding to one of the tensile strength tests. 36
Fig. 12. Comparison between the experimental results of Neiriz rock and the predictions of the new approach for the fracture toughness ratio.
37
Fig. 13. Comparison between the experimental results of Ghorveh marble and the predictions of the new approach for the fracture racture toughness ratio.
38
Fig. 14. Analytical predictions of K ΙΙxc / K Ιyc plotted against ax / rc for different values of (a) a y / rc , (b) S/R, (c) a/R.
39
Highlights:
• Specimen size effect on KIIc of brittle materials is investigated experimentally • A new analytical approach is proposed to estimate size- and geometry-dependent KIIc •
Good estimates can be provided for size- and geometry-dependent KIIc
40