Initiation of ductile fracture in mixed-mode I and II aluminum alloy specimens

Engineering Fracture Mechanics 93 (2012) 189–203

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Initiation of ductile fracture in mixed-mode I and II aluminum alloy specimens Xudong Qian ⇑, Wuchao Yang Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore

a r t i c l e

i n f o

Article history: Received 21 September 2011 Received in revised form 11 June 2012 Accepted 25 June 2012

Keywords: Fracture initiation Mixed mode Ductile fracture Strain detection Energy release rate

a b s t r a c t This study investigates the initiation of mixed-mode crack extensions in four-point bend and shear specimens fabricated from aluminum alloys. The determination of fracture initiation utilizes a new strain detection method, which provides a uniform criterion for specimens over the complete mixed-mode I–II loading range. The 0.2 mm offset method for mode I specimens and the physical features on the fracture surface of mode II dominant specimens validate this method. The total measured fracture toughness shows a minimum value for the pure mode I case, a maximum for the pure mode II case and an oscillated trend over the mixed-mode I–II loads. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The existing guidelines [1,2] to measure the critical fracture resistance for ductile metallic materials include a few common types of laboratory-scaled specimens, e.g., the compact tension [C(T)], the single-edge-notched bend specimen [SE(B)], and the disk-shaped compact [DC(T)] specimen. These specimens yield the fracture toughness value at the onset of the ductile crack extension, often measured by the critical energy release rate, under pure mode I loadings. However, realistic flaws in engineering structures often experience mixed-mode loads, which may cause significant variations in the critical fracture resistance for the metallic materials [3–5]. Previous experimental investigations over the last three decades on the critical fracture resistance under mixed-mode I and II loadings for ductile metals lead to two contradicting conclusions. A few groups of researchers [5–11] demonstrate independently that the critical fracture resistance under mixed-mode I and II loadings remains lower than that under mode I loading, while other works [4,12–14] indicate that the mixed-mode fracture resistance may exceed that under the pure mode I condition. The lack of a universally agreed testing standard to measure mixed-mode I and II fracture resistance leads partially to the divergent observations reported by the previous researchers [15]. Each individual research group defines a separate criterion to determine the mixed-mode fracture initiation and the corresponding fracture resistance. The measurement of the critical mixed-mode fracture resistance requires a simple and reliable method to determine the initiation of the ductile crack extension, applicable to the entire mixed-mode I and II loading range. Fracture toughness tests for mixed-mode I and II specimens face critical challenges in engineering laboratories. Typical difficulties arise in monitoring the local deformation near the crack plane, controlling the anticipated mode-mixity, and calculating the mixed-mode energy release rates from the measured load and deformation parameters. The mixed-mode I and II fracture tests often utilize two common set-ups, i.e., the four-point loading on a single-edge-notched specimen [8,13,16] and ⇑ Corresponding author. Tel.: +65 6516 6827; fax: +65 6779 1635. E-mail address: [email protected] (X. Qian). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.06.018

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Nomenclature a a0 ai anotch b B BN E FV J Jel Ji JI JIc JII Jpl JT Ki M P Pc Pi rp S0 S W beq dL dR dV dVi e e\ e//

cCMOD gCMOD h

r ry ru t

crack depth initial crack depth current crack depth machined crack depth remaining ligament in a specimen thickness of a specimen net thickness of a specimen elastic modulus shear force energy release rate elastic energy release rate energy release rate at fracture initiation mode I energy release rate critical energy release rate for mode I specimens based on ASTM E 1820 mode II energy release rate plastic energy release rate total energy release rate mode I or II stress-intensity factor for i ¼ I or II respectively bending moment applied load critical load corresponding to JIc applied load at fracture initiation for mixed-mode specimens plastic rotation factor distance between load line and crack plane for mixed-mode specimens distance between load line and the nearest support width of a specimen equivalent mode-mixity angle displacement parallel to the crack plane on the left of the crack plane displacement parallel to the crack plane on the right of the crack plane shear displacement between two crack planes shear displacement between two crack planes at crack initiation strain strain perpendicular to the crack plane near the crack tip strain parallel to the crack plane near the crack tip correction factor to calculate the energy release rate correction factor to calculate the energy release rate rotation between the two crack planes stress yield stress ultimate strength Poisson’s ratio

the compact tension shear specimen with an S-shape grip or the Arcan specimen [17–19]. Variations in the applied shear force and bending moment on the crack plane enable the specimens to reach an anticipated mode-mixity ranging from the pure mode I loading to the pure mode II condition. The fracture resistance of the mixed-mode specimen then derives from the single specimen method [14,15], the multiple specimen method [8] or the hybrid method [20]. Tohgo and Ishii [8] report the J–R curves measured from multiple specimens with a different initial crack length fabricated from the aluminum alloy 6061-T651 subjected to the same mixed-mode ratio. Their study separates the total energy release rate for the mixed-mode I and II specimens into a mode I component, JI, and a mode II component, JII. Bhattacharjee and Knott [13] determine the local shear strain near the crack tip corresponding to the crack initiation in different C(T) specimens with different mode-mixity for the HY100 steel. Kamat and Hirth [14] apply the unloading compliance approach to determine the mixedmode J–R curve, similar to the procedures outlined in ASTM E1820 [1]. Smith et al. [21] calculate the energy release rates (Jvalues) in specimens made of A508 class 3 forged steels using the experimental records of the load and displacement, coupled with the geometrical functions determined from finite element (FE) analysis. Previous researches have proposed various methods to determine the fracture initiation for mixed-mode I and II experiments. Tohgo and Ishii [8] utilize the J-values at zero crack extension (Da = 0) to represent the condition for the crack initiation. Aoki et al. [19] propose a local fracture mechanics parameter, the critical stretch zone length, which depends on the numerically computed J-values. Hallback and Nilsson [5] determine the critical J-values by the intersection between the J–R curve and an 0.2 mm offset construction

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line, similar to the method for mode I specimens in ASTM E1820 [1]. Smith et al. [15] investigate the effect of the equivalent mode-mixity angle (beq), on the J-values at Da = 0.2 mm based on the crack extension measured on the surface of the specimens through a traveling microscope. Similarly, Pirondi and Dalle Donne [10,11] determined the fracture toughness at crack initiation for a ferritic steel tested under mixed-mode I and II loadings by using the J-values at Da = 0.2 mm. This experimental study aims to examine the effect of mode-mixity on the fracture resistance, measured by J-values, at the initiation of the ductile crack extension in the aluminum alloy 5083-H112 material. The determination of the fracture initiation utilizes a simple strain detection method, which detects the fracture initiation by a sharp reduction in the measured strain values near the crack tip. The strain detection method is implemented on mixed-mode I/II, four-point bend and shear specimens. The study covers a wide range of mode mixity including the pure mode I, mixed-mode I and II, and pure mode II specimens. The critical J-values at the fracture initiation for mixed-mode specimens are calculated with the method proposed by Tohgo and Ishii [8]. The experimental study examines the variation of the critical energy release rate at the fracture initiation with respect to the mode-mixity angle. This paper starts with an introduction on the experimental procedure of the mixed-mode I and II tests. The following section describes the mathematical details to calculate J-values for both SE(B) specimens and mixed-mode I and II specimens using the experimentally measured parameters. The next section verifies the strain detection method to determine the initiation of ductile crack extensions in mixed-mode I and II specimens. The subsequent section presents the experimental results for the mixed-mode specimens. The last section summarizes the main conclusions obtained from this study. 2. Experimental program The experimental program includes three types of tests, the tensile test, the pure mode I SE(B) test and the mixed-mode I and II test on four-point bend and shear specimens, all conducted at an ambient room temperature under static loads. The SE(B) test consists of both side-grooved SE(B) specimens and plane-sided SE(B) specimens. The mixed-mode I and II test comprises of four-point bend and shear specimens with different initial crack depths, a0/W  0.2 for shallow cracks and a0/W  0.5 for deep cracks. Table 1 summarizes the fracture specimens tested in this study. Table 1 Summary of the mode I SE(B) and mixed-mode I and II, four-point load specimens. Specimen type

Specimen name

anotch (mm)

a0 (mm)

a0/W

Side-grooved SE(B)

S1-A S1-B S2-A S2-B

5.2

8.5 8.0 18.5 18.4

0.236 0.222 0.513 0.511

– – – –

16

S0 (mm)

beq (°)

No. of specimens

B (mm)

90

1 1 1 1

18.2

Plane-sided SE(B)

M0-A M0-B

16 5.2

18.8 8.0

0.522 0.222

– –

Deep cracked mixed-mode

M1 M2 M3 M4 M5 M6

16

18.6 18.5 18.7 18.8 18.9 18.9

0.517 0.514 0.518 0.522 0.526 0.526

20 10 5 3 2 0

75 60 45 30 20 0

2 2 2 2 2 2

Shallow cracked mixed-mode

M7 M8

5.2

8.0 8.0

0.221 0.221

15 7

75 60

1 1

(a)

1 1

(b)

Fig. 1. (a) Configuration of the coupon specimen; and (b) uniaxial stress–strain relationships for the aluminum alloy 5083-H112.

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2.1. Tensile test Mechanical properties of the aluminum alloy 5083-H112 derive from the tensile tests performed at the room temperature, as outlined in ASTM E8M-04 [22]. Fig. 1a shows the geometry of the coupon specimen. The displacement-controlled loading has a loading rate of 0.1 mm/min. Fig. 1b presents the engineering stress versus engineering strain curve and the true stress versus true strain curve measured from coupon specimens with the thickness equal to 15 mm. The solid circle in the Fig. 1b indicates the true stress computed from the maximum engineering stress. The Young’s modulus E equals 69 GPa, the Poisson’s ratio t = 0.35, the yield stress ry = 243 MPa, the ultimate stress ru = 347 MPa and the elongation at fracture over a 50 mm gage length is 21.85%. 2.2. SE(B) test The mode I SE(B) tests include two types of specimens, the side-grooved specimens and the plane-sided specimens. The plane-sided SE(B) specimen represents the reference mode I condition for the mixed-mode tests, which consist of

(a)

(b)

(c)

(d)

Fig. 2. Configuration of mode I: (a) side-grooved SE(B) specimens, and (b) plane-sided SE(B) specimens; and the test set-ups for: (c) side-grooved SE(B) specimens, and (d) plane-sided SE(B) specimens.

(a)

(b)

Fig. 3. Test set-up and instrumentation on the mixed-mode I and II, four-point bend and shear specimens.

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plane-sided four-point bend and shear specimens. The tests on both the side-grooved and the plane-sided specimens follow the procedures prescribed in ASTM E1820 [1]. All SE(B) specimens have a thickness of B = 18.2 mm, a width of W = 36 mm and a span of 144 mm between the two roller supports, as illustrated in Fig. 2. The plane-sided SE(B) specimens, which correspond to the pure mode I specimens in the entire mixed-mode test program, have the same total length of 280 mm as the mixed-mode I and II specimens. The side-grooved SE(B) specimens entail a 20% reduction in the thickness, with the groove angle equal to 90°. The machined notch depths equal anotch = 5.2 mm and anotch = 16 mm for SE(B) specimens with a shallow crack and those with a deep crack, respectively. Each SE(B) specimen has a pre-crack length of approximately 2–3 mm, as recommended in ASTM E1820 [1]. The deep-cracked SE(B) specimen creates high plasticity constraints near the crack front and the shallow-crack SE(B) specimen generates low plasticity constraints along the crack front. The test procedure measures the crack mouth opening displacement (CMOD) and the applied load (P) for side-grooved specimens. For the plane-sided SE(B) specimens, additional strain gauges near the tip of the fatigue crack measures the strains in a direction perpendicular to the crack plane. The test procedure also includes the unloading and re-loading cycles to monitor the change in the specimen compliance with an extending crack. At the end of each fracture test, the experimental procedure applies a cyclic fatigue loading to mark the end of the ductile crack extension. Fig. 2c and d shows the test setup for SE(B) specimens. 2.3. Mixed-mode I and II test The mixed-mode I and II test follows the method recommended by Tohgo and Ishii [8]. Fig. 3 sketches the mixed-mode test setup, together with the required measurement on the specimen necessary to derive the mixed-mode energy release rate. All mixed-mode specimens in this study share the identical thickness and width as illustrated in Table 1. All mixedmode specimens contain a plane-sided free surface. The experimental procedure applies a mode I cyclic loading to create a sharp fatigue pre-crack of approximately 2–3 mm ahead of the machined notch prior to the mixed-mode test. The distance between the loading point and its nearest support remains fixed at S = 40 mm for all specimens. The variation of the distance between the loading point and the crack plane, S0, generates different mode mixity at the tip of the fatigue pre-crack. Fig. 3b illustrates schematically the deformed crack tip under mixed-mode I and II loading, which includes a sharpened side and a blunted side. The current experimental study uses a strain detection approach to determine the initiation of the ductile crack extension in mixed-mode specimens. Fig. 4 illustrates the strain measurement near the crack tip. Unlike pure mode I specimens, mixed-mode I and II specimens contain different strain measurement for mode I dominant specimens and for mode II dominant specimens, as illustrated in Fig. 4a and b, respectively. For mode I dominant fracture specimens, the crack extension releases the opening stress/strain perpendicular to the crack plane. The experimental instrumentation thus attaches two strain gauges on both sides of the crack plane to monitor the strain in the direction perpendicular to the crack plane, denoted as e\. The strain gauges attached on these specimens utilize the post-yield strain gauge with the measurable strain limit equal to 15%. The center of the strain gauges (represented by the arrows with dotted lines in Fig. 4) is located directly above the tip of the fatigue pre-crack, as shown in Fig. 4a. For mode II dominant fracture specimens, the instrumentation measures both the strains in a direction perpendicular to the crack plane and that parallel to the crack plane (e//), as shown in Fig. 4b. The test procedure mounts the strain gauges at a 1 mm offset from the crack tip to prevent the adverse effects caused by plastic wings formed ahead of the crack tip on the specimen surface under mode I dominant, plastic deformations. This test measures the CMOD by mounting a crack opening displacement gauge at the mouth of a crack, similar to the mode I specimens shown in Fig. 2c and d. Four machined holes with inner threads drilled near the edges of the crack plane, as shown in Fig. 3b, facilitate the attachment of four stiff cantilever bars on both sides of the crack. The setup in Fig. 3 enables the measurement of the displacement parallel to the crack plane on the left side of the crack plane, dL, and that on the right side of the crack plane, dR, using four displacement transducers symmetrically located on both surfaces of the specimen. The shear displacement of the crack plane thus equals,

(a)

(b)

Fig. 4. Strain instrumentation for: (a) mode I dominant specimens; and (b) mode II dominant specimens.

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dV ¼ dR  dL

ð1Þ

The loading procedure includes a displacement-controlled load applied at 0.1 mm/min, with multiple unloading–reloading cycles to monitor the change in the specimen’s compliance as the crack extends. Each unloading–reloading cycle has a loading range of approximately 30% of the load level prior to the start of the unloading. At the end of the mixed-mode test, the test procedure applies a mode I cyclic load to generate distinctive surface characteristics, marking the end of the fracture test. The post-test examination breaks each specimen and scans the fracture surface using an optical microscope. The shear force (FV) versus the relative shear displacement (dV) and the bending moment (M) versus the rotation angle (h) at the crack plane derive then from the measured load (P), the shear displacement (dV = dR  dL) and the CMOD. The strain value obtained by the strain gauges anticipates a sharp reduction as the crack extension initiates. The corresponding load level defines the fracture initiation load, Pi. The critical J-values (JIi, JIIi and JTi) at the fracture initiation are then calculated from the local deformation at Pi. Section 4 presents the detailed discussion and results for the above procedure. 3. Energy release rate and mode mixity This section introduces the mathematical details to calculate the J-values for SE(B) specimens and for four-point load, mixed-mode I and II specimens, followed by a description on the numerical procedure to determine the mode-mixity of the fracture specimens. 3.1. J-value for SE(B) specimen The traditional approach to determine the J–R curve for a single SE(B) specimen often utilizes the elastic unloading compliance method, which requires accurate determination of load-line displacement (LLD), the magnitude of the applied load (P) and the CMOD. The unique relationship between the compliance of the P-CMOD data and the crack length enables the determination of the crack extension Da. The area under the load versus the load-line displacement corresponding to different crack lengths allows the calculation of the energy release rate, J-value [23]. Zhu et al. [24] propose an incremental approach to obtain the J–R curve for SE(B) specimens directly from the load versus the CMOD data. This study utilizes this CMOD based J–R curve method for SE(B) specimens. At any loading point n, the total J-integral separates into two parts,

J n ¼ J elðnÞ þ J plðnÞ

ð2Þ

where the elastic component Jel(n) calculates from the linear-elastic stress-intensity factor (SIF),

J elðnÞ ¼

½K n ðan Þ2 ð1  t2 Þ E

ð3Þ

where t is the Poisson ratio. The plastic component Jpl(n) follows:

J plðnÞ ¼



Jplðn1Þ þ

gn1 CMOD bn1 BN

An1;n CMODpl

  cn1 1  CMOD ðan  an1 Þ bi1

ð4Þ

where BN denotes the net thickness of specimen, An1;n CMODpl refers to the incremental area under the load versus the plastic CMOD curve,

An1;n CMODpl ¼

ðPn þ Pn1 ÞðV npl  V n1 pl Þ 2

ð5Þ

where Vpl defines the plastic CMOD. Zhu et al. [24] propose a modified expression of the energy correction factor gCMOD,

gCMOD ¼ 3:677  2:199ða=WÞ þ 0:437ða=WÞ2 ; 0:1 6 a=W 6 0:7

ð6Þ

and the crack length correction factor cCMOD follows:

cCMOD ¼ 0:131 þ 2:131ða=WÞ  1:465ða=WÞ2 ; 0:25 6 a=W 6 0:7

ð7Þ

ASTM E-1820 [1] determines the material fracture toughness JIc based on the intersection between the J–R curve and a 0.2 mm offset of the construction line, which follows:

J ¼ 2ry Da

ð8Þ

3.2. J-value for mixed-mode I and II specimens Tohgo and Ishii [8] propose a method to determine J-integral values for four-point bend and shear specimens by measuring the local deformation at the crack plane. They separate the total energy release rate, JT, into,

J T ¼ J I þ J II

ð9Þ

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195

where JI is the opening-mode energy release rate,

2 bB

JI ¼

Z

h

Mdh

ð10Þ

0

where b denotes the length of the remaining ligament and JII corresponds to the shear-mode energy release rate,

J II ¼ J eII þ

2 bB

Z

dV

0

1 F V dd  FdV 2

 ð11Þ

where the elastic J eII follows:

J eII ¼

K 2II ð1  t2 Þ F 2II pað1  t2 Þ 2 ¼ FV E EðWBÞ2

ð12Þ

where FV refers to the shear force applied at the crack plane and KII denotes the mode II stress-intensity factor determined from FE results,

F II ¼

K II WB pffiffiffiffiffiffi F V pa

ð13Þ

The above method to calculate the J-value depends on the accurate measurement of the rotation angle and the shear displacement at the crack plane. Tohgo and Ishii [8] have developed a deformation gauge for the four-point load, mixed-mode specimen, with the assumption that the center of rotation in the crack plane remains fixed for specimens with the same initial crack depth under different mode mixity. The current study follows a similar assumption that the center of rotation depends only on the initial crack depth (a0), and the rotation of the crack plane (h) becomes,

h¼

CMOD a0 þ rP ðW  a0 Þ

ð14Þ

where rp represents the plastic rotation factor and equals to 0.44 as suggested in ASTM E1820 for SE(B) specimens [1]. The comparison between the final rotation angle measured on the specimen with Eq. (14) confirms the accuracy of the assumed rp value. Substituting Eq. (14) into Eq. (10) reveals that the mode I energy release rate, JI, operates on the area under the load versus the CMOD curve, consistent with the approach proposed in [24] and used in the current study to determine the energy release rate for mode I specimens.

(a)

(b)

Fig. 5. (a) Typical FE mesh used to compute linear-elastic, mixed-mode stress-intensity factors; and (b) variations of the mixed-mode angle beq with respect to the distance S0 for mixed-mode specimens with two different crack depths.

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(a)

(b)

Fig. 6. The load versus the measured strains: (a) e\–P for mode I dominant specimens; and (b) e//–P for mode II dominant specimens.

Table 2 Comparison of the critical load and the energy release rates measured in side-grooved and plane-sided SE(B) specimens. Specimen name

a0/W

Pi (kN)

Pc (kN)

JIc (kJ/m2)

S1-A S1-B S2-A S2-B M0-A M0-B

0.214 0.201 0.491 0.485 0.522 0.222

– – – – 9 26

26.0 25.0 10.2 10.9 11.4 29.0

39.0 40.5 34.2 32.0 39.0 41.5

Table 3 The critical energy release rates and the local crack-plane deformation for the mixed-mode specimens. No.

beq (°)

JIi (kJ/m2)

JIIi (kJ/m2)

JTi (kJ/m2)

dVi (mm)

CMODi (mm)

Pi (kN)

M0-A M0-B M1 M2 M3 M4 M5 M6 M7 M8

90 90 75 60 45 30 20 0 75 60

27.0 30.0 43.7 53.5 23.2 8.7 4.5 0.0 45.6 34.2

– – 3.0 2.7 29.6 28.8 30.9 61.3 2.15 23.4

27.0 30.0 46.7 56.2 52.8 37.5 35.4 61.3 48.4 57.6

– – 0.13 0.19 0.32 0.35 0.33 0.50 0.07 0.28

0.28 0.25 0.57 0.65 0.48 0.32 0.22 0.00 0.43 0.48

9 26 64 120 138 141 144 150 142 178

3.3. Mode mixity Hallback and Nilsson [5] define the equivalent mode-mixity angle as,

beq ¼ tan1 ðK I =K II Þ

ð15Þ

In Eq. (15), a beq angle of 90° represents the pure mode I loading and beq = 0° corresponds to the pure mode II loading. The current study utilizes the interaction-integral approach [25] to compute the linear-elastic, mixed-mode stress-intensity factors, necessary to determine the mixed-mode angle for varying locations of the crack plane in the mixed-mode I and II specimens. Fig. 5a shows a typical FE mesh for the mixed-mode specimens used in computing the mixed-mode SIFs. The FE model has the identical in-plane geometry as the tested specimens and consists of one-layer of 8-node hexahedral elements in the thickness direction, with all nodes constrained against the out-of-plane displacement to simulate a plane-strain condition. The crack front elements employ the collapsed 8-node hexahedral elements, as shown in Fig. 5b, similar to the approach adopted by Subramanya et al. [26]. Fig. 5c shows the variation of beq versus S0 for two crack depth ratios, together with the theoretical solution [27,28] for four-point bend and shear specimens. Table 1 summarizes the S0 values utilized in the current study.

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(a)

197

(a)

Fig. 7. (a) The e\–P curve for plane-sided mode I SE(B) specimens; and (b) the corresponding load versus the CMOD curve.

(a)

(b)

(c)

(d)

Fig. 8. (a) The e//–P curve for the mode II dominant specimen M5; (b) the corresponding load versus the CMOD relationship; (c) the fracture surface of M5; and (d) striations on the fracture surface of M5.

4. Verification of the strain detection method This section describes the verification of the strain detection method to determine the initiation of ductile fracture for both mode I dominant and mode II dominant specimens. A shape decrease in measured strain occurs at the initiation of the ductile crack extension. 4.1. Measured strain response Fig. 6a and b shows the measured strain perpendicular to the crack plane, e\, versus the applied load for the opening-type, mode I dominant specimen and the measured strain parallel to the crack plane, e//, versus the applied load for the shear-type, mode II dominant specimen, respectively. The solid cycles in both figures highlight the peak strains measured in the strain

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gauge. The subsequent decrease in the measured strain implies the local unloading near the crack tip, caused by the extension of the crack. The peak strain thus denotes the onset of the ductile crack extension. The corresponding load, Pi, defines the fracture initiation load, as listed in Table 2 for the mode I specimens and in Table 3 for all mixed-mode specimens. For both the mode I dominant specimen and the mode II dominant specimens, the measured strain decreases sharply at crack initiation as assumed in Section 2.3. 4.2. Verification of fracture initiation For mode I dominant specimens, the current study verifies the strain detection method by comparing both the calculated JIc values based on the ASTM E1820 [1] and the corresponding applied loads by using SE(B) specimens. For mode II specimens, the verification of the strain detection method predicates on the first striation observed in the aluminum fracture surface produced by the multiple unloading–reloading cycles. The plane-sided mode I SE(B) specimens, M0-A with a0/W = 0.522 and M0-B with a0/W = 0.222, utilize the same strain instrumentation as the mode I dominant specimens shown in Fig. 4a. The horizontally oriented strain gauges monitor the variation of strains perpendicular to the crack plane near the crack tip. Fig. 7a and b presents the evolution of the e\ versus the load and the load versus CMOD for the plane-sided SE(B) specimens. The initiation of the fracture, as indicated by the sharp reduction in the measured strain value, occurs at Pi = 9 kN for the specimen M0-A and at Pi = 26 kN for the specimen M0-B. The J-integral values calculated at the same load levels (Ji) equal 27 kJ/m2 for M0-A and 30 kJ/m2 for M0-B, while the critical fracture toughness determined by the 0.2 mm offset method as stated in ASTM E1820 [1] yields JIc = 39 kJ/m2 for M0A and JIc = 40.5 kJ/m2 for M0-B. The critical load levels, Pc, corresponding to these JIc values record a value of 11.4 kN for M0-A and 29 kN for M0-B. The strain detection method indicates a 20% lower initiation fracture toughness than does the 0.2 mm offset method in ASTM E1820 [1], which measures the fracture toughness with a small amount of prior crack extension, i.e., Da > 0.2 mm. The smaller Ji values, compared to the JIc values, prove that the strain detection method provides a reasonable indication of the ductile crack initiation for the mode I specimens and mode I dominant specimens, which experience a similar fracture mechanism as the pure mode I specimens. The post-test examination reveals an interesting feature observed on the fracture surface of mode II dominant specimens. Each unloading–reloading cycle after the fracture initiation introduces a striation mark on the fracture surface of the mode II dominant specimens. The number of striations imprinted on the fracture surface equals to the number of unloading–reloading cycles after the fracture initiation indicated by the peak strain. Fig. 8a and b shows the measured strain versus the load and the load versus the CMOD for the specimen M5, which fractures in the shear mode. At the peak strain, the corresponding load Pi records a value of 144 kN, as shown in Fig. 8a. Fig. 8a indicates five unloading–reloading cycles beyond the fracture initiation load level Pi for the specimen M5. Fig. 8c shows the fracture surface of M5 together with a graduated rule having a master unit of centimeters. Fig. 8d shows an enlarged view of the cracked surface near the mid-thickness of the specimen observed under an optical microscope. The block arrows in Fig. 8c and d indicates the direction of the crack extension. The dark area in the middle of Fig. 8d represents the crack extension surface, and the bright and flat surface on the right side of this figure corresponds to the fatigue pre-cracked surface. The dash lines indicate the striations and the solid white line denotes the first visible striation. The first striation generated by the 6th unloading–reloading cycle has a distance around 0.3 mm away from the initial fatigue pre-cracked front, implying the fracture initiation occurs prior to the 6th unloading– reloading cycle. This observation confirms the validity of the strain detection method in determining the initiation of the fracture for mode II dominant specimens. All mode II dominant specimens exhibit the similar phenomenon, which is assumed as the physical basis to determine the initiation of the ductile crack extension and to derive the complete J–R curve for mixed-mode I and II specimens [29].

(a)

(b)

Fig. 9. (a) The load versus the CMOD curves; and (b) the J–R curves; for mode I specimens.

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M1 - M2

M3 - M6 Fish-scale surface

Fatigue precrack Fatigue precrack

Crack extension

(a)

Crack extension

(b)

Fig. 10. Typical fracture surface for: (a) mode I dominant specimens; and (b) mode II dominant specimens; scanned under an optical microscope.

(a)

(b)

(c)

(d)

Fig. 11. (a) The M–h curves and (b) FV–dV curves, for deep-crack mixed-mode specimens; and (c) M–h curves and (d) FV–dV curves, for shallow-crack mixedmode specimens.

5. Results and discussion 5.1. Mode I SE(B) specimen Fig. 9a and b presents the load versus the CMOD relationship and the J–R curves for all six SE(B) specimens. Four of the SE(B) specimens contain side-grooved surfaces and the other two contain plane-sided surfaces. The plane-strain fracture toughness JIc values derive from the intersection between the J–R curves and the 0.2 mm offset of the construction line,

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J = 2ryDa, as outlined in ASTM E-1820 [1]. The very close load-CMOD responses and the measured J–R curves between sidegrooved specimens with an approximately equal initial crack length confirm the repeatability of the experimental approach. The initial load-CMOD response for the plane-sided SE(B) specimen remains close to that of the side-grooved specimen with a similar crack depth. The fracture toughness JIc value for the plane-sided SE(B) specimen, obtained from the intersection of the 0.2 mm offset construction line and the J–R curve, does not indicate significant differences from the JIc values measured from the side-grooved SE(B) specimen, despite that the plane-sided SE(B) specimen sustains a strong rise in the J–R curves as the crack extends further. Table 2 summarizes the JIc values and the corresponding critical load levels for the mode I SE(B) specimens. The low plasticity constraints near the front of shallow-crack SE(B) specimens lead to slightly higher JIc values than those in the deep-crack SE(B) specimens, which experience high plasticity constraints along the crack front, in line with the conclusions drawn from previous researches [30].

5.2. Mixed-mode I and II specimens Mode I dominant specimens and mode II dominant specimens exhibit distinctive features on the fracture surface made of this aluminum alloy. These distinctive features confirm the mode of fracture in addition to the mode-mixity angle. Fig. 10 illustrates the fracture surfaces scanned using an optical microscope for two types of specimens, the opening-type, mode I dominant specimen (M1 and M2) and the shear-type, mode II dominant specimen (M3–M6). The fracture surfaces shown in Fig. 10 are located near the mid-thickness of specimens. In Fig. 10, the block arrows denote the direction of crack extension and the dash lines represent the initial crack front produced by the fatigue pre-cracking. The mode I dominant loading on specimens M1 and M2 triggers dimples on the fracture surface, typically observed in the ductile fracture mechanism. In contrast, the strong shearing action in specimens M3–M6 generates a flat fracture surface, which resembles the fish-scale fracture morphology observed in aluminum alloys by other researchers [31]. Therefore, M1 and M2 experience mode I dominant loads and M3–M6 undergo mode II dominant actions, as foreseen by the mode-mixity angle in Table 1. Fig. 11 shows the evolution of the bending moment at the crack plane (M) with respect to the rotation of the crack plane (h), and the relationship between the shear force at the crack plane (FV) with respect to the shear displacement (dV), for all mixed-mode I and II specimens. The solid circles in Fig. 11 indicate the fracture initiation determined by the strain detection method. The shear-type, mode II loading drives more significant plastic deformations near the crack tip than does the mode I loading with an equal magnitude, measured by the stress-intensity factors, as reflected by the Irwin’s estimation on the plastic zone size [32]. The presence of the mode II loading in the mode I dominant specimen, therefore, redistributes the very high near-tip opening stresses driven by the mode I loading to the adjacent elastic materials, leading to an increasing local deformation of the crack plane at the fracture initiation. The crack planes in the mode I dominant specimens M1 and M2 thus sustain an increasingly large rotation before the opening stress near the crack tip reaches a sufficiently large magnitude to initiate the crack extension. The critical energy release rate at the fracture initiation, for the mode I dominant specimens, depends primarily on the area under the M–h curve prior to the fracture initiation indicated by the solid symbols in Fig. 11a. An increasingly large h with the increase in the mode II loading implies that the critical energy release rate increases with a raised mode II loading, for mode I dominant specimens. Table 3 summarizes the critical energy release rates (Ji) for all specimens, corresponding to the fracture initiation event determined by the strain detection method. A small amount of mode II loading (beq = 75° and beq = 60°) elevates substantially the total energy release rate at the fracture initiation, compared to the JTi value for pure mode I specimens. The mode II dominant specimens experience significant increases in the shear load in the crack plane prior to the fracture initiation, compared to the shear action in mode I dominant specimens. Over the mode II dominant loading range

(a)

(b)

Fig. 12. (a) Variation of the critical Ji values with beq; and (b) variations of the CMOD and shear deformation with respect to beq.

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45° 6 beq 6 20° (specimens M3–M5), however, the increase in the mode II loading does not generate significantly larger shear actions on the crack plane, as reflected by the FV–dV curves in Fig. 11b. In contrast, these specimens indicate pronounced reductions in the applied moment on the crack plane, and hence significant reductions in the area under the M– h curve. The comparison of the FV–dV curves and the M–h curves for M3–M5 specimens suggests that these mode II dominant specimens do not experience a significant variation in the mode II energy release rate at the fracture initiation. However, the total energy release rate decreases as the mode II loading increases due primarily to the reduction in the mode I energy release rate, as confirmed by the J-values listed in Table 3. As highlighted by Pirondi and Dalle Donne [10], the yielding in the shear mode is concentrated along a strip of material ahead of the crack tip and there is less material to dissipate energy by plasticity. Therefore, a reduction of JT value occurs with the increasing mode II component. The pure mode II specimen M6 sustains an increased shear deformation driven by the large shear action on the crack plane in the absence of the opening stress perpendicular to the crack plane. The large shear deformation in M6 elevates the area under the FV–dV curve and leads consequently to an increased total energy release rate at the fracture initiation, as indicated in Table 3. This conforms with the physical explanation reported in [10]. The presence of a small amount of mode I loading in the mode II dominant specimens causes the void growth and coalescence and facilitates the crack extension. This leads subsequently to a lower toughness value for the mode II dominant specimens than the toughness values for the pure mode II specimens. Fig. 12a presents the variation of the critical energy release rate at the fracture initiation with respect to the mode-mixity angle, beq. The critical energy release rates are computed from the area under the M–h curve and that under the FV–dV curve corresponding to the initiation of the crack extension determined by the strain detection method. For the deep-crack specimens with beq P 60°, the critical total energy release rate (JTi) derives primarily from the mode I energy release rate (JIi), implying a negligible mode II energy release rate. The total energy release rate (J Ti ) remains the minimum at beq = 90° (pure mode I loading) and increases as the mode II load increases, consistent with the observation made by Aoki et al. [19], who present the fracture resistance for the aluminum alloy 5083-O specimens with the mixed-mode angle 30° 6 beq 6 90°. The critical JTi-value rate reaches a local peak at beq  60°, followed by a gradual decrease caused by the sharp reduction in the critical JIi-value with an increasing mode II loading. This decrease in the JIi-value over 20° 6 beq 6 60° couples with the drastic reduction in the CMOD value at the initiation of the fracture defined by the strain detection method, as indicated in Fig. 12b. Both the critical mode II energy release rate JIIi (in Fig. 12a) and the shear force applied on the crack plane (in Fig. 11b) over the mixed-mode range 20° 6 beq 6 60° maintain approximately constant values, which infer that the critical shear displacement remains at a similar level over the mode-mixity range, 20° 6 beq 6 60°, as reflected in Fig. 12b. The constant critical shear displacement, dVi  0.33 mm for the material studied here, suggests that the local shear deformation or the local shear strain can emerge as a convenient criterion to determine the initiation of crack extension in mode II dominant specimens, consistent with the observation in [10,12,13]. The critical JTi-value reaches a local minimum at about beq  20°, and reaches an absolute maximum at beq  0° under the pure mode II loading. The critical JTi-values measured for shallow-crack specimens follow a similar trend as the critical JTi-values for deep-crack specimens under the mode I dominant loading. As the mode II loading increases, however, the shallow-crack specimens initiate a reduction in the critical mode I energy release rate, JIi, at a higher mode-mixity angle than do the deep-crack specimens, as shown in Fig. 12a. Shallow-crack, four-point load specimens experience significant constraint loss ahead of the crack tip, compared to the deep-crack specimens with the same geometry. The presence of mode II loading further promotes plastic deformations in the near-tip materials. Consequently, the shallow-crack mixed-mode specimens initiate a shear-type failure at a higher mode-mixity ratio than that in the deep-crack specimens.

6. Summary and conclusions This study investigates experimentally the effect of mode-mixity on the initiation of ductile crack extensions for the aluminum alloy 5083-H112. The experimental procedure measures the local deformation at the crack plane, including the rotation of the crack plane and the shear deformation between the two crack planes. The determination of the fracture initiation utilizes a new, strain detection method for both the mode I and mixed-mode I and II specimens, depending on the physical mechanism that the initiation of crack extension triggers the local unloading in the near-tip material and therefore leads to a reduction in the measured strain value. The critical energy release rate at the initiation of the ductile crack extension is computed from the area under the M–h curve and the FV–dV displacement curve measured for the crack plane. The present investigation supports the following conclusions derived for the aluminum alloy 5083-H112: (1) The strain detection method provides a criterion to determine the fracture initiation corresponding to the physical unloading near the crack tip detectable by a strain gauge mounted nearby. The strain detection method provides a value of the initiation of the ductile crack extension lower than the 0.2 mm offset method outlined in the ASTM E1820 [1], since the 0.2 mm offset method yields a small amount of crack extension in the specimen (Da > 0.2 mm). The critical energy release rate, Ji, at the fracture initiation determined by the strain detection method, corresponds to a J-value immediately after the change of slope in the experimentally measured J–R curve. The comparison of the load levels, the CMOD and J-values corresponding to the fracture initiation determined by the strain detection method and those based on the 0.2 mm offset method validates the proposed strain detection method for mode I dominant specimens.

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(2) For mode II dominant specimens, the post-test examination reveals distinctive striation marks on the fracture surface of aluminum alloy corresponding to each unloading–reloading cycle in the test after the initiation of the crack extension. The first striation mark beyond the fracture initiation therefore validates the strain detection method in determining the initiation of the crack extension. The strain detection method thus delivers a uniform criterion to determine the fracture initiation over the complete mixed-mode I and II loading range, i.e., 0° 6 beq 6 90°, for the aluminum alloy specimens studied. (3) This study utilizes consistent approaches to evaluate the energy release rate JI for pure mode I specimens and the JIvalue for mixed-mode I and II specimens, both operating on the load versus the CMOD curve. The fracture toughness JIc measured for the side-grooved SE(B) specimens remains slightly smaller than the JIc values obtained from the planesided specimens, due to the varying and lower plasticity constraints experienced by the crack front in the plane-sided specimens. The shallow-crack SE(B) specimens indicate larger JIc and Ji values than those for the deep-crack SE(B) specimens, due to the elevated crack-front constraints in deep-crack specimens. The side-grooved specimens present a more manifested effect of the crack-front constraints on the JIc values than do the plane-sided specimens. (4) The critical energy release rate corresponding to the fracture initiation determined by the strain detection method shows an oscillated fracture toughness over the complete mixed-mode I and II loading range, i.e., over 0° 6 beq 6 90°. (5) The JTi value for the pure mode I loading remains the smallest over the entire mixed-mode I and II loading, while the JTi measured for the pure mode II loading indicate the maximum value, for the deep-cracked specimens considered. The weakest fracture resistance in the pure mode I specimens, therefore, confirms that mode I cracks are the most likely cracking condition for the aluminum alloy specimens studied. (6) The mode I dominant specimens with beq P 60° exhibit an increasing fracture resistance, JTi, with the increase in the mode II loading. The presence of shear actions on the crack plane allows the near-tip material to sustain an increased plastic deformation before the initiation of crack extension takes place. (7) The mixed-mode specimens with 20° 6 beq 6 60° exhibit an approximately constant critical mode II energy release rate and a corresponding, approximately constant shear deformation near the crack tip. The JTi values over 20° 6 beq 6 60° decrease with the increase of mode II loading, due primarily to the significant reduction in the JIi values.

Acknowledgement The research scholarship provided by the National University of Singapore to the second author is gratefully acknowledged. References [1] American Society of Testing and Materials (ASTM). Standard test method for measurement of fracture toughness. E1820-11. West Conshohocken, (PA, United States): ASTM International; 2011. [2] International Organization for Standardization. Metallic materials – determination of plane-strain fracture toughness. ISO 12737; 2010. [3] Laukkanen A. Analysis of experimental factors in elastic–plastic small specimen mixed-mode I–II fracture mechanical testing. Fatigue Fract Engng Mater Struct 2001;24(3):193–206. [4] Roy YA, Narasimhan R, Arora PR. An experimental investigation of constraint effects on mixed mode fracture initiation in a ductile aluminium alloy. Acta Mater 1999;47(5):1587–96. [5] Hallback N, Nilsson F. Mixed-mode I/II fracture behavior of an aluminum-alloy. J Mech Phys Solids 1994;42(9):1345–74. [6] Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fracture 1972;16:519–27. [7] Yoda M. The effect of the notch root radius on the J-integral fracture-toughness under modes I, II and III loadings. Engng Fract Mech 1987;26(3):425–31. [8] Tohgo K, Ishii H. Elastic plastic fracture-toughness test under mixed-mode I–II loading. Engng Fract Mech 1992;41(4):529–40. [9] Shi YW, Zhou NN, Zhang JX. Compasrison of mode-I and mode-II elastic–plastic fracture-toughness for 2 low-alloyed high-strength steels. Int J Fracture 1994;68(1):89–97. [10] Pirondi A, Dalle Donne C. Characterisation of ductile mixed-mode fracture with the crack-tip displacement vector. Engng Fract Mech 2001;68(12):1385–402. [11] Dalle Donne C, Pirondi A. J-integral evaluation of single-edge notched specimens under mixed-mode I/II loading. J Test Eval 2001;29:239–45. [12] Maccagno TM, Knott JF. The mixed-mode I–II fracture-behavior of lightly tempered HY130 steel at room-temperature. Engng Fract Mech 1992;41(6):805–20. [13] Bhattacharjee D, Knott JF. Ductile fracture in HY100 steel under mixed mode-I/mode-II loading. Acta Metall Mater 1994;42(5):1747–54. [14] Kamat SV, Hirth JP. Mixed mode I/II fracture toughness of 2034 aluminum alloys. Acta Mater 1996;44(1):201–8. [15] Smith DJ, Swankie TD, Pavier MJ, Smith MC. The effect of specimen dimensions on mixed mode ductile fracture. Engng Fract Mech 2008;75(15):4394–409. [16] Maccagno TM, Knott JF. The fracture-behavior of PMMA in mixed mode-I and mode-II. Engng Fract Mech 1989;34(1):65–86. [17] Richard H, Benitz K. A loading device for the creation of mixed mode in fracture mechanics. Int J Fracture 1983;22(2):55–8. [18] Banks-Sills L, Arcan M. A compact mode II fracture specimen. ASTM Spec Tech Publ 1986;17:347–63. [19] Aoki S, Kishimoto K, Yoshida M, Sakata M, Richard HA. Elastic plastic fracture behavior of an aluminum-alloy under mixed-mode loading. J Mech Phys Solids 1990;38(2):195–213. [20] Qian X, Yang W. A hybrid approach to determine fracture resistance for mode I and mixed mode I and II fracture specimens. Fatigue Fract Engng Mater Struct 2011;34:305–20. [21] Smith RS, Tsui TY, Ho PS. Effects of ultraviolet radiation on ultra-low-dielectric constant thin film fracture properties. J Mater Res 2009;24(9):2795–801. [22] American Society of Testing and Materials (ASTM). Standard test methods for tension testing of, metallic materials. E8M-04; 2004. [23] Zhu X, Joyce J. Revised incremental J-integral equations for ASTM E1820 using crack mouth opening displacement. J Test Eval 2009;37(3):205–14.

X. Qian, W. Yang / Engineering Fracture Mechanics 93 (2012) 189–203

203

[24] Zhu X, Leis B, Joyce A. Experimental estimation of JR curves from load-CMOD record for SE(B) specimens. J ASTM Int 2008;5(5). [25] Gosh M, Moran B. An interaction energy integral method for computation of mixed mode cracks via two-state integrals. Engng Fract Mech 2002;69:299–319. [26] Subramanya HY, Viswanath S, Narasimhan R. A three-dimensional numerical study of mixed mode (I and II) crack tip fields in elastic–plastic solids. Int J Fracture 2005;136(1–4):167–85. [27] He MY, Hutchins JW. Asymmetric four-point crack specimen. J Appl Mech 2000;67(1):207–9. [28] Murakami Y. Handbook of stress intensity factors. Oxford: Pergamon Press; 1987. [29] Yang W, Qian X. Fracture resistance curve over the complete mixed-mode I and II range for 5083 aluminum alloy. Engng Fract Mech, submitted for publication [30] Joyce JA, Link RE. Effects of constraint on upper shelf fracture toughness. ASTM Spec Tech Publ 1995:142–77. [31] Cui XG, Cui CY, Cheng XN, Xu XJ, Lu JZ, Hu JD, et al. Microstructure and tensile properties of the sub-micro and nano-structured Al produced by laser surface melting. Mater Sci Engng A – Struct Mater Prop Microstruct Process 2010;527(27–28):7400–6. [32] Irwin GR. Plastic zone near a crack and fracture toughness. Mechanical and metallurgical behavior of sheet materials. In: Proceedings seventh Sagamore ordinance materials conference; 1960(IV). p. 63–78.