Fracture resistance curve for single edge notched tension specimens under low cycle actions

Engineering Fracture Mechanics 211 (2019) 47–60

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Fracture resistance curve for single edge notched tension specimens under low cycle actions

T

⁎

Tianyao Liua,b, Xudong Qiana, , Wei Wangb,c, Yiyi Chenb,c a

Department of Civil and Environmental Engineering, Centre for Offshore Research and Engineering, National University of Singapore, Singapore 117576, Singapore State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China c Department of Structural Engineering, Tongji University, Shanghai 200092, China b

A R T IC LE I N F O

ABS TRA CT

Keywords: J-R curve η approach Cyclic fracture resistance Single edge notched tension specimen hybrid method

This paper investigates monotonic and cyclic fracture behaviors of aluminum alloy 6061-T6511 and analyzes the cause of different fracture behaviors. This study adopts the η approach and a modified η approach to obtain monotonic and cyclic J-R curves respectively. The cyclic loaded SE (T) specimens with zero load ratio have similar J-R curves with monotonically load SE(T). In contrast, a tensile residual stress field at the crack front introduced by negative load ratios promotes a faster crack extension, which leads to a lower load resistance at the same CMOD level. The strain energy corresponding to the area under the P-CMOD envelope becomes lower, which induces the degradation of cyclic J-R curve with negative load ratios.

1. Introduction Large-scale engineering structures (e.g., bridges, pipelines, offshore platforms, high-rise buildings, etc.) experience severe lowcycle actions caused by extreme environmental loadings (waves and seismic loadings) and other accidental loadings [1–2]. Low cycle actions with high loading magnitudes can cause unstable and brittle fracture failure, especially in structures with pre-existing cracklike defects. Compared to the materials and structures under monotonic loadings, experimental evidences [3–4] have revealed a clear decrease in the fracture resistance under cyclic actions. To prevent such abrupt structural failures, a detailed understanding on the fracture resistance of the material under cyclic loading becomes necessary. The crack extension resistance curve, R-curve, has found wide applications in the fracture assessments on various engineering materials and structures, such as high-strength steels [5] and unidirectional glass/epoxy composites [6]. The material testing standard ASTM E1820 [7] provides guidelines to determine the Rcurve of metallic materials. A number of approaches to measure the fracture energy release rate, often measured the J-value, and the material fracture resistance curve, also known as the J-R curve, have evolved over the last few decades, including the single specimen approach [7], the multiple specimen method [8], among others. The ASTM E1820 [7] recommends a compliance-based single specimen technique, originally proposed by Rice et al. [9], to measure the fracture resistance curve for metallic materials under monotonic loading condition. The single specimen approach calculates the J-R curve utilizing the area under the load versus deformation curve (U ) and the dimensionless parameters η and γ . This method determines the J value via dividing ηU by the remaining ligament area, and corrects this J value by a correction factor based on γ [7]. For materials and structures under cyclic loading, however, a widely recognized approach has yet to be developed for the integrity assessment of engineering structures under low-cycle and ultra-low ⁎

Corresponding author. E-mail address: [email protected] (X. Qian).

https://doi.org/10.1016/j.engfracmech.2019.02.001 Received 27 June 2018; Received in revised form 15 January 2019; Accepted 2 February 2019 Available online 05 February 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 211 (2019) 47–60

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P

P

(a)

(b)

Area

Area

į

į crack opening point

(c)

P

Area

į crack opening point

Fig. 1. The area used in calculating the cyclic J integral: (a) area above the crack opening point in each cycle; (b) total area above the zero-load axis; and (c) total area above the crack opening point.

cycle actions. The reversal cycle in the cyclic loading creates unloading to the materials near the crack front and impinges on the fundamental validity of the J -integral, which predicates on the deformation theory of plasticity instead of the incremental theory of plasticity. To address this challenge, Dowling and Begley [10], Dowling [11] have proposed a cyclic J to characterize the driving force for fatigue under elastic-plastic deformations. The calculation of the cyclic J employs the same equation proposed by Rice et al. [9] and uses the area under the load-deformation curve above the crack opening point where the crack opens fully as shown in Fig. 1a. Based on the concept of the cyclic J , Landes and McCabe [12] have introduced the cyclic J-R curve by similar calculation method using the area between the upper envelope of the force versus displacement curve and the zero-load axis as indicated in Fig. 1b. Marschall and Wilkowski [13] highlight that the cyclic J-R curve is not an inherent material property and may vary for different loading conditions. To explore the effect of cyclic loadings on the fracture resistance, a few researchers have carried out experiments using the compact tension, C(T), specimen under a cyclic displacement-controlled loading which increases the peak displacement in every cycle. Seok et al. [3–4] report a deleterious decrease in the measured J-R curve for negative load ratios. Singh et al. [14] conclude that a higher loading rate degrades the fracture resistance further. Besides, their work proves that the cyclic J-R curves with non-negative load ratios vary insignificantly compared to monotonic J-R curve. Roy et al. [15] investigate the effect of cyclic loading on the base metal and the weldment. Their experimental results demonstrate that the lower bound of cyclic J-R curves occurs at a load ratio R = −1, and a negative R with a higher magnitude increases the fracture resistance slightly. All the above works indicate a decrease in the cyclic J-R curve when the load ratio is negative. This implies that the fracture assessment for structure under cyclic loading becomes more critical than structure under monotonic loading due to the reduced fracture resistance for a negative load ratio. In parallel, other researchers have extended the cyclic J concept to compute the elastic–plastic J-value under cyclic loading [16], to analyse the fatigue crack propagations [17], and to predict the fatigue limit based on cyclic R-curve [18]. Engineering testing standards [7] document the J-R curve measurement procedure for standard fracture specimens with high crack-front constraints corresponding to a positive T-stress, e.g., the C(T) specimen and the single-edge-notched bend, SE(B), specimen. The high crack-front constraints limit the amount of plastic deformation near the crack tip, which leads to a conservative J-R curve. The single-edge-notched tension, SE(T), specimen provides the fracture resistance value under relatively low plasticity constraints, usually corresponding to a negative T-stress. The J-R curve measured from SE(T) has thus become widely accepted in the fracture assessment for structures with low crack-tip constraints, e.g., cracked pipes and structural components with embedded or shallow surface cracks [19]. A number of previous efforts have addressed the J-R curve measurement using SE(T) specimens. Ruggieri [20] evaluates the η factor for the SE(T) specimen made of homogeneous materials. Paredes and Ruggieri [21] calculate the η factor for the SE(T) specimen with a weld centerline crack. Both works demonstrate the validation of the η approach for the J measurement of SE(T) specimens. Wang et al. [22] develop a complementary η approach incorporating both strain hardening and thickness correction in the J estimation. They demonstrate a good agreement between their work and conventional η results. Cravero and Ruggieri [23] calibrate the η and γ factors for clamped and pin-loaded SE(T) specimens. Subsequently, they apply these parameters to measure J-R curves to for SE(T) specimens with a growing crack [24]. Parool et al. [25] introduce a hybrid method to estimate fracture resistance curves for SE(T) specimens, which agree closely with the J-R curve based on the η approach. 48

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The current study aims to measure the J-R curve for the clamped SE(T) specimens made of aluminum alloy 6061-T6511 under low cycle loadings. This study extends the cyclic J by Dowling [11] to SE(T) specimens and determines the η and γ values required in measuring the J value through combined numerical and experimental approaches. The J-R curve for the clamped SE(T) specimen derives from a modified η approach under different load ratios. The numerical study analyzes the stress distribution near the crack tip to illustrate the load ratio effect on the measured cyclic J-R curve. The next section introduces two methods to measure the J-R curve for metallic materials. The subsequent section describes the experimental program including the material property test and the monotonic and cyclic fracture test. The following section presents the numerical work that calibrates η and γ parameters to support the J value measurement. The next section compares and discusses monotonic and cyclic J-R curves through the η approach and discusses the cyclic fracture response. The last section summaries key findings in this study. 2. Methods to measure J-R curve 2.1. Single specimen technique The ASTM E1820 [7] has implemented a compliance-based single specimen approach, originally proposed by Rice et al. [9], to measure the fracture resistance curve. This method divides the J-integral into a linear elastic part and a plastic part,

(1 − v 2) KI2 + Jpl E

J=

(1)

The linear-elastic part of the J value for the SE(T) specimen depends on the stress intensity factor, KI ,

a KI = f ⎛ ⎞ σ πa ⎝W ⎠

(2)

( ) a

where f W denotes a dimensionless function depending on the loading condition and the geometry of the specimen. The measurement of the plastic J value entails an η approach for an extending crack in the following form,

Jpl =

∫0

δpl

ηP dδpl − BN b0

∫0

a

γJpl0 b0

da

(3)

where η is the dimensionless factor to determine the plastic energy release rate from the area under the load-plastic deformation curve [26], γ is the dimensionless parameter to correct the J value for crack growth [27], BN denotes the net thickness that equals the distance between the roots of the side grooves, b0 represents the initial remaining ligament, δpl refers to the plastic deformation, and Jpl0 indicates the first term in Eq. (3). The first term in Eq. (3) calculates the energy release rate based on the strain energy, while the second term corrects for the crack growth [7]. Cravero and Ruggieri [24] verify that both the load-line displacement (LLD) and the crack mouth opening displacement (CMOD) can provide a reliable J value result for SE(T) specimen. This study adopts the load versus the CMOD data to determine the J value. The above η approach provides a procedure to build the monotonic J-R curve directly based on the load versus CMOD data. A number of researchers have extended the η approach to measure the J-R curve for specimens under cyclic loading. Chowdhury et al. [27] extend the method developed by Joyce et al. [28] to calculate the J resistance value under cyclic loading using the area above the crack opening point as shown in Fig. 1c, while other researchers [14–15] advocate that the J calculation should use the area surrounded by the envelope of cyclic load versus deformation curve above the zero-load axis. The former approach assumes that the applied energy starts contributing to the crack extension once the crack becomes fully opened despite the presence of a compressive global load. The latter treats the envelope of the cyclic load versus deformation curve as a special monotonic case. Chowdhury et al. [27] compare the J-R curves measured by both methods using the same set of experimental data. They point out that both methods can predict the decrease of fracture resistance under cyclic loading and the cyclic J-R curve by the latter method is more conservative. Roy et al. [15] recommend the latter method since it can reflect the effect of cyclic loadings on the monotonic J-R curve directly. The current study therefore utilizes the area surrounded by the envelope of the cyclic load versus deformation curve and the zero-load axis to determine the cyclic J-R curve. The crack extension measurement during the fracture test often deploys the compliance-crack size relationship,

C = f (a/ W )

(4)

where C denotes the compliance of the specimen,

C=

Δδ ΔP

(5)

The change of the deformation, Δδ , in Eq. (5) refers to the change of either LLD or CMOD. Zhu and Joyce [29] verify that the crack size estimated by the CMOD based compliance is more accurate than the LLD based compliance. The fracture test includes a partially unloading–reloading procedure at different CMOD intervals to measure the specimen compliance under monotonic loadings. The compliance calculation for the SE(T) specimen under cyclic loading makes use of the partial unloading-reloading record in-between cycles and the full unloading CMOD record in different cycles. 49

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P

(a)

a1 a2 a3 a4 a5

(b)

U

test

dU da

į

a1

a2

a4

a3

a5

a

(c)

J

a1- a0 a2-a0

a3-a0

a4-a0

a5-a0 ǻa

Fig. 2. The hybrid approach: (a) intersections between a single experimental P-δ curve and multiple numerical P-δ curves for different crack sizes; (b) variation of the strain energy with respect to the crack size; and (c) J-R curve based on the slope in (b).

2.2. Hybrid method The hybrid method combines a single experimental specimen with a growing crack and multiple finite element (FE) models each with a stationary crack of different sizes to determine J-R curve [30]. This method has the same principle as multiple specimen method [8], which measures the J value using,

J=−

1 dU B da

(6)

The multiple specimen approach estimates the derivative in Eq. (6), dU / da, by plotting U measured from specimens with different crack sizes against the crack size a . Fig. 2 illustrates the procedure of the hybrid method. Fig. 2a plots the load-deformation curve of the experimental record and numerical results. Qian and Yang [30] assume the crack size of the experimental specimen at the intersection point between the experimental load-deformation record and the numerical curve equals approximately the crack size in the corresponding FE model in the hybrid method. The strain energy, U , for each crack size at the intersection point thus derives from the area under the numerical load-displacement curve. Fig. 2b presents the variation of U with respect to the crack size at fixed displacement levels. The slope of the U versus a curve in Fig. 2b allows the determination of the J value using Eq. (6). Coupling the calculated J with the crack extension, Δa = ai − a0 , leads to the J-R curve shown in Fig. 2c. Qian and Yang [30] validate the hybrid method against the single specimen approach for the mode I J-R curve reported by Zhu and Joyce [31] and the mixed-mode fracture tests reported by Tohgo and Ishii [32]. Parool et al. [25] have extended the hybrid method for SE(T) specimen and pipes with a circumferential surface crack. 3. Experimental program This study encompasses an experimental program to investigate the fracture behavior of aluminum alloy 6061-T6511 under cyclic loads. Table 1 presents the chemical composition of the aluminum alloy. The experimental program includes: (1) coupon specimens to measure the mechanical properties of the specimen under uniaxial tension and uniaxial cyclic loading; (2) SE(T) specimens to measure the fracture resistance curve under monotonic loading; and (3) SE(T) specimens to measure the fracture resistance curve Table 1 Chemical compositions of aluminum alloy 6061-T6511. Weight (%)

Al

Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

6061-T6511

97.632

0.720

0.191

0.300

0.023

1.050

0.067

0.005

0.012

50

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(a)

(b)

450mm 40mm 225mm

75mm

50mm 12mm

R = 145mm

ı (MPa)

400

(c)

300 200 ıys = 280 MPa ıu = 340 MPa E = 68 GPa

100 0

0

0.02

0.04

İ

0.06

0.08

0.10

Fig. 3. (a) Configuration of the tension test; (b) set-up of the tension test; and (c) true σ-ε curve.

under cyclic loading. 3.1. Material characterization Fig. 3a shows the configuration of the tension specimen according to ASTM E8/E8M [33]. Fig. 3b presents the setup for the tension test. The experimental procedure applies a displacement-controlled loading at the rate of 0.02 mm/min. An extensometer of a gauge length of 100 mm monitors the axial strain of the tension specimen together with the post-yield strain gauge. Fig. 3c plots the true stress versus true strain curve. The material has a Young’s modulus E of 68 GPa with a Poisson’s ratio v of 0.3. The yield strength of the aluminum alloy equals to 280 MPa, and the ultimate strength is 340 MPa. Fig. 4a shows the specimen geometry for the uniaxial cyclic test based on ASTM E606/E606M [34]. Fig. 4b illustrates the test setup for the cyclic test. The experimental procedure follows the displacement-controlled loading with the displacement monitored by an extensometer of a gauge length of 50 mm. Fig. 4c describes the displacement-controlled loading, where the strain, ε , refers to the applied strain based on the extensometer measurement. The cyclic strain has an initial amplitude of 0.3% in the first cycle, with 0.1% increment in each subsequent cycle. The cyclic uniaxial test terminates when the strain amplitude reaches 1.6%. Fig. 4d plots the cyclic stress-strain curve. The material cyclic hardening property includes an isotropic part and a kinematic part [35]. The isotropic hardening part describes the evolution of the radius of the yield surface by Eq. (7),

k (ε p) = k 0 + Q (1 − exp[−bε p])

(7)

where Q denotes the difference between the yield strength and the ultimate strength, and b is the rate at which the ultimate tensile stress is reached with respect to the accumulated plastic strain, ε p . The kinematic hardening evaluates the translation of the yield surface through the backstress with the Hu in units of stress and the dimensionless γu by Eq. (8),

αk =

p p Hu 1 − e−γu ε + αk,1 e−γu ε γu

(

)

(8)

where αk and αk,1 denote the current backstress and the backstress at the beginning of current cycle. Table 2 shows the cyclic material property with respect to the hardening model. 3.2. Fracture test This study utilizes the SE(T) specimen for the fracture tests. Fig. 5a shows the configuration of the SE(T) specimen. The specimen has a span length S = 120 mm and a widthW = 30 mm. The thickness, B , is 15 mm, and the net thickness after side grooving, BN , equals 12 mm. The SE(T) specimen has an initial machined notch of aN / W = 0.35. Each specimen experiences a cyclic pre-cracking 51

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(a)

(b)

520mm 40mm 200mm

80mm 60mm

20mm Side view

Front view

R =32.5mm

(c)

İ

400

ı (MPa)

1.6%

(d)

TEST FE

200

0.3%

0 -0.3%

-200 -1.5%

-1.6%

-400 -0.02

Time

-0.01

0.00

İ

0.01

0.02

Fig. 4. (a) Configuration of the cyclic uniaxial test specimen; (b) set-up of the cyclic test; (c) strain-controlled loading; and (d) comparison between the experimental and numerical true σ-ε curves. Table 2 Summary of cyclic material property. E (GPa)

v

σys (MPa)

Hu (MPa)

γu

68

0.3

280

3800

68

loading as prescribed in ASTM E1820 [7]. The fatigue pre-cracking creates a fatigue crack extension of 2–4 mm ahead of crack front. The measurement of the J value from the specimen requires input on the η and γ values. The η value for SE(T) specimen derives from the numerical analyses presented in Section 4.2. The γ value considers the effect of crack extension on the measured J value and derives from the hybrid approach in this study (see discussions in Section 4.3). To validate a γ value for different crack sizes, the experimental program includes SE(T) specimens with another machined notch of aN / W = 0.5. Table 3 summary details of SE(T) specimens. The initial crack size listed in Table 3 includes the fatigue pre-cracking size. The displacement-controlled loading for the monotonic and cyclic fracture tests has a fixed loading rate of 0.05 mm/min. The monotonic fracture test includes an unloading-reloading cycle at every increment of 0.05 mm in the CMOD value to monitor the compliance of the specimen. Fig. 5c illustrates the displacement loading imposed on the SE(T) specimen for the cyclic fracture test. The LLD increases by 0.08 mm in each applied cycle until fracture failure occurs. The cyclic test entails three load ratios, R = 0, −0.5 and −1. Fig. 5d illustrates the test setup for the fracture test. A crack opening displacement (COD) gauge attached to the knife edges of the SE(T) specimen (see Fig. 5b) measures the crack mouth opening displacement during the test. Fig. 6 shows the load versus CMOD, measured from the monotonic load tests and cyclic tests with different load ratios.

4. Finite element analysis The determination of the J-R curve from the data presented in Fig. 6 requires input on the stress intensity factor in Eq. (2), the η and γ values in Eq. (3), and the compliance function in Eq. (4). This section describes the finite element analysis to: (1) calculate the linear-elastic stress intensity factor and the specimen compliance; 2) determine the η value for the SE(T) specimen; and 3) estimate the γ value from the hybrid method. Fig. 7a shows a quarter finite element model for the SE(T) specimen, which consists of a local crack front mesh generated in FEA CRACK [36] and the global mesh generated in MSC PATRAN [37] to generate 3D models for the SE(T) specimen. The numerical model consists of 20-node hexagonal elements with the reduced integration algorithm implemented in the research code WARP3D [38]. Since the initial crack size of the fatigue pre-cracked SE(T) specimen is around 14.4 mm, the 52

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Top view

(a) 15mm

1.5mm reduction in each side

1mm

75mm

120mm

Clamped bar

(b)

60°

45°

a0

0.8mm

30mm

Front view

(d)

(c)

į Pmax = 0.08 mm

PmaxxR

Time Fig. 5. (a) Configuration of the SE(T) specimen; (b) details of the knife edge; (c) displacement-controlled cyclic loading; and (d) set-up of the fracture test. Table 3 Summary of SE(T) specimens. Specimen No

Fabricated crack size (mm)

Initial crack size (mm)

Load ratio

Specimen No

Fabricated crack size (mm)

Initial crack size (mm)

Load ratio

SET10M-1 SET10M-2 SET10M-3 SET15M-1 SET15M-2 SET15M-3 SET10R0-1

10.5 10.5 10.5 15 15 15 10.5

14.41 14.44 14.40 17.28 17.26 17.23 14.25

monotonic monotonic monotonic monotonic monotonic monotonic 0

SET10R0-2 SET10R0-3 SET10RN05-1 SET10RN05-2 SET10RN1-1 SET10RN1-2 SET10RN1-3

10.5 10.5 10.5 10.5 10.5 10.5 10.5

14.20 14.41 14.61 14.23 14.11 14.18 14.17

0 0 −0.5 −0.5 −1 −1 −1

numerical models include a range of crack sizes from 14.4 mm to 21.9 mm with an increment of 0.5 mm. 4.1. KI and compliance The calculation of the elastic J -integral requires the linear-elastic KI solution. Hammond and Fawaz [39] have derived the function of the stress intensity factor for plain-sided SE(T) specimens and compare their results with the work of other researchers [40–42]. However, the KI solution in [39] does not apply to SE(T) specimens with a side groove. This study thus re-calculates the mode I stress intensity factor for the side-grooved SE(T) specimens. The remote tensile stress for the side-grooved SE(T) specimen depends on the effective thickness,

σ=

F W BBN

(9)

The non-dimensional geometric factor f (a/ W ) used in Eq. (2) derives from a nonlinear fit to the numerically computed KI values for different a/ W ratios, and has the following form,

a a 3 a 2 a f ⎛ ⎞ = 3.6597 ⎛ ⎞ − 4.3396 ⎛ ⎞ + 3.5716 ⎛ ⎞ + 0.4969 ⎝W ⎠ ⎝W ⎠ ⎝W ⎠ ⎝W ⎠

(10)

Fig. 8a demonstrates the close agreement between Eq. (10) and the FE results for the SE(T) specimen with a fixed H / W = 4.0 investigated in this study. The CMOD-based compliance shows a unique relationship with the crack size for a SE(T) specimen with a fixed H / W = 4.0 ratio. This study computes the CMOD-based compliance from linear-elastic analysis of the SE(T) specimen. The regression analysis of the computed compliance (in mm/kN) and the crack depth ratio a/ W leads to the following relationship, 53

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70

P (kN)

(a)

70

Monotonic

60

(b) R=0

60

50

50 a0= 14.4 mm

40

40

30

30 a0= 17.28 mm

20

20 10

10 0

P (kN)

0

0.2

0.4

0.6

0.8

0 0

1.0

0.2

0.4

CMOD (mm) 60

0.6

0.8

1.0

1.2

CMOD (mm)

P (kN)

(c)

60

R = -0.5

P (kN)

(d) R = -1

40

40

20 20

0

0

-20 -40

-20 0

0.2

0.4

0.6

CMOD (mm)

0.8

-60 -0.2

1.0

0

0.2

0.4

0.6

0.8

1.0

CMOD (mm)

Fig. 6. P-CMOD curves measured from SE(T) specimens for: (a) monotonic tests with the initial crack sizes of 14.4 mm and 17.3 mm; (b) cyclic tests with R = 0; (c) cyclic tests with R = –0.5; and (d) R = –1.

(a)

(b)

Clamped nodes

Crack tip mesh at symmetric plane

Symmetric plane

(d)

(c)

Clamped nodes Crack tip mesh in 2D model Fig. 7. (a) 3D quarter-symmetric model; (b) crack front mesh of 3D model; (c) 2D half-symmetric model; and (d) crack front mesh of 2D model.

a 3 a 2 a C = 0.0391 ⎛ ⎞ − 0.0289 ⎛ ⎞ + 0.0197 ⎛ ⎞ − 0.00234 ⎝W ⎠ ⎝W ⎠ ⎝W ⎠

(11)

Fig. 8b shows the good agreement between Eq. (11) and the FE results. 54

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2.3

f(a/W)

(a)

H/W = 4.0 B = 15mm BN = 12mm

2.2 2.1 2.0

0.016

C (mm/kN) H/W = 4.0 B = 15mm BN = 12mm

0.012

1.9

(b)

0.008

1.8 1.7 1.6 1.5 0.45

0.90

0.004

FE Eq. (10) 0.50

0.55

0.60

0.65

a/W

0.70

0 0.45

0.75

0.50

0.55

0.60

(c)

0.6

FELLD Eq. (14)

Ȗ

0.75

(d)

0.2

Eq. (16)

0.75 FECMOD Eq. (15)

Average hybrid Ȗ

0 -0.2

0.65 0.60 0.45

0.70

0.4

0.80

0.70

0.65

a/W

Ș

0.85

FE Eq. (11)

0.50

0.55

0.60

a/W

0.65

0.70

-0.4 0.45

0.75

0.50

0.55

0.60

0.65

0.70

0.75

a/W

Fig. 8. comparison between FE result and function of (a) f(a/W); (b) compliance; (c) ηLLD and ηCMOD; and (d) comparison between average hybrid γ and load separation γ.

4.2. η Factor The elastic-plastic J value depends on the area under the load versus the plastic deformation curve, as indicated in ASTM E1820 [7],

J=

ηApl (1 − v 2) KI2 + E BN b0

(12)

Eqs. (12) provides a direct relationship to determine the η value based on the computed J-integral and the area under the load versus plastic deformation curve. The numerical work defines 6 domains to calculate the through-thickness average J value in the side-grooved SE(T) model. Fig. 9a and b show the paths surrounding the user specified domains for the SE(T) model with a/ W = 0.563. Fig. 9c compares the J values computed in different domains at various loading levels. Negligible variation among the domains indicates a stable J result which is adequate in the η calibration. However, η is not a strictly constant value at different loading levels. Dramatic fluctuations occur at small load levels as the near-tip stress field reaches towards a steady state condition. At a large load level, the plastic zone near the crack tip will increase beyond the limit of the small-scale yielding. ASTM E1820 [7] indicates the maximum J -integral capacity for a specimen should remain smaller than b0 σY /10 or BσY /10 , which provides an upper limit to estimate the η value. The effective yield strength, σY , here refers to the average value of the yield strength and the ultimate strength. As for the lower limit, the plastic J-integral becomes significant in the total J-integral when J value is around b0 σY /50 or BσY /50 . Consequently, the current study calculates the η value using the following range of J values,

Max{b0 σY /50, BσY /50} < J < Min{b0 σY /10, BσY /10}

(13)

Eqs. (14) and (15) show functions of LLD based and CMOD based η respectively,

a 2 a ηLLD = 1.3529 ⎛ ⎞ − 1.7933 ⎛ ⎞ + 1.3808 ⎝W ⎠ ⎝W ⎠

(14)

a 2 a ηCMOD = 1.6755 ⎛ ⎞ − 1.7697 ⎛ ⎞ + 1.1197 W ⎝ ⎠ ⎝W ⎠

(15)

Fig. 8c compares the equations and numerical results of LLD based and CMOD based η. 55

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(a)

(b)

60mm

path4

0.05

60mm

path2

path3

path5

path1

J/b0ıY

path6

(c) LLD = 0.4mm LLD = 0.5mm LLD = 0.6mm LLD = 0.7mm

0.04 0.03 0.02 0.01 0 1

2

3

4

Path No.

5

6

Fig. 9. (a) definition on domain 1–4 for J-integral computation; (b) definition on domain 5–6 for J-integral computation and (c) comparison among J value computed from different domains at various load levels.

4.3 γ. Factor The factor γ usually derives from the load separation method by the following equation,

γ = −1 + ηLLD −

′ b ηLLD W ηLLD

(16)

Eq. (16) indicates that the factor γ derives from the LLD-based η . Substituting Eq. (16) into Eq. (14) leads to an expression for γ ,

(

a

) (2.7058 ( ) − 1.7933) ( ) − 1.7922 ( ) + 1.3808

−1 W a 2 a γ = 1.3529 ⎛ ⎞ − 1.7933 ⎛ ⎞ + 0.3808 + a ⎝W ⎠ ⎝W ⎠ 1.3529 W

a W

2

a W

(17)

The parameter γ represents a crack growth correction factor in the η approach and remains positive. However, the value of γ in Eq. (17) becomes negative when a/ W exceeds 0.534 as shown in Fig. 8d, which violates the physical meaning of γ . To solve this issue, this study determines the γ value from the hybrid method using the steps detailed below: Step 1: Apply the hybrid method to calculate J-R curve with the maximum crack extension of 0.25b0 . In this step, the crack size in the experimental specimen at the intersection point between the test data and the FE result follows the crack size in the FE model; Step 2: Calculate the factor γ at crack sizes corresponding to the loading steps based on the ASTM E1820 [7] approach,

Jpl (i) = [Jpl (i − 1) +

ηpl (i − 1) Apl (i) − Apl (i − 1) b(i − 1)

BN

][1 − γpl (i − 1)

a(i) − a(i − 1) ] b(i − 1)

(18)

or,

γ(i − 1) =

Jpl (i) b(i − 1) ·(1 − ηpl (i − 1) Apl (i) − Apl (i − 1) ) ai − a(i − 1) Jpl (i − 1) + ( b ) B (i − 1)

(19)

N

where Apl denotes the area under load versus plastic CMOD curve; Step 3: Set all negative γ as zero; Step 4: calculate the average γ of all the positive and modified γ in step 3. The average γ value thus calculated equals 0.393. This study utilizes the average γ as a constant factor instead of setting γ as a function of the crack size for the SE(T) specimen. To validate the average γ value based on the above hybrid method, this study compares the J-R curves from the η approach with that from the hybrid method for SE(T) specimens with initial crack sizes of 14.4 mm and 17.3 mm. Fig. 10a and 10c show numerical loadCMOD curves and experimental data corresponding to specimens with initial crack sizes of 14.4 mm and 17.3 mm respectively. 56

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80

P (kN)

(a) a1 = 14.4 mm a2 = 14.9 mm a3 = 15.4 mm a4 = 15.9 mm a5 = 16.4 mm a6 = 16.9 mm a7 = 17.4 mm a8 = 17.9 mm a9 = 18.4 mm a10 = 18.9 mm a11 = 19.4 mm a12 = 19.9 mm test

70 60 50 40 30 20 10 0 0

60

0.2

0.4

0.6

0.8

1.0

CMOD (mm)

1.2

200

(b)

Hybrid method approach

160 120 80 40 0 0

1.4

1

2

3

4

5

6

ǻa (mm)

P (kN)

(c) a1 = 17.4 mm a2 = 17.9 mm a3 = 18.4 mm a4 = 18.9 mm a5 = 19.4 mm a6 = 19.9 mm a7 = 20.4 mm a8 = 20.9 mm a9 = 21.4 mm a10 = 21.9 mm test

50 40 30 20

120

J (kJ/m 2 )

100

(d)

Hybrid method approach

80 60 40 20

10 0 0

J (kJ/m 2 )

0.2

0.4

0.6

0.8

0 0

1.0

CMOD (mm)

1

2

3

ǻa (mm)

4

5

Fig. 10. Hybrid procedure for SE(T) with initial crack size of 14.4 mm (a) experimental and numerical load versus CMOD curves; (b) comparison between J-R curves from hybrid approach and η approach; hybrid procedure for SE(T) with initial crack size of 17.28 mm (c) experimental and numerical load versus CMOD curves; and (d) comparison between J-R curves from hybrid approach and η approach.

Fig. 10b and 11d compare the J-R curve based on the hybrid approach and the J-R curve based on the η and γ values for the two SE(T) specimens with different initial crack sizes. The crack extension for the η approach derives from the compliance and crack size relationship in Eq. (11). The γ value calculated in Eq. (19) remains sensitive to the variation of J value in different steps. To examine the effect of the γ value on the measured J-R curve for the SE(T) specimen, this study compares a number of different γ values in deriving the J-R curve: (1) a zero γ value which ignores the correction in J for the crack growth; (2) the average γ value from the hybrid method, γ = 0.393; (3) twice the average γ value from the hybrid method, γ = 0.786; and (4) a linear fit of the hybrid γ value against the crack size, which follows,

a γ = −3.4834 ⎛ ⎞ + 2.4538 ⎝W ⎠

(20)

Fig. 11a and b compare the J-R curves from the hybrid method and the η approach with different γ for specimens with initial crack sizes of 14.4 mm and 17.3 mm respectively. Fig. 11 confirms that the changes in the γ value does not lead to noticeable variations in the J-R curve. A larger γ value leads to a slightly lower J-R curve, as shown in Fig. 11 and reflected in Eq. (3). The subsequent analysis 200 160 120

J (kJ/m 2 )

(a)

120

Hybrid result =0 = 0.393 = 0.786 fitting

100 80

J (kJ/m 2 )

(b)

Hybrid result =0 = 0.393 = 0.786 fitting

60 80

0 0

40

a0/W = 0.48 B = 15mm BN = 12 mm

40 1

2

3

ǻa (mm)

4

5

a0/W = 0.576 B = 15mm BN = 12 mm

20 0 0

6

1

2

3

ǻa (mm)

4

5

Fig. 11. sensitivity analysis in effect of γ on J resistance of the specimen with initial crack size of (a) 14.4 mm; and (b) 17.28 mm. 57

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200 160

J (kJ/m 2 )

(a)

200 160 120 80

80 SET10M-1 SET10M-2 SET10M-3

40

120 100

1

2

3

ǻa (mm)

4

5

SET10R0-1 SET10R0-2 SET10R0-3

40 0 0

6

1

2

3

4

J (kJ/m 2 )

(c)

120

R = -0.5

100 80

60

60

40

J (kJ/m 2 )

1

2

3

6

(d)

R = -1

40 SET10RN05-1 SET10RN05-2

20

5

ǻa (mm)

80

0 0

(b)

R=0

Monotonic

120

0 0

J (kJ/m 2 )

4

5

SET10RN1-1 SET10RN1-2 SET10RN1-3

20 0 0

6

ǻa (mm)

1

2

3

4

5

6

ǻa (mm)

Fig. 12. J-R curves for the case of (a) monotonic loading; (b) R = 0; (c) R = −0.5; and (d) R = −1.

reported in this paper thus utilizes the average γ value due to its simplicity. 5. Fracture resistance curve under cyclic loading The cyclic fracture test includes three load ratios of 0, −0.5 and −1. Fig. 12 shows the cyclic J-R curve constructed from the upper envelope of the cyclic P-CMOD record. This study assumes the same η and γ values for the SE(T) specimen under cyclic loading and the monotonic loading, since η and γ depend primarily on the specimen geometry. Fig. 13 compares the lower-bound J-R curve for each load ratio. The zero-load ratio introduces little change to the J-R curve compared with the monotonic J-R curve. It indicates the insignificant effect of non-negative load ratio on the cyclic J-R curve. In contrast, a negative load ratio leads to significant deterioration in the fracture resistance curve. To understand the reason for the degradation in J-R curve caused by the negative load ratio, this study investigates the near tip stress field using a 2D plane-strain model with a stationary crack shown in Fig. 7c and d. The half symmetric 2D model has the same in-plane dimension as the SE(T) specimen under the cyclic load tests. The numerical procedure applies a monotonic displacement load and cyclic displacement load with a load ratio of 0, −0.5 and −1 as shown in Fig. 14a. The cyclic simulation adopts the kinematic hardening model shown in Table 2. Fig. 14b presents the near-tip opening stresses computed from the plane-strain SE(T) model corresponding to the zero load at

Fig. 13. comparison among lower bounds of different loading conditions. 58

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(a)

į (mm) 0.40 0.32 0.24 0.16 0.08 0

yy

3

/

(b)

ys

R=0 R = -0.5 R = -1

2

Pmax

1 0 A

-1

Time

-2 -3

Pmax x R

-4 0

0.1

0.2

0.3

0.4

0.5

x / b0 6

ǻa (mm)

5 4

(c)

70

Monotonic R=0 R = -0.5 R = -1

(d)

60 50 40

3

30

2

Monotonic R=0 R = -0.5 R = -1

20

1 0 0

P (kN)

10 0.2

0.4

0.6

CMOD (mm)

0.8

0 0

1.0

0.2

0.4

0.6

CMOD (mm)

0.8

1.0

Fig. 14. (a) displacement control mode in numerical analysis; (b) near-tip opening stresses corresponding to the zero load at point A; (c) comparison among crack extension versus CMOD curves of different load ratios; and (d) the lower-bound envelopes from the duplicated SE(T) specimens from each load ratio.

point A in Fig. 14a, which defines the starting point of the 5th cycle. Fig. 14b indicates a severe compressive residual stress in the immediate vicinity of the crack tip for the SE(T) specimen under a zero load ratio. The presence of this compressive residual stresses ahead of the crack tip delays the crack extension in SE(T) specimens under a zero load ratio. The SE(T) specimens with a negative load ratio, on the other hand, experiences a tensile residual stress immediately ahead of the crack tip, followed by a compressive residual stress field further away from the crack tip. This tensile residual stress facilitates a faster crack extension in SE(T) specimens under a negative load ratio than that under the zero load ratio. The evolution of the compliance determined crack extension and the CMOD value in Fig. 14c confirms the faster crack extension in SE(T) specimens under a negative load ratio. The crack size presented in Fig. 14c corresponds to the CMOD value at the upper envelope of the load-CMOD curve for different SE(T) specimen. As indicated in Fig. 14c, a negative load ratio facilitates the crack extension, which leads to a larger crack size at the same CMOD value for R < 0 than that for R = 0 and the monotonic loading. The larger crack size causes a lower load resistance at the corresponding CMOD value, as evidenced in Fig. 14d, which compares the upper P-CMOD envelope for the cyclic loaded SE(T) specimen and the P-CMOD for the monotonic loading condition. The P-CMOD curves presented in Fig. 14d represent the lower-bound envelopes from the duplicated SE(T) specimens from each load ratio. The P-CMOD curves for the SE(T) specimens with the negative load ratios encompasses a smaller area than the P-CMOD for R = 0 and the monotonic loading, which leads to a smaller strain energy U and hence a smaller J value at the same CMOD value. The smaller J value, coupled with a large crack extension at the corresponding CMOD value (shown in Fig. 14c), leads to a lower J-R curve for the SE(T) specimen with a negative load ratio than the J-R curve for R = 0 and the monotonic loading. 6. Summary and conclusions This study reports an experimental effort to measure the J-R curve for SE(T) specimens under monotonic loading, and low cycle actions with three different load ratios. The J-R curve for the cyclic loaded SE(T) specimen derives from the upper envelope of the PCMOD curve recorded in the experiment. The measurement of the J value makes use of a numerically determined η function and an average γ value for different crack sizes determined from a hybrid method. The determination of the crack extension in the SE(T) specimen utilizes the compliance versus the crack size relationship computed from a linear-elastic analysis of the 3D SE(T) model. The above work focuses on the clamped SE(T) specimens and supports the following conclusions: (1) The J-R curves exhibit marginal dependence on the magnitude of the γ value for SE(T) specimens under a monotonic loading. The J-R curve measured from the η approach using the averageγ value agrees closely with that determined using the hybrid method. (2) The fracture resistance curve measured from the SE(T) specimen remains similar to that measured from the cyclic loaded SE(T) specimen with a zero load ratio (R = 0 ) due to the similar P-CMOD envelopes and CMOD-Δa relationships. 59

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(3) The negative load ratios (R = −0.5 and − 1.0 ) creates a tensile residual stress field immediately ahead of the crack tip at the zero load level, which promotes apparently a faster crack extension in the SE(T) specimens, as reflected by the evolution of the ΔaCMOD relationship for the SE(T) specimens. (4) The faster crack extension in the cyclically loaded SE(T) with a negative load ratio causes a lower load resistance of the specimen at the same CMOD level than the SE(T) specimen under a zero load ratio or a monotonic loading. The smaller strain energy corresponding to the area under the P-CMOD envelope leads to a lower J-R curve for the cyclic loaded SE(T) specimens with a negative load ratio. Acknowledgement The research scholarship provided by the National University of Singapore to the first author is gratefully appreciated. 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