Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 29, No. 5, October, 2016 ISSN 0894-9166

Experimental and Numerical Study of Shear Fracture inBrittle Materials with Interference of Initial Double

Cracks⋆⋆

Hadi Haeri1⋆ Vahab Sarfarazi2 Mohammad Fatehi Marji3

Ahmadreza Hedayat4 Zheming Zhu5

(1Young Researchers and Elite Club, Bafgh Branch, Islamic Azad University, Bafgh, Iran)(2 Department of Mining Engineering, Hamedan University of Technology, Hamedan, Iran)

(3Associate Prof., Head of Mine Exploitation Engineering Department, Faculty of Mining and Metallurgy,

Institution of Engineering, Yazd University, Yazd, Iran)(4Department of Civil and Environmental Engineering, Colorado School of Mines,

Golden, Colorado 80401, USA)

(5College of Architecture and Environment, Sichuan University, Chengdu 610065, China)

Received 19 August 2015, revision received 17 December 2015

ABSTRACT A simultaneous experimental and numerical study of shear fracture of concrete-likematerials is carried out using Brazilian disc specimens with initial double edge cracks and fourpointbending beam specimens with double edge-notches. The interference effects of two cracks/notchesare investigated through varied ligament angles and crack lengths. It is shown that shear fracturingpaths change remarkably with the initial ligament angles and crack lengths. The cracked specimensare numerically simulated by an indirect boundary element method. A comparison between thenumerical results and the experimental ones shows good agreement.

KEY WORDS double edge cracks, concretelike specimens, crack propagation, indirect shear load-ing, overlapped cracks

I. IntroductionConcrete is one of the most widely used construction materials because of its relatively high compres-

sive strength, high durability, long-term stability, low permeability and good fire resistance properties.A concrete structure, either plain or reinforced can be readily handled and placed into any form andcast into any desired shape, and is therefore very flexible and common in use. Cracks that occurbefore hardening are usually the result of settlement within the concrete mass, or shrinkage of thesurface (plastic-shrinkage cracks) caused by rapid loss of water while the concrete is still plastic. Inreinforced concrete the settlement cracks may develop over embedded items, such as reinforcing steelor adjacent to forms or hardened concrete as the concrete settles or subsides. In plain concrete, thesettlement cracking results from insufficient consolidation (vibration) and high slumps (overly wet con-crete). Plastic-shrinkage cracks are relatively short cracks that may occur before the final finishing ondays with high winds, low humidity, and high temperature. Surface moisture evaporates faster than itcan be replaced by rising bleed water, causing the surface to shrink more than the interior concrete.

⋆Corresponding author. E-mail: haerihadi@gmail.com

⋆⋆Project supported by Iran’s National Elites Foundation (INEF). Partial supports of center of excellent for structures

and earthquake engineering at Sharif University of technology is greatly appreciated.

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As the interior concrete restrains shrinkage of the surface concrete, stresses develop and exceed theconcrete’s tensile strength, resulting in surface cracks. Cracks that occur after hardening are usuallythe result of drying shrinkage, thermal contraction, or subgrade settlement. While drying, hardenedconcrete will shrink. To accommodate this shrinkage and control the location of cracks, joints are placedat regular intervals within the concrete structure. However, the presence of different cracks (which areusually produced in the form of internal cracks within the concrete texture) may extremely reducethe strength of concrete structures especially in initial hours of casting and even before it is placedunder loading. Concrete may break due to both shear and tension under various loading conditions.The shear fracturing of concrete may be studied by investigating the propagation mechanism of thesedefects (internal cracks). The shear fracture in pure Mode II (shearing mode) may occur under somepeculiar conditions but the mixed mode fracture (Mode I or opening mode and Mode II) may occur inmany circumstances in concrete structures[1].

The mechanical behavior of pre-notched concrete structures under shear loading condition hasattracted the attention of researchers and practical engineers over the last three decades. Extensivework on the coalescence pattern and shear resistance of pre-existing cracks in solid materials havebeen performed in many experimental and theoretical studies[2]. The mechanical behavior of concretemay depend on the extension of pre-existing cracks (notches) which in turn depends on the geometricdistribution and intensity of these flaws, their sizes, positions, orientations, and loading conditions.

Kaplan[3] investigated the fracturing behavior of notched concrete beams using the linear elasticfracture mechanics (LEFM) theory for three and fourpoint bending tests conducted on concrete androck materials. He suggested some models for predicting the fracture behavior of concretes such as theFictitious Crack Model (FC-model)[4], the Crack Band model[5,6] and the Two Parameter Model[7].Ozcebe et al.[8] studied the breakage of T-type beams and Ruiz and Carmona[9] considered the effectof the shape of crack propagation paths on rectangular and T-type beams. On the other hand, Ruiz etal.[10] theoretically analyzed the initiation and propagation cracks at the head of the beam. Savilahti etal.[11] carried out direct shear tests on specimens of jointed plaster material containing non-overlappedand overlapped joints. Wong et al.[12] conducted direct shear tests on specimens of plaster material andnatural rocks containing open non-persistent joints to investigate the shear strength and breakage ofbrittle solids. According to their findings, the breakage pattern of the specimens mainly occurred by jointseparation. In addition, Gehle and Kutter[13] studied the breakage and shear behavior of intermittentrock joints under direct shear loading conditions. It has been concluded that the shear resistance ofpre-cracked rock specimens may be affected by crack inclination. Many experimental studies havealso been devoted to studying the initiation, propagation, interaction and coalescence of pre-existingcracks in specimens of various solids, including natural rocks or rock-like materials under compressiveloading[14–18].

In the present work, the crack propagation mechanism of pre-cracked concrete specimens is studiedby carrying out some indirect shear tests on the specially prepared samples (with a proper mixture ofPortland Pozzolana Cement (PPC), fine sand and water) in a geotechnical laboratory. The mechanismof crack propagation in these pre-cracked specimens is also simulated numerically by a modified higherorder displacement discontinuity method. A computer code is provided by using a cubic variation ofdisplacement discontinuities along each boundary element (i.e. each boundary element contains fourequal sub-elements). The Mode I and Mode II stress intensity factors are computed based on the LEFMconcepts. The σ-criterion is implemented in this computer code to predict the possibility of crackpropagation and estimate the crack initiation direction.

In this study, the double edge-cracked Brazilian disc and double edge-notched beam specimens aretested to study the crack propagation process of brittle concrete specimens. It is observed that the tensilecracks (wing cracks) are commonly produced in indirect shear tests (Brazilian disc and fourpoint flexurebending tests). These wing cracks are generated from the original tips of the pre-existing notches. Insome cases more than one crack (single crack) may occur in concretes. As the shearing behavior of pre-cracked specimens can be studied using various cases the design of some proper specimens with differentgeometries and different loading conditions can further increase the knowledge and understanding ofthe fracturing behavior of concrete.

The specimen with double notches (cracks) represents more practical crack orientation compared tothe specimen containing a single crack. Moreover, the effects of ligament angles and crack lengths on

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the shearing breakage path of beam and disc specimens are also investigated. Results of the numericalanalysis are compared with the corresponding experimental ones. This comparison shows that resultspredicted using the numerical approach fit the experimental ones with adequate accuracy.

II. Experimental StudiesShear fractures may occur in brittle materials such as concrete under various loading conditions.

Although fibers may be added to increase the ductility of concrete[19], there is a great interest indetermining the shear behavior of concrete. There are different types of specimen geometry and testingprocedure which can be used to produce shear stress in a pre-determined failure plane such as notched,pre-cracked/jointed specimens (e.g. in a shear-box under direct shearing or in other instruments usedfor diametric and compressive loading conditions)[20]. The Brazilian disc test and fourpoint bendingflexural test are shown in Fig.1[20]. The shear tests on the Z-type Brazilian discs (Fig.1(a)) and on theN-type edge-notched beams (Fig.1(b)) are mostly used to determine the shear strength of concrete[20].These specimens are basically in the form of a shear prism including two L-type blocks connectedthrough a plane on which the shear stresses can be induced by applying a compressive line load. Thesetest configurations allow the propagation of cracks under mode I, mode II and mixed mode I/II loadingaccording to the concepts of fracture mechanics[21].

Fig. 1. (a) Z-type specimen, (b) N-type specimen for shear tests.

In this study, the Z-type (double edge-cracked Brazilian discs) and the N-type (double edge-notchedbeam) specimens are used for all shear tests carried out on the concrete.

2.1. Preparation of the pre-cracked specimens and their mechanical properties

The pre-cracked Brazilian disc and the notched beam specimens of quasi-brittle materials (concrete)with 100×30 mm (100 mm in diameter and 30 mm in thickness) and 50×10×10 cm (50 cm in length,10 cm in width and 10 cm in thickness) are specially prepared from a mixture of PPC, fine sand andwater. Table 1 shows the mix preparations and mechanical properties of control specimens with nointentional pre-cracking.

Table 1. Mix preparations and mechanical properties of the control specimens with no intentional pre-cracking

Mix preparations (%) Mechanical properties

Tensile Uniaxial Fracture ElasticPoisson’s

PP cement Fine sand water strength compressive toughness modulusratio

(MPa) strength (MPa) (MPa m1/2) (GPa)44.5 22.5 33 3.81 28 0.3 17 0.21

For the pre-cracked Brazilian disc specimens where each contains two pre-existing edge cracks, theedge cracks were created by inserting thin metallic shims with different widths (indicating the pre-existing crack lengths) and a constant thickness (2 mm) into the specimens when they were cast in themold (Fig.2(a)).

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Fig. 2. Typical molds: (a) for edgecracked Brazilian disc specimen (b) for edge-notched beams.

The pre-notched beam specimens were also prepared by mixing PPC, fine sand and water and thenpouring them into a steel mold with an internal dimension of 50 × 10 × 10 cm (Fig.2(b)). The moldcontained four steel sheets bolted together plus two fiberglass plates which were placed at the top andbottom of the mold (Fig.2(b)). The liquid mixture was fed into the mold through two orifices on the topplate. The upper and lower surfaces of the mold had slits with 1 mm in thickness and the length andwidth of the mold were both 10 mm. Through these slits, notches or cracks were created by inserting twothin metallic shims (2 mm in thickness) before casting the specimens (Fig.2(b)). Grease was brushedon the shim to facilitate future removal. After about 7 hours; the steel shims were removed from themolds, and the specimens were kept in laboratory environment for 28 days.

After the mixtures were placed and hardened within the molds, each shim left an open crack inthe specimen through the thickness which was perpendicular to the front and back of the specimen. Itshould be noted that the grease prevented damage of the crack during the shim being removed.

The edge-cracked disc and the edge-notched beam specimens containing two edge cracks of differentligament angles, Ψ and crack lengths, b were prepared and tested in a geotechnical laboratory. Thecompressive line loading, F was applied and the loading rate was kept constant (at about 0.5 MPa/s)during the tests.

Two edge cracks with different locations (non-overlapped/overlapped cracks) for the double edge-cracked disc specimens are shown in Figs.3. The crack geometry is defined by two parameters namelyΨ showing the Ligament angle (which is the counterclockwise angle between the ligament length andshear axis) and b denoting the lengths of the two cracks, respectively. For different ligament angles, thelengths of edge cracks are different. The ligament angles at Ψ = 70◦, 90◦, 96◦, 105◦, 110◦, 130◦ and thecrack lengths at b = 44, 45, 47, 50, 55, 63 mm are used throughout this research.

Two cracks with different locations (non-overlapped/overlapped cracks) for the double edgenotchedbeam specimens are shown in Fig.4. Different ligament angles at Ψ = 0◦, 20◦, 40◦, 45◦, 65◦, 163◦ anddifferent crack lengths at b = 15, 17, 33, 38, 50, 58 mm are used in this study.

2.2. Shear fracturing behavior of the double edge-cracked Brazilian disc specimens

The Brazilian disc tests were carried out on the Z-type disc specimens to study the shear fracturebehavior of concrete. These specimens (with different ligament angles and crack lengths) were speciallyprepared and tested in the laboratory. Each specimen contained two eccentric notches and was testedunder compression. The results of these tests were used to analyze the fracturing behavior of concretespecimens It has been observed that the tensile cracks (wing cracks) immediately and quasi-staticallyinitiated at the tips of the original edge cracks (Fig.5). The development and coalescence of wing cracksin the bridge area (the area in between the two pre-existing edge cracks) may be the main cause of thefractured paths in the concrete discs.

It can be seen that when the ligament angle Ψ = 70◦ (Fig.5(a)), the wing cracks may initiate at tipsof the pre-existing cracks and propagate in curved paths so that the specimen may fail due to the twofracture paths of the two propagating edge cracks (the wing cracks are originated from the tips of the

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Fig. 3. Crack geometry showing different ligament angles, Ψ and crack lengths, b for the double edge-cracked disc specimens.

upper and lower edge cracks). In some cases coalescence may not occur at tips of the propagating edgecracks. When the ligament angles are taken as Ψ = 90◦ and Ψ = 96◦ (Fig.5(b), (c)), the wing cracksmay initiate at tips of the edge cracks and propagate in the bridge area. These cracks may also coalescewith each other at tips of the propagating cracks. When the ligament angles are chosen as Ψ = 105◦

and Ψ = 110◦ (Fig.5(d), (e)), the wing cracks may initiate at the tips of both cracks and then the lowerwing crack of the upper crack and the upper wing crack of the lower crack may extend and coalescewith each other.

It should be noted that for the case shown in Fig.5(f) (Ψ = 130◦) the cracks may not propagatefrom the tips but propagate from the side-walls, therefore, the specimen may fail due to axial splittinglike the un-cracked specimen in a conventional Brazilian test.

2.3. The shear behavior of double edge-notched beam specimens

The fourpoint bending (flexural) test may generate the conditions of mixed mode I/II loading andallow the propagation of cracks due to both tensile and shear loadings. Various shear tests of the doubleedge-notched beam (DENB) specimens (or N-type beam specimens) were carried out in the laboratoryunder bending stresses (central loading). The pre-existing cracks are propagated and the failure of

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Fig. 4. Crack geometry showing different ligament angles, Ψ and crack lengths, b for the double edge-notched beam

specimen.

N-type beam specimens normally occurs in the flexure failure mode rather than the shear failure mode.The flexure failure mode is usually a function of the eccentricity of the applied compressive load[21].

Figures 6(a)-(g) illustrate the fracture paths of non-overlapped cracks (for angles Ψ = 0◦, 20◦, 40◦,45◦, 65◦) and overlapped cracks (for angle Ψ = 163◦). For the case shown in Fig.6(a) (Ψ = 0◦), thewing cracks initiated at the crack tips and continued growing in a direction almost perpendicular to thedirection of the maximum compression throughout the bridge area. This fracturing configuration showsthat the bridge area may fail with a single breakage surface. For the non-overlapped cracks shown inFigs.6(b), (d), (e) (i.e. for Ψ = 20◦, 45◦, 65◦), the wing cracks initiated at the crack tips and coalescedwith each other. Also, for the case of non-overlapped cracks shown in Fig.6(c) (i.e. Ψ = 40◦), the wingcracks initiated at tips of the two original cracks and propagated in a curved path until they coalescedwith the tip of the other crack.

For the case of overlapped cracks shown in Fig.6(e) (i.e. for Ψ = 163◦), the wing cracks initiatedat the walls of the two original cracks and propagated in a direction (approximately) perpendicular tothe direction of the compressive line load. These wing cracks were developed in the intact portion (i.e.not in the bridge area) and then coalesced with the opposite crack tips.

The failure load of the double edge-cracked Brazilian disc specimens with different ligament anglesand crack lengths was measured by testing specimens in the laboratory. The same tests were alsoperformed for the double edge-notched beam (DENB) specimens. Table 2 presents the wing crackinitiation and failure loads for the double edge-cracked Brazilian disc and double edge-notched beam(DENB) specimens (shown in Figs.3 and 4). Initiation and propagation of the wing cracks (from theupper and lower cracks) were observed with naked eyes (i.e. starting from 340 N to 710 N for the double

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Fig. 5. Cracking patterns in double edge-cracked disc specimens with different ligament angles and crack lengths.

Fig. 6. Fracture patterns in the double edge-notched beam specimens with different ligament angles and crack lengths.

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edge-cracked Brazilian disc specimens and starting from 410 N to 920 N for the double edge-notchedbeam (DENB) specimens). These specimens completely failed at 870-1250 N in the case of doubleedge-cracked Brazilian disc specimens and at 970-1640 N for the double edge-notched beam (DENB).

Table 2. The wing crack initiation and failure loads for the double edge-cracked Brazilian disc and double edge-notched beam(DENB) specimens

Double edge-cracked Brazilian disc specimens

crack geometriesWing crack initiation load (N)

Breakage load (N)Upper crack Lower crack

(a) Ψ = 70◦, b = 45 mm 710 690 1250(b) Ψ = 90◦, b = 47 mm 670 650 1180(c) Ψ = 96◦, b = 50 mm 570 530 1090(d) Ψ = 105◦, b = 46 mm 350 340 870(e) Ψ = 110◦, b = 55 mm 640 610 1140(f) Ψ = 130◦, b = 63 mm 420 390 1020

Double edge-notched beam (DENB) specimens

Crack geometriesWing crack initiation load (N)

Breakage load (N)Upper crack Lower crack

(a) Ψ = 0◦, b = 50 mm 560 540 1020(b) Ψ = 20◦, b = 33 mm 720 710 1260(c) Ψ = 40◦, b = 15 mm 860 860 1310(d) Ψ = 45◦, b = 38 mm 810 810 1460(e) Ψ = 65◦, b = 17 mm 920 890 1640(f) Ψ = 163◦, b = 58 mm 420 410 970

III. Numerical Simulation of the Experimental TestsA displacement version of the indirect boundary element in two dimensions known as displacement

discontinuity method (DDM) is modified so that a higher order displacement discontinuity variationalong each boundary element can be used[22–27] This method is employed to numerically simulate thetesting specimens used in this study. It has been shown that the higher order displacement discontinuitymethod gives accurate solution of the normal displacement discontinuity (crack opening displacement)and the shear displacement discontinuity (crack sliding displacement) near the crack ends[26,27].

Based on the LEFM principles, the stress intensity factors (SIFs) of Mode I and Mode II can beestimated based on these displacement discontinuities[28]. This method gives relatively accurate resultswhen the special crack tip elements can be used to account for the singularities of stress and displacementfields near the crack ends[26]. This method also reduces the boundary meshes (elements) as the twosides of line cracks are simultaneously discretized with similar boundary conditions[26].

3.1. Higher order displacement discontinuity method (HODDM)

The displacement discontinuities along the boundary of the problem can be accurately estimatedby using higher order displacement discontinuity (HODD) elements (e.g. quadratic or cubic HODDelements) in the solution of boundary value problems[26].

In the present research, a cubic DD element (Dk(ε)) is used. In this approach, each boundary elementis divided into four equal sub-elements (each containing a central node) for which the nodal displacementdiscontinuity (DD) is numerically evaluated (i.e. the opening displacement discontinuity Dy and slidingdisplacement discontinuity) (Fig.7).

Dk(ε) =

4∑

i=1

Πi(ε)Dik (k = x, y, i = 1, 2, 3, 4)

Dx(ε) = Π1(ε)D1x + Π2(ε)D

2x + Π3(ε)D

3x + Π4(ε)D

4x

Dy(ε) = Π1(ε)D1y + Π2(ε)D

2y + Π3(ε)D

3y + Π4(ε)D

4y

(1)

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Fig. 7. Cubic collocations for the DD variation.

where D1k (i.e. D1

x and D1y), D2

k (i.e. D2x and D2

y), D3k (i.e. D3

x and D3y) and D4

k (i.e. D4x and D4

y) arethe cubic nodal displacement discontinuities,and,

Π1(ε) = −

3a31 − a2

1ε − 3a1ε2 + ε3

48a31

, Π2(ε) =9a3

1 − 9a21ε − a1ε

2− ε3

16a31

Π3(ε) =9a3

1 + 9a21ε − a1ε

2− ε3

16a31

, Π4(ε) = −

3a31 + a2

1ε − 3a1ε2− ε3

48a31

(2)

are the cubic collocation shape functions using a1 = a2 = a3 = a4. A cubic element has 4 nodes, whichare the centers of its four sub-elements as shown in Fig.7.

The singularities of the stresses and displacements near the crack ends may reduce their accuracies.Therefore, special crack tip elements can be used to increase the accuracy of the DDs near the cracktips[26]. The DD variation for three nodes can be formulated using a special crack tip element containingthree nodes (or having three special crack tip sub-elements).

Dk(ε) = [ΓC1(ε)]D1k(a) + [ΓC21(ε)]D

2k(a) + [ΓC3(ε)]D

3k(a) (k = x, y) (3)

where the crack tip element has the length of a1 = a2 = a3.Considering a crack tip element with three equal sub-elements (a1 = a2 = a3), the shape functions

ΓC1(ε), ΓC2(ε) and ΓC3(ε) can be obtained as

ΓC1(ε) =15ε1/2

8a1/21

−

ε3/2

a3/21

+ε5/2

8a5/21

, ΓC2(ε) =−5ε1/2

8a1/21

+3ε3/2

2√

3a3/21

−

ε5/2

4√

3a5/21

ΓC3(ε) =3ε1/2

8√

5a1/21

−

ε3/2

2√

5a3/21

+δ5/2

8√

5a5/21

(4)

Based on the LEFM principles, the Mode I and Mode II stress intensity factors KI and KII (expressedin MPa m1/2) can be written in terms of normal and shear displacement discontinuities and be estimatedfor the last special crack tip element as[22]

KI =µ

4(1 − ν)

(

2π

a1

)1/2

Dy(a1), KII =µ

4(1 − ν)

(

2π

a1

)1/2

Dx(a1) (5)

where µ is shear modulus and ν is Poisson’s ratio of the brittle material.In the simulation of double edge-notched beam specimens, the discretization of boundaries have

been accomplished by using 38 cubic elements along the beam specimens and 8 cubic elements alongeach pre-existing crack. In the numerical modeling, the ratio of crack tip element length, L to the totalcrack length, b is taken as 0.2 (L/b = 0.2) and three special crack tip elements are used.

3.2. Numerical simulation of the experimental work

Although the tests carried out in this study are threedimensional the numerical simulation of thesetests is accomplished by a two-dimensional higher order displacement discontinuity. This numericalanalysis is based on the twodimensional plain strain conditions which are more usual in the literatureof fracture mechanics.

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The double edge-cracked Brazilian disc (shown in Figs.5(a)-(e)) and the double edge-notched beamspecimens (shown in Figs.6(a)-(e)) are simulated by the indirect boundary element method and the crackpropagation paths are illustrated in Figs.8(a)-(e) and Figs.9(a)-(e). In the present numerical simulation,the Mode I and Mode II stress intensity factors (SIFs) proposed by Irwin[28] are estimated based on theLEFM approach. A boundary element code is provided using the maximum tangential stress criteriongiven by Erdogan and Sih[29]. An iterative algorithm is provided based on a stepwise procedure so thatthe propagation paths of wing cracks are estimated. As shown in Figs.8 and 9, the simulated propagationpaths are in good agreement with the corresponding experimental results obtained through performingvarious laboratory tests.

Fig. 8. Numerical simulation of crack propagation paths and crack coalescence for the double edge-cracked Brazilian disc

specimens under compression.

IV. Discussions and ConclusionsThe crack propagation process of brittle solids under various shear loading conditions has been

experimentally and numerically studied in this research. Various types of double edge-cracked Braziliandisc and double edge-notched beam (DENB) specimens were tested to study the crack propagationprocess of pre-cracked brittle concrete These specimens with different ligament angles and crack lengthswere tested in a geotechnical laboratory.

Ahigher order displacementdiscontinuitymethodwas especiallymodified tomodel the crack propaga-tion process and crack coalescence in the double edge-cracked Brazilian disc and the double edge-notchedbeam (DENB) specimens, respectively.

It is concluded that in double edge-notched beam specimens containing non-overlapped cracks withrelatively high ligament angles, the cracks are initiated from both crack tips and elliptical breaking

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Fig. 9. Numerical simulation of crack propagation paths and crack coalescence for the double edge-notched beam (DENB)

under four-point bending.

surface may occur in the specimens. On the other hand for lower ligament angles, the cracks maypropagate and coalesce with each other at the propagating crack tips and the specimen may fail at asingle surface.

In the double edge-notched beam specimens containing overlapped cracks, by increasing the ligamentangles, the cracks seem to be initiated from both walls of the cracks and the elliptical breaking surfacemay occur in these specimens. In the double edge-crackedBrazilian disc specimens containing overlappedcracks with relatively high ligament angles, the cracks may be initiated from the two crack tips andthen propagate towards the crack walls. In the case of cracks with lower ligament angles, the cracksmay propagate and coalesce with each other at the propagating crack tips and the specimen may failat a single surface. In double edge-cracked Brazilian disc specimens with non-overlapped cracks, byincreasing the ligament angles, the cracks are initiated from both tips of cracks and the specimen mayfail at the two surfaces.

The numerical analyses of the models (made for the same tested specimens) illustrate that the shearbehavior of double edge-cracked Brazilian disc and double edge-notched beam specimens are producingthe cracks propagation paths (produced by the coalescence phenomenon of the non-overlapped andoverlapped cracks). Comparison of the numerical results with some of the corresponding experimentalresults shows that there is a good agreement between the numerical and experimental results.

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