An extended edge-notched disc bend (ENDB) specimen for mixed-mode I+II fracture assessments

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An extended edge-notched disc bend (ENDB) specimen for mixed-mode I+II fracture assessments A. Bahmani , M.R.M. Aliha , M. Jebalbarezi Sarbijan , S.S. Mousavi PII: DOI: Reference:

S0020-7683(20)30058-5 https://doi.org/10.1016/j.ijsolstr.2020.02.017 SAS 10615

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International Journal of Solids and Structures

Received date: Revised date: Accepted date:

28 November 2019 2 February 2020 9 February 2020

Please cite this article as: A. Bahmani , M.R.M. Aliha , M. Jebalbarezi Sarbijan , S.S. Mousavi , An extended edge-notched disc bend (ENDB) specimen for mixed-mode I+II fracture assessments, International Journal of Solids and Structures (2020), doi: https://doi.org/10.1016/j.ijsolstr.2020.02.017

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An extended edge-notched disc bend (ENDB) specimen for mixed-mode I+II fracture assessments A. Bahmani1,2,3, MRM. Aliha3*, M. Jebalbarezi Sarbijan3,4, SS. Mousavi3,5 1

Department of Mechanical Engineering, McGill University, Montreal, QC H3A 2K6, Canada.

2

Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Ave. West, Waterloo N2L 3G1, Canada. 3

Welding and Joining Research Center, School of Industrial Engineering, Iran University of

Science and Technology (IUST), Narmak, 16846-13114, Tehran, Iran. 4

Asphalt Mixtures and Bitumen Research Center (ABRC), Iran University of Science and Technology (IUST), Narmak, 16846-13114 Tehran, Iran. 5

School of Mechanical Engineering, Iran University of Science and Technology (IUST),

Narmak, 16846-13114, Tehran, Iran. * Corresponding Author E-mail address: [email protected]

Abstract A new test specimen for mixed-mode I+II fracture studies is proposed. An extended version of edge notch disc bend (ENDB) specimen, which was recently proposed to assess mixed-mode I+III fracture toughness, is designed and developed for introducing opening-in-plane-shear sliding mode. This new specimen that is called asymmetric ENDB specimen (A-ENDB), was numerically analyzed to determine the stress intensity factors and T-stress along the crack front for a wide range of geometric and loading conditions. The finite element results demonstrated that (i) the contribution of mode III component is negligible under asymmetric loading of ENDB specimen and; (ii) the full modes I and II mixities including pure mode I, pure mode II and intermediate mode mixities can be achieved by changing loading support distances. The capability of the proposed asymmetric ENDB specimen was then experimentally assessed by conducting several

mixed-mode I+II fracture toughness tests on a brittle model polymer (PMMA). The mode II fracture toughness value obtained from the A-ENDB specimen was noticeably greater than KIc value. Although the fracture trajectory of all A-ENDB specimens was nearly self-similar and along the plane of initial crack, different fracture surface morphologies were observed for mixed-mode I+II. The fracture surface consists of mirror area and ridge markings. While, towards to pure mode II, both density and length of the ridge marking were increased. It is shown that by using the same shape test specimen, loading type, and test configuration, both mixed-mode I+II and I+III can be studied. Keywords: Mixed-mode I+II and I+III; edge-notched disc bend (ENDB) specimen; fracture parameters; fracture surface 1. Introduction Cracked components in the engineering applications are subjected to complex loading conditions, and in most cases, a combination of tensile-shear or tensile-tear deformations may result in catastrophic failure of defected parts. In order to analyze the reliability of cracked components, it is necessary to understand the mixed-mode fracture mechanism. For this assessment, the experimental and theoretical framework of fracture mechanics is usually used. From the experimental viewpoint, crack growth mechanism, and the onset of unstable fracture is usually studied using suitable laboratory-scale test specimens. The selection of test specimen geometry for fracture toughness measurements indeed depends on the material type and its manufacturing processes. For instance, rectangular beams can be an easy and suitable candidate for metallic, ceramic, and polymeric materials (Zhu and Joyce, 2012).

On the other hand, disc and cylindrical shape specimens are among the favorite choices in the case of geomaterials such as rocks, cement concretes, and asphalt mixtures because they can be easily extracted from the cylindrical cores (Aliha et al., 2014, 2016a; Aliha et al., 2017a, 2017b; Aliha and Bahmani 2017; Mirsayar et al. 2017). Several test specimens with different cylindrical and disc shape geometries, as well as various loading setups, were used to measure the fracture toughness of various materials subjected to mixed-mode I+II and I+III (Aliha et al., 2019a, 2018a, 2016b; Aliha et al., 2017c; Saghafi et al., 2013). Examples of these specimens for mixed-mode I+II measurement include Brazilian disc specimen containing straight crack or chevron notch (Akbardoost et al., 2014; Aliha et al., 2017a; Wei et al., 2018) flattened Brazilian disc specimen (Wang and Wu, 2004), center cracked ring-shape specimen (Aliha et al., 2008), which all of them were loaded under diametrical compression. In addition, the edge-cracked semi-circular disc (Aliha et al., 2014, 2016b; Aliha et al., 2017a; Saghafi et al., 2013), short rod specimen (Barker, 1981; Ingraffea et al., 1984), and edge-cracked cylindrical specimen (Ouchterlony, 1990) are other frequently used test geometries for investigating the mode I and mixed-mode I+II fracture mechanism in rock materials. However, many drawbacks, including difficulties in the preparation of center crack or chevron notch in the disc specimens are seen in practice when some of the mentioned test specimens are used in laboratory-scale fracture toughness experiments. Various test specimens such as anti-plane four-point bend specimen (Rao and Liao, 2005), antiplane punch-through shear specimen (Bažant and Prat, 1988), asymmetric abnormal three-point bend specimen (Bažant and Prat, 1988), torsional circumferentially notched cylinder (Berto et al., 2013, 2012), edge-cracked plate loaded by a special pin loading fixture (Ayatollahi and Saboori, 2015), edge‐notched diametrically compressed disc (Bahmani et al., n.d.), and inclined edge cracked three-point bend specimen (Ahmadi-Moghadam and Taheri, 2013) were used to measure

the mixed-mode I+III fracture toughness value of different materials; however, most of these specimens are not able to cover a full range of mixed-mode I+III (Ahmadi-Moghadam and Taheri, 2013; Bažant and Prat, 1988; Rao and Liao, 2005). Also, some of them require complicated test configurations for testing (Berto et al., 2013, 2012). To fulfill this demand, the edge-notched disc bend (ENDB) specimen was successfully used to conduct a full range of mode I+III fracture toughness experiments from pure mode I to pure mode III for a verity of brittle and quasi-brittle engineering materials (Aliha et al., 2015a, 2015b, 2016a, 2016c, 2017c,2018a, 2019; Aliha and Bahmani, 2017; Bahmani et al., 2017). In several recently published studies, the effects of test specimen geometry and loading type (i.e., tension, compression, and bending) were studied (Aliha et al., 2018b; Aliha et al., 2017a; Aliha et al., 2017b) on the fracture toughness value of various brittle materials. The fracture toughness indeed is an inherent property of the material; however, interestingly, the type of test specimen geometry and loading dose influence the measuring of this material property (Aliha et al., 2018b; Aliha et al., 2017a; Aliha et al., 2017b). Accordingly, using a similar test specimen geometry and loading type for measuring the fracture toughness of materials under different mixed-mode loading conditions (i.e., mixed-mode I+II and I+III) has yet to be assessed. None of the mentioned test specimens are able to cover a full range of both mixed-mode I+II and I+III with the same geometry and loading type. Therefore, for each mixed-mode case (i.e., I+II and I+III) a particular test specimen with different geometry and loading type have been used. This can affect the experimental test data and their consistency. In this paper, the applicability of the ENDB specimen for being a new test specimen for studying mixed-mode I+II fracture toughness is numerically and experimentally investigated. Indeed, a new

test specimen that can be used for both mixed-mode I+II and I+III fracture studies without changing the geometry and loading type is introduced. 2. Asymmetric ENDB (A-ENDB) specimen Geometry and loading configuration of the suggested asymmetric ENDB (A-ENDB) specimen is illustrated in Fig. 1. The specimen is a disc with an edge crack at one side and along the diameter of the disc. The radius and height of specimen are denoted by R and B, respectively. While a represents crack depth. The specimen is loaded using a conventional three-point bend setup. If the span supports are symmetric relative to the crack plane (i.e., S1=S2), the specimen is subjected to pure opening mode (i.e., pure mode I) as indicated earlier by Tutluoglu and Keles (Tutluoglu and Keles, 2011) and Aliha and coworkers (Aliha et al., 2019a, 2018a, 2017c,2016a, 2016c, 2015a, 2015b; Aliha and Bahmani, 2017; Bahmani et al., 2017). However, by fixing one of the span supports (e.g., S1) in a constant distance relative to the crack, and decreasing the distance of other span support (e.g., S2) towards the crack plane, the influence of in-plane shear mode would also contribute to the opening mode of crack deformation. According to the finite element results, which will be shown later in section 4, S2 distance can be decreased until a dominant in-plane shear or sliding crack deformation occurs, and hence, the effect of opening mode or normal crack deformation would be zero or negligible. A full range of mode I+II mixities from pure mode I (i.e., opening) to pure mode II (i.e., in-plane sliding) can be attained simply by altering the position of one of the span supports.

Fig. 1. The geometry and loading configuration of asymmetric edge-notched disc bend (AENDB) specimen; when S1 = S2, opening deformation (i.e., pure mode I) occurs in the crack flank; when S2 < S1 the combination of tensile-in-plane shear (i.e., mixed-mode I+II) can happen, and when S2 << S1 or S2 is adequately close to the crack plane only in-plane shear deformation (i.e., pure mode II) can be attained. Crack tip stress/strain field in the A-ENDB specimen is described by singular and non-singular stress terms. The modes I and II stress intensity factors (i.e., KI and KII) are related to the singular terms, while the first non-singular term is called T-stress. Both these terms in the A-ENDB specimen are the functions of specimen geometry (a, B, and R values), loading configuration (P, S1, and S2 values), and crack front location (z/R) which are written as follows:

K I  Y I  a a / B , S 1 / R , S 2 / R , z / R 

(1)

K II  Y II  a a / B , S 1 / R , S 2 / R , z / R 

(2)

T  T

*

a / B , S 1 / R , S 2 / R , z

/R

(3)

where YI and YII are the normalized geometry factors corresponding to opening and shear crack deformation modes, which define the state of mode I and mode II mixity in the A-ENDB specimen. T* is also the non-dimensional form of T-stress; σ is reference stress which is defined via dividing the bending moment (M) of disc geometry to its section modulus (AM) as follows:

PS 1S 2 (S  S ) 3PS 1S 2 M   1 22  2RB AM RB 2 (S 1  S 2 ) 6

(4)

where P is applied load. Thus, KI , KII and T-stress are defined using following expressions:

KI 

3PS 1S 2Y I  a a / B , S 1 / R , S 2 / R , z / R  R B 2 (S 1  S 2 )

(5)

K II 

3PS 1S 2Y II  a a / B , S 1 / R , S 2 / R , z / R  RB 2 (S 1  S 2 )

(6)

T 

3PS 1S 2T * a / B , S 1 / R , S 2 / R , z / R  R B 2 (S 1  S 2 )

(7)

A suitable mixed-mode I+II test specimen should be able to produce the full range of mode mixities from pure mode I to pure mode II. Therefore, in the next section, the ability of A-ENDB specimen in this regard is numerically examined. 3. Finite element modeling A large number of A-ENDB models were analyzed using the commercial finite element (FE) code ABAQUS to compute mixed-mode I+II fracture parameters using the J-integral method. In these FE models, two different variables, including crack depth over disc height ratio (a/B) and span support distances over disc radius ratios (S1/R and S2/R), were considered. However, other geometrical features, including disc radius and height, were constantly defined as R = 37.5 mm and B = 15 mm. The crack depth over disc height ratio (a/B) was also varied from 0.2 to 0.6. In

order to produce tensile and in-plane shear mode mixities in the crack flanks, the right-hand span support (i.e., S2 distance) was varied from symmetric loading case (i.e., S1= S2) towards the crack plane. The left-hand span support (i.e., S1 distance) was meanwhile fixed for all FE models. By altering S2 distance, the influence of both shear and opening components would contribute to crack tip deformations. The span support distances over disc radius ratios (S1/R and S2/R) were varied from 0.7 to 0.9 for S1/R and from 0.9 to approximately 0.05 for S2/R. Fig. 2 shows a zoomed view of generated singular elements in the crack tip, as well as the total mesh scheme of the A-ENDB specimen. The total number of solid C3D20 elements for each case was around 53000. For the purpose of applying three-point bend boundary conditions, rigid body contact was assumed between loading and supporting spans and specimen surfaces. A general hard contact was assumed in the models, and both span supporters were fixed. A constant uniformly distributed reference load of P = 1000 N was thereafter applied on the top surface of the specimen along the disc diameter. The isotropic elastic properties of a typical brittle material such as polymethylmethacrylate (PMMA) were considered as E (tensile modulus) = 2950 MPa and ν (Poisson’s ratio) = 0.3 for these FE analyses.

Fig. 2. Mesh scheme of the A-ENDB specimen subjected to three-point bend loading using rigid element parts, and a zoomed view of the singular element at the crack tip.

4. Finite element results The FE results of analyzed A-ENDB specimens are illustrated and discussed in this section. 4.1. Variation of fracture parameters through the crack front Since the crack front in the A-ENDB specimen has been extended along a straight line through the disc diameter, depending on the crack front location, different stress intensity factors were obtained for each point. Figs. 3 (a-d) show the variation of normalized stress intensity factors along the crack front location defined by z/R parameter for a typical and constant crack length (i.e., a/B = 0.4). z parameter also has been illustrated in Fig. 1, and varies from z = 0 (i.e., the mid-section of the A-ENDB specimen) to z = ±R (i.e., the free surface regions of the A-ENDB specimen). In Fig. 3 (a), when S1 is equal to S2 (i.e., for symmetric specimen), only normalized mode I stress intensity factor exists in the crack front and both normalized shear stress intensity factors are zero in the entire crack front, revealing that the specimen is under pure mode I (i.e., pure tensile) loading condition. The variation of normalized KI along the crack front can also be seen in this Figure. Since the maximum value of normalized KI is attained at the mid-section (i.e., z/R=0), it is expected that the fracture of the A-ENDB specimen would be initially started from this critical point under pure mode I condition and then extended along the other points of the crack front. Fig. 3 (b) also shows the variation of normalized stress intensity factors for a typical mixed-mode I+II case (i.e., S1/R=0.9 and S2/R=0.4); by moving the right-hand span support (S2) towards the crack plane, in addition to the opening (or mode I) component, in-plane shear-mode component also contributes to fracture process of the A-ENDB specimen. In this loading condition, the value of normalized KI is higher than the in-plane shear-mode (i.e., normalized KII). By further decreasing S2/R to 0.2 (as shown in Fig. 3 (c)), the difference between the normalized KI and KII values becomes smaller,

demonstrating the increase of mode II contribution. Fig. 3 (d) illustrates that at a particular S2/R value, which here is 0.053, the value of normalized KI becomes zero in the range of

0.7  z R  0.7 , while the normalized KII values in this range are positive, indicating dominantly pure mode II condition exists in the A-ENDB specimen. Note that in all these loading conditions, the significance of induced mode III component (i.e., normalized KIII) relative to the normalized modes I and II stress intensity factors is very small. By moving from the free-surface regions (i.e.,

z R  0.8 ) towards the mid-section (i.e., z/R = 0), the value of KIII tends to be negligible and zero at the mid-point, concluding that a full range of mode I+II mixities (from pure mode I to pure mode II) can be attained using the A-ENDB specimen.

(a)

(b)

(d) (c) Fig. 3. Variations of normalized stress intensity factors (i.e., KI/KIm, KII/KIm, and KIII/KIm) along the crack front (i.e., z/R) for different mode mixities from pure mode I to pure mode II when the distance of left-hand span is a constant value of S1/R=0.9, and the distance of right-hand span (i.e., S2/R) is variable. In order to further assess the capability of the A-ENDB specimen to produce mixed-mode I+II deformations, KI and KII were both normalized via KIm, which is the corresponding value of mode I stress intensity factor at the mid-section of the A-ENDB specimen. The variation of KI/KIm, KII/KIm , and KIII/KIm along the crack front location (i.e., z/R) were demonstrated in Figs.4 (a-c) for different S2/R values ranging from 0.9 to 0.1. Except the free-surface regions of the A-ENDB specimen (i.e., z R  0.8 ), the variations of normalized stress intensity factors along the crack front for each S2/R value is relatively constant implying that the produced mixed-mode I+II deformations in this specimen is not sensitive to the crack front position which can be considered as a practical advantage for the A-ENDB specimen.

(a)

(b)

(c) Fig. 4. Variations of (a) KI/KIm, (b) KII/KIm, and (c) KIII/KIm versus the crack front (i.e., z/R) for different mode mixities when S1/R=0.9, and S2/R is variable.

In a three-dimensional cracked part, which is subjected to the multi-axial loading, the in-plane shear behavior can generally produce the out-of-plane shear deformations (Aliha and Saghafi, 2013; Pook et al., 2015; Sukumar et al., 2000). Therefore, in order to show the negligibility of induced mode III compared with modes I and II, the normalized KIII/KI and KIII/KII along z/R are also presented for different S2/R values. Figs. 5 (a-b) demonstrate that the mode III normalized values along the entire crack front, particularly in the vicinity of mid-section seems to be negligible. At the mid-section of the A-ENDB specimen, the magnitude of normalized stress intensity factors for modes I and II reach their maximum values, while normalized stress intensity factors for mode III is zero at this location. Therefore, it can be concluded that the A-ENDB specimen would experience merely tensile-in-plane mixed-mode I+II deformations at the onset of fracture wherein expected crack would be initiated at this critical point (i.e., mid-section of the AENDB specimen). Note that the computed results for free-surface regions (i.e., z /R  0.8 ) are not valid and must be disregarded due to variations in the crack tip singularity in these regions.

(a)

(b) Fig. 5. Variations of (a) KIII/KI and (b) KIII/KII versus the crack front (i.e., z/R) in a constant value of S1/R=0.9 and the variable value of S2/R from pure mode I to pure mode II. The variations of T* for a typical constant value of S1/R=0.9, a/B = 0.4, and different S2/R values versus z/R is illustrated in Fig. 6. The magnitude and sign of T* in the A-ENDB specimen is sensitive to S2/R value. The variation of T* in the range of z R  0.8 is nearly consistent.

Fig. 6. Variations of normalized T-stress (i.e., T*) along the crack front (i.e., z/R) for various mode mixities from pure mode I to pure mode II when S1/R = 0.9 and S2/R is variable.

4.2. Computed normalized parameters The computed geometry factors (YI and YII) and normalized T-stress (T*) at the mid-section point of the A-ENDB specimen are presented in this section. These parameters are calculated using Eqs. (5-7). Figs. 7 (a-c) and (d-f) illustrate the variations of normalized stress intensity factors (i.e., YI and YII) versus S2/R for different crack length ratios (a/B) and various S1/R distances. By decreasing S2/R, the magnitudes of YI values tend to zero for each case, and it is vice versa for YII values representing the full range of mixed-mode I+II mixities. Based on these figures, the variations of both YI and YII curves are not sensitive to S1/R parameter; but by increasing the crack depth (i.e., a/B ratio) the magnitudes of geometry factors (especially the mode I component) become greater for any given S1/R and S2/R. This insensitivity of the A-ENDB specimen to the position of span supports can be assumed as one of its advantages.

(a)

(d)

(b)

(e)

(f) (c) Fig. 7. Variations of (a-c) mode I and (d-f) mode II geometry factors (i.e., YI and YII) versus the different distances of right-hand span (i.e., S2/R) at the mid-section of A-ENDB specimen (i.e., z/R = 0) for different crack lengths and a constant distance of left-hand span (i.e., S1/R). Figs. 8 (a-c) also show the variation of T* versus S2/R for different crack length ratios (a/B) and various S1/R distances. Similar to geometry factors, T* is not significantly affected by the variation of S1/R. However, increasing the crack depth results in increasing the absolute value of T-stress parameter, particularly for dominantly pure mode II loading conditions. The sign of T-stress for all obtained results is negative (Figs. 8 (a-c)). Indeed, it was shown that negative T-stress can significantly influence on the load-bearing capacity of the A-ENDB specimen, and increases the mixed-mode I+II fracture toughness value (Smith et al., 2006).

(a)

(b)

(c)

Fig. 8. (a-c) Variations of normalized T-stress (i.e., T*) versus the different values of S2/R at the mid-section of A-ENDB specimen (i.e., z/R = 0) for different crack lengths (i.e., a/B) and a constant value of S1/R. Fig. 9 represents the size and shape of the von-Mises process zone along the crack front obtained from the numerical results of the A-ENDB specimen under pure mode I, mixed-mode I+II, and pure mode II. It is seen from this Figure that the shape of crack tip process zone is significantly affected by the mode mixity, such that the stress contour rotates about the crack plane from the symmetric condition by moving from mode I to mode II (as explained in other papers (e.g. Aliha et al. 2019b)). The contours illustrate the variations of process zone size. The minimum size of the process zone is seen at the mid-section of the specimen, and it tends to be increased toward the free surface regions (i.e., z /R  0.8 ). The reason for this variation of process zone size, stems from existing stress triaxiality constraints. Hence, the obtained mixed-mode I+II fracture toughness at the mid-section of the A-ENDB specimen is close to actual plane-strain fracture toughness. The A-ENDB specimen benefits from a long crack front compared with other proposed mixed-mode I+II test specimens (Aliha et al. 2016b, 2017a, 2018b, 2019b; Akbardoost et al. 2014). Therefore, it is expected that the obtained fracture toughness at the mid-section of the A-ENDB specimen can be a reliable representative for the minimum fracture toughness of a material.

Fig. 9. Shape and size of the von Mises stress contour for the crack front of A-ENDB specimen computed under pure mode I, mixed-mode I+II, and pure mode II. All three mode I, mode II, and mode III stress intensity factors generally existed in the crack front of the AENDB specimen under mixed-mode I+II and pure mode II loading conditions. The effect of mode III stress intensity factor is remarkable for the corner regions and free surfaces (i.e.,

z /R  0.8 ). Hence, it is essential to precisely define the onset of fracture, and the location of critical fracture point in the A-ENDB specimen under mixed-mode I+II and pure mode II loading conditions. Several semi-empirical fracture models proposed under multi-axial loading and exhibited that the critical location wherein crack initiates and propagates is along the direction of maximum equivalent stress intensity factor (Keq) (Richard et al., 2003; Hosseini-Toudeshky et al. 2013; Salimi-Majd et al. 2016; Sajjadi et al., 2015, 2016). These models are usually derived in terms of the normal and shear stress/strain components or directly from the modes I, II and III

stress intensity factors (KI, KII and KIII). For instance, Richard et al. (2003) and HosseiniToudeshky et al. (2013) used the following relation to determine the onset of fracture:

K eq-initiation 

KI K I2 2 2   1K II    2K III  2 4

(8)

where δ1= KIc / KIIc and δ2= KIc / KIIIc. Salimi-Majd and coworkers (Salimi-Majd et al. 2016; Sajjadi et al., 2015, 2016) proposed a local equivalent stress intensity factor for a branched crack (defined by Keq-branched) which can consider the effects of in-and out-of-plane shear fracture toughness values in addition to the tensile fracture resistance of a cracked material. The Keq-branched is written as follows:

K eq-branched 

K 

* 2 I

 1  K II*   2  K III*  2

2

(9)

where K I* , K II* , and K III* are the mode I, mode II and mode III local stress intensity factors (SIFs) for the branched crack, respectively that can be obtained from the global SIFs of the main crack. λ1 and λ2 are material constants defined to consider the contribution of each local SIFs and the influence of material characteristics in fracture phenomenon. λ1 and λ2 are 0.1 and (KIc/ KIIIc)2, respectively (Salimi-Majd et al., 2016). The required material constants (i.e., KIc, KIIc, and KIIIc) of the PMMA material were obtained from Aliha et al. (2015a, 2016b, 2016c, 2017c), then the equivalent stress intensity factors (i.e., Keqinitiation and

Keq-branched) were computed from Eqs. (8) and (9) along the crack front of the A-ENDB

specimen for different mode mixities (Fig. 10). It is seen that by moving from corner regions and free surfaces (i.e., z /R  0.8 ) towards the mid-section, the magnitudes of both equivalent stress intensity factors (i.e. Keq-initiation at the onset of fracture and Keq-branched at the beginning of

propagation stage) increases such that their maximum values are obtained at z/R = 0 for any mode mixties.

(a)

(b) Fig. 10. The variation of equivalent stress intensity factor through the crack front of the A-ENDB specimen for different mode mixities when S1/R=0.9, and S2/R is variable. (a) equivalent stress intensity factor for initiation computed using Eq. 8; (b) equivalent stress intensity factor for branching computed using Eq. 9.

5. Experimental tests In order to experimentally assess the capability of the proposed A-ENDB specimen under mixedmode I+II condition, a set of experiments were performed using polymethylmethacrylate (PMMA) as a well-known model brittle material in laboratory-scale studies. Several circular discs with the radius of R=37.5 mm and thickness of B=15 mm were cut from a PMMA sheet. A rotary diamond saw blade with a thickness of 0.3 mm was then used to create an artificial crack with the initial depth of slightly less than 6 mm through the diameter of the specimen. A razor blade thereafter was used to create a sharp pre-crack and make the final length of crack equal to 6 mm. Twenty AENDB specimens were prepared to conduct four different mixed-mode I+II mixities, which each case repeated four times to accurately measure the fracture toughness value. These tests were implemented using a three-point bend fixture installed on a universal servo-hydraulic test machine (GALDABINI, Italy) with a displacement control monotonic rate of 1 mm/min at room temperature. Fig. 11 (a) shows the A-ENDB specimen under pure mode II condition before fracture. Fig. 11 (b) also exhibits the typical load-displacement curves for different tested mode mixities (from pure mode I to pure mode II). This figure shows the brittle and linear behavior of tested PMMA material. Due to the brittle behavior of the PMMA, the peak load of each curve was used as P in Eqs. (5, 6) to measure the critical stress intensity factors.

(a)

(b)

Fig. 11. (a) A-ENDB specimen subjected to pure mode II using asymmetric three-point bend configuration; (b) typical load-displacement curves for each tested mode mixities. A mode mixity parameter was also defined as follows:

M 12e 

K  arctan  I    K II  2

(8)

This parameter was used to determine the contribution of each mode (i.e., mode I and mode II) in the crack tip deformation. M 12e  1 and M 12e  0 denote pure mode I and pure mode II loading conditions, respectively. In order to have consistency in the tests and mode mixities, the left-hand span support was fixed at S 1 / R  0.8 (i.e., S1=28 mm), and the right-hand span support (i.e., S1) was varied from pure mode I to mixed-mode I+II, and then pure mode II. The length of right-hand span support (i.e., S1) and its corresponding mode mixity (i.e., M 12e ) for conducting mixed-mode fracture toughness experiments on the PMMA is indicated in Table 1.

Table.1. The distances of left-hand span support for each tested mode mixity.

M 12e

S 2 (mm)

1.00

28.00

0.80

7.00

0.4

3.50

0.2

2.75

0.00

2.00

6. Experimental results Table. 2 indicates the fracture toughness test results for each mode mixity and corresponding fracture load, the length of right-and left-hand span support, and geometry factors. The average value of fracture toughness under mode I (KIc) measured form the A-ENDB specimen is 1.45 MPa√m. This value is also reported in the literature ranging from 1 to 2.1 MPa√m (Aliha et al., 2016b, 2016c, 2017c; Ayatollahi et al., 2019, 2011, 2006; Ayatollahi and Aliha, 2009; Ayatollahi and Saboori, 2015; Davenport and Smith, 1993; Maccagno and Knott, 1989; Razavi and Berto, 2019; Saghafi et al., 2013; Zeinedini, 2019). The average value of pure mode II fracture toughness (KIIc) in this specimen is 2.60 MPa√m, while this value have been reported in the range of 1-2 MPa√m (Aliha et al., 2016b, 2016c, 2017c; Ayatollahi et al., 2019, 2011, 2006; Ayatollahi and Aliha, 2009; Ayatollahi and Saboori, 2015; Davenport and Smith, 1993; Maccagno and Knott, 1989; Razavi and Berto, 2019; Saghafi et al., 2013; Zeinedini, 2019).

Table 2. Details of tested the A-ENDB specimen under mixed-mode I+II including four different mode mixities and corresponding span supports distances, geometry factors, maximum peak loads, as well as fracture toughness values.

M 12e

S1 (mm)

S2 (mm)

YI

YII

Pcr (N)

KIc (MPa√m)

KIIc (MPa√m)

1.00

28.00

28.00

1.20

0.00

1652

1.35

0.00

1.00

28.00

28.00

1.20

0.00

1921

1.57

0.00

1.00

28.00

28.00

1.20

0.00

1721

1.41

0.00

1.00

28.00

28.00

1.20

0.00

1793

1.47

0.00

0.80

28.00

7.00

0.92

0.29

4535

1.14

0.35

0.80

28.00

7.00

0.92

0.29

5282

1.32

0.41

0.80

28.00

7.00

0.92

0.29

4166

1.04

0.33

0.80

28.00

7.00

0.92

0.29

4980

1.25

0.39

0.40

28.00

3.50

0.55

0.92

9034

0.75

1.26

0.40

28.00

3.50

0.55

0.92

11456

0.95

1.60

0.40

28.00

3.50

0.55

0.92

12755

1.06

1.78

0.40

28.00

3.50

0.55

0.92

10323

0.86

1.44

0.20

28.00

2.75

0.27

1.48

9023

0.29

1.63

0.20

28.00

2.75

0.27

1.48

10255

0.33

1.85

0.20

28.00

2.75

0.27

1.48

11100

0.36

2.00

0.20

28.00

2.75

0.27

1.48

12880

0.42

2.33

0.00

28.00

2.00

0.00

2.11

15803

0.00

3.03

0.00

28.00

2.00

0.00

2.11

14015

0.00

2.69

0.00

28.00

2.00

0.00

2.11

13297

0.00

2.55

0.00

28.00

2.00

0.00

2.11

11165

0.00

2.14

The A-ENDB specimens were fractured at the crack front without local damage in the span support region. The fracture trajectory of all tested A-ENDB specimens under different mode mixities (i.e., from pure mode I to pure mode II) was nearly straight and extended along the initial crack (even for pure mode II case). Fig. 12 shows the fracture path of the tested A-ENDB specimen for the different mode mixities, including pure mode I ( M12e  1 ), mixed-mode I+II ( M12e  0.4 ), and pure mode II ( M12e  0 ); none of these cases does have any significant curvature or kinking trajectory. Although the fracture trajectory of different mode mixities were nearly similar, their fracture surfaces were entirely different.

M12e  1 (mode I)

M12e  0.4 (mixed-mode I+II)

M12e  0 (mode II)

Fig. 12. Nearly straight fracture trajectory observed for the tested A-ENDB specimens under different mode mixities (i.e. M12e  1 , 0.4, and 0) Fig. 13 shows the fracture surface for the A-ENDB specimens tested under different mode mixities. The fracture surfaces in these specimens consist of two different zones: (i) mirror area (i.e., very flat and smooth surface) and (ii) area of ridge markings. For M12e  1 and 0.8 in which pure mode I is dominant, a mirror area was seen at the fracture surface. While the very small ridges with the length of approximately 1.70 mm ( M12e  1 ) and 2.95 mm ( M12e  0.8 ) were observed

along the entire crack front. By increasing mode II contribution (i.e., M12e  0.4 ), both density and length of ridges are grown along the entire crack front as well as, a semi-elliptical region is formed in the fracture surface. The highest length of ridges is seen at the mid-section of the crack front; thus, the fracture initiation and propagation could be started at this point and extends along the radial direction to form a thumbnail shape surface, which is consisted of ridges. The ridges are extended towards the upper loading roller. The length of ridges also tends to be enlarged by moving from pure mode I towards pure mode II loading condition. Thus, the highest ridge length is seen when the A-ENDB specimen is subjected to pure mode II. The fracture surface of pure mode II had the most density of parallel smooth ridges inside the semi-elliptical zone ahead of the crack front. In addition, some rough and long ridges were also seen in the corner regions ( z R  0.8 ). The formation of such different ridge markings can be attributed to the edge, and corner effects in the tested A-ENDB specimens. Fig. 13 shows the schematic representation of fracture morphology observed for different mode mixities. No hyperbolic markings were created for the fracture surface of tested A-ENDB specimens, and only the ridge markings were seen in the whole specimens under different mode mixities. The main reason for this observation can be related to the negative sign of T-stress that exists along the entire crack front of A-ENDB specimen and for the full range of mode mixities (as shown in Figs. 6 and 8). The relation obtained for the shape of ridges in the fracture surface and the sign of T-stress is in agreement with the conclusions of previous works performed on the fractured surface morphology of PMMA material (Bhattacharjee and Knott, 1995; Gong and Bandyopadhyay, 2007; Lach and Grellmann, 2017; Ono and Allaire, 2000; Smith et al., 2006).

Fig. 13. Realistic and schematic morphology of fracture surface observed for different mode mixities of tested PMMA using the A-ENDB specimen. 7. General discussion and conclusion The average value of KIIc/KIc is 1.80, with a ±25% deviation. Not only the mode II fracture toughness of the A-ENDB specimen is remarkably higher than pure mode I fracture toughness, but also it is greater than other proposed specimens. The reason for this higher mode II fracture toughness, stems from the negative sign of T-stress through the crack front in the A-ENDB specimen; in addition, the relatively self-similar fracture trajectory of the A-ENDB specimen under mixed-mode I+II and pure mode II loading conditions. Several theoretical models have been proposed to predict the fracture parameters of mixed-mode I+II. Examples of these well-established theoretical models include maximum tangential stress (MTS) (Erdogan and Sih, 1963), strain energy density (SED) (Berto and Gomez, 2017; Sih, 1974),

maximum tangential strain (MTSN) (Chang, 1981), generalized maximum tangential stress (GMTS) (Smith et al., 2001), and extended maximum tangential strain (EMTSN) (Mirsayar, 2015; 2017; 2018, Mirsayar et al. 2016; 2018a,b). The prediction of the MTS model for pure mode II condition (i.e., KIIc/KIc) is constant and around 0.87 when the Poisson's ratio effect is not considered (Mirsayar, 2015). In addition, the prediction of the GMTS model for KIIc/KIc of the PMMA for specimens with negative T-stress values is in the range of 1-1.20 (Aliha et al., 2016b; Ayatollahi et al., 2011; Saghafi et al., 2013). For the PMMA material, which its Poisson's ratio is in the range of 0.30-0.35, KIIc/KIc prediction of the MTSN and EMTSN is around 0.50 (Mirsayar, 2015). There are other test specimens, including punch-through shear test (PTS) (Backers and Stephansson, 2012; Yoon and Jeon, 2004) and shear-box test of the cubic specimen (Rao et al., 2003) in which the value of KIIc is remarkably higher than KIc, and hence, these well-established fracture models are not able to predict pure mode II conditions. The common characteristics between the A-ENDB specimen (presented in this work), punch-through shear test (PTS), and shear-box test of the cubic specimen are that in all of them, first, the fracture path is almost selfsimilar, and crack is not noticeably kinked relative to the crack plane under the entire range of mixed-mode I+II. Second, the T-stress is negative. These specimens can produce a particular configuration that mixed-mode crack does not follow the direction of maximum hoop stress/strain, and due to excessive constraint, the crack is forced to propagate along a certain and predefined direction. Thus, the value of KIIc is remarkably higher than KIc, and hence, these theoretical models which have been written based on maximum hoop stress cannot provide a good correlation or prediction for the mode II or mixed-mode I+II loading conditions.

The high values of KIIc/KIc have been reported particularly for geo-materials in oil and gas reservoirs as well as mines (Backers and Stephansson, 2012; Rao et al., 2003; Yoon and Jeon, 2004). A theoretical model is indeed required to accurately predict this high ratio of KIIc/KIc for mode II case and also other mixed-mode I+II mixities in such high constraint cracked specimens. However, empirical elliptical and parabolic equations are fitted with experimental data in Fig. 14. 2

 K I   K II      1,   0.83  K Ic    K IIc  

(9)



 K I   K II       1,   0.77,   1.50 K K  Ic   IIc 

(10)

Fig. 14. Normalized experimental data (KII/KIIc versus KIc/KIc), and empirical fitted equations. This characteristic of the A-ENDB specimen may make it a practical and suitable test specimen, particularly for geo-materials society. Both mixed-mode I+III and I+II fracture toughness can also be captured using the ENDB specimen and its asymmetry form (i.e., A-ENDB specimen), which is proposed in this study. Both ENDB and A-ENDB specimens are identical under pure mode I. The only difference between these two specimens is that in the ENDB specimen, the crack

inclination angle alters from 0° to 62° degree relative to loading axis to produce mixed-mode I+III (0° for pure mode I and 62° for pure mode III) (Aliha et al., 2019a, 2018a, 2016a, 2016c, 2015a, 2015b; Aliha and Bahmani, 2017; Bahmani et al., 2017), while it is shown that to capture mixedmode I+II, the ENDB specimen can be subjected to an asymmetry three-point bend loading when the crack inclination angle is constant at 0° degree; however, one of the bottom loading supports moves toward the crack plane to alter the state of mode I and II mixity. This form of the ENDB specimen is termed A-ENDB in this study. Since the simplicity in test specimen geometry, preparation process, loading configuration, as well as its cost-efficiency, are essential parameters in experiments, the proposed A-ENDB specimen benefits from all these parameters. Accordingly, the combination of similar disc shape specimen and three-point bend loading is an advantage for measuring both mixed-mode I+II and I+III fracture toughness of brittle materials consistently. The main concluding remarks of this study are presented as follows: -

A simple disc shape test specimen with easy loading configuration termed asymmetry edge-notched disc bend (A-ENDB) for studying mixed-mode I+II crack deformations was proposed. The fracture parameters of this specimen were numerically computed using finite element analyses, and it was shown that this specimen is able to capture the full range of mixed-mode I+II.

-

Several experimental tests were conducted using the A-ENDB specimen to measure the mixed-mode I+II fracture toughness of a typical brittle material (PMMA). The experimental results showed that the pure mode II fracture toughness measured by AENDB specimen is remarkably higher than the previous data reported in the literature for PMMA.

-

The fracture surface of the A-ENDB specimen under different mode mixities was relatively self-similar along the pre-crack plane from pure mode I to pure mode II. The fracture surface and morphology of fracture were different for mode I and mixed-mode I+II cases. The length and density of ridge markings were increased by moving from dominant mode I conditions towards dominant mode II loading conditions.

-

A full range of mixed-mode I+II was captured using A-ENDB specimen, while a full range of mixed-mode I+III was previously captured via the symmetry form of this specimen. It was shown that using same geometry and loading type (i.e., three-point bending), both mixed-mode I+II and I+III fracture toughness, can be assessed for brittle materials.

Declaration of interests - The authors declare that there is no conflict of interest. This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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