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Stress intensity factor for clamped SENT specimen containing non-straight crack front and side grooves
Accepted Manuscript Stress Intensity Factor for Clamped SENT Specimen Containing Non-Straight Crack Front and Side Grooves Yifan Huang, Wenxing Zhou PII: DOI: Reference:
S0167-8442(17)30047-2 http://dx.doi.org/10.1016/j.tafmec.2017.07.011 TAFMEC 1916
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
25 January 2017 22 April 2017 12 July 2017
Please cite this article as: Y. Huang, W. Zhou, Stress Intensity Factor for Clamped SENT Specimen Containing Non-Straight Crack Front and Side Grooves, Theoretical and Applied Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.tafmec.2017.07.011
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Stress Intensity Factor for Clamped SENT Specimen Containing NonStraight Crack Front and Side Grooves Yifan Huanga, b, * and Wenxing Zhouc a
Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By
Drive, Ottawa, Ontario K1S 5B6, Canada b
School of Engineering Science, Simon Fraser University, 8888 University Drive, Burnaby,
British Columbia V5A 1S6, Canada c
Department of Civil and Environmental Engineering, The University of Western Ontario, 1151
Richmond Street, London, Ontario N6A 5B9, Canada *: Corresponding author, Email: [email protected]
Abstract:
Three-dimensional (3D) finite element analyses (FEA) of clamped single-edge
notched tension (SENT) specimens are performed to evaluate the stress intensity factor (SIF). Both the average and local SIF values over the crack front are evaluated. Plane-sided (PS) specimens containing both straight and curved crack fronts and side-grooved (SG) specimens containing straight crack fronts are considered including six average crack lengths (i.e. aave/W = 0.2 to 0.7). PS specimens with three thickness-to-width ratios (i.e. B/W = 0.5, 1 and 2) are analyzed. The curved crack front is assumed to be bowed symmetrically and characterized by a power-law expression with a wide range of curvatures. For SG specimens, one relative thickness (B/W = 1) and ten depths of side groove are considered. The 3D SIF are compared with the present two-dimensional (2D) SIF. It is observed that the 2D SIF are sufficient as the difference between the 2D and 3D SIFs is in general within 10% if the crack front straightness requirements reported in the literature is satisfied.
An empirical 3D SIF solution is developed that can
potentially serve as a driving force for an assessment in actual structures that has similar crack geometries and loading configurations.
Keywords:
Stress intensity factor (SIF); Three-dimensional finite element analysis (3D
FEA); Single-edge notched tension (SENT); Crack front curvature; Side groove
Nomenclature a
crack length
aave
average crack length over the specimen thickness
az = 0, az = ±B/2
crack lengths at the mid-plane and surfaces of the specimen
B
gross thickness of the specimen
BN
net thickness of the specimen
CT
compact tension
CTOD
crack tip opening displacement
dsg
depth of the side groove on each side
E
elastic (Young’s) modulus
2D
F
normalized SIFs associated with K2D
F3Dloc, F3Dave
normalized SIFs associated with K3Dloc and K3Dave
H
daylight length (distance between grips)
J
J-integral, nonlinear energy release rate
Jave
average J evaluated over the crack front
Jloc(i)
the local J value at the ith layer
K2D
SIF in two-dimensional analyses
K3Dloc
local SIF in three-dimensional analyses
K3Dave
average SIF over the crack front in three-dimensional analyses
Mij, Nij
fitting coefficients of Eqs. (9) and (10)
m
non-dimensional factor in Eq. (13)
P
applied load
p
shape parameter used in Eq. (3)
PS
plane-sided
Qi
fitting coefficients of Eq. (11)
q
fitting coefficients of Eq. (12)
SENB
single-edge notched bend
SENT
single-edge notched tension
SG
side-grooved
SIF
stress intensity factor
u
applied displacement on clamped surface
W
the width of the specimen
non-dimensional factor in Eq. (14)
parameters used to characterize the crack front curvature
n
remote nominal tensile stress
Poisson’s ratio
CR, SG, *
modification factors to construct K3Dloc solutions
CR, SG, *
modification factors to construct K3Dave solutions
non-dimensional factor in Eq. (13)
ratio between the net and gross specimen thicknesses
1. Introduction The single-edge notched tension (SENT) specimen [1-6] is one of the commonly used specimens in fatigue and fracture toughness tests of metallic materials. Based on the notch configuration, the SENT specimens can be classified into two subsets. The first group of SENT specimens contain a surface notch (i.e. partly-through-thickness notch) and are generally used to evaluate the fatigue life of the materials [5, 6]. The notch is usually elliptical and requires the calculation of the stress intensity factor (SIF) along the three-dimensional (3D) crack front (e.g. [7, 8]).
The second group of SENT specimens are through-thickness notched and the
corresponding SIF can be determined from the simplified two-dimensional (2D) analyses [9-13]. Figure 1 schematically shows the configuration of a typical SENT specimen including the width (W), thickness (B), crack length (a) and the daylight distance (distance between the clamped surfaces, H) of the specimen. Cross sections with surface notch and through-thickness notch are also shown in the figure. The through-thickness-notched SENT specimens usually need to be fatigue pre-cracked to simulate natural cracks before the subsequent ductile tearing test. In the recent years, the use of the through-thickness-notched SENT specimen with clamped ends in the
experimental evaluation of ductile fracture toughness, such as the J-integral resistance (J-R) or crack tip opening displacement resistance (CTOD-R) curve, has gained significant research interests [14-19] especially in the oil and gas pipeline industry. Because the crack-tip stress and strain fields of such specimens are similar to those of the full-scale pipe containing surface cracks under longitudinal tension and/or internal pressure [10, 20], the fracture toughness determined from the SENT specimen can lead to less-conservative design and assessment of pipelines with respect to cracks compared with the typical standard single-edge notched bend (SENB) and compact tension (CT) specimens [21, 22]. The present paper is focused on the through-thickness notched SENT specimens with clamped ends, which will be referred to as the SENT specimen hereafter for simplicity. Testing on plane-sided (PS) through-thickness-notched specimens generally leads to a curved crack front and shear failure of the trailing edge [23] (see Fig. (1b)). These phenomena are caused by the difference in the states of stress along the crack front. At the region near the center of the PS specimen, the stress state is close to the plane-strain condition with higher SIF, which promotes a relatively fast crack growth. On the other hand, the stress state near the side surface is close to the plane-stress condition with lower SIF, therefore causing relatively slow crack growth and the shear lips near the free surface [23]. The SIFs over the crack front in this case are similar to the ones in surface-notched specimens. The side-grooved (SG) specimens are used in the J(CTOD)-R curve testing to achieve relatively straight crack fronts. As shown in Fig. 1(c), one groove is machined into each lateral side of the specimen to remove the low-SIF region. The net thickness between the side grooves (BN) can be calculated as BN = B – 2dsg with dsg being the depth of each side groove. The present authors [18] have previously investigated the impact of the BN/B ratio on the evaluated SIF and proposed a modification factor to account for this effect.
Numerical and analytical studies concerning the calculation of SIF for SENT specimens are reported in the literature [9-13, 24]. For example, Cravero and Ruggieri [9] and Shen et al. [10] respectively proposed the SIF solutions for SENT specimens with the crack length (a) over specimen width (W) ratio in the range of 0.1 ≤ a/W ≤ 0.7 and 0.05 ≤ a/W ≤ 0.95 based on 2D plane-strain finite element analyses (FEA). By using 2D plane-stress FEA, John and Rigling [11] developed a 6th-order polynomial SIF solution for SENT specimens with 0.1 ≤ a/W ≤ 0.9. Zhu [12, 13] proposed an analytical SIF solution for SENT specimens with 0 < a/W < 1. Jin and Wang [24] conducted 3D FEA to evaluate the local SIF distribution along the straight crack front for PS SENT specimens with a/W = 0.2, 0.4, 0.6 and 0.8 and the thickness-over-width ratio (B/W) equal to 0.1, 0.2, 0.5, 1, 2 and 4. It is noted that all of the existing SIF solutions [9-13] are based on the results from 2D analytical or numerical methods and 3D FEA of plane-sided SENT specimens with straight crack fronts [24]. There is a lack of a systematic study that validates the applicabilities of these solutions to the SENT specimens with curved crack fronts or/and side grooves. Such a study is valuable in that it provides guidance in the standardization of the testing procedure for the SENT specimen. Note that there are currently no crack front straightness criteria specified in any SENT test standards.
The objective of the present study was to carry out a systematic
investigation of the SIF for SENT specimens containing curved crack fronts or side grooves. We carried out 3D FEA of both PS and SG SENT specimens with wide ranges of the average crack length (aave) over specimen width ratio (aave/W), B/W, crack front curvature and side-groove depth to evaluate the local SIF distributed along the crack front. The focus of the present study is the clamped SENT specimen with a daylight-over-width ratio H/W = 10 because the crack-tip stress fields of such specimens correspond closely to those of the full-scale pipes containing
circumferential cracks [10]. The impacts of the crack front curvature and side-groove depth on the 3D SIF are studied. Both the local and average 3D SIF are compared with the 2D SIF and characterized by two parameters and ψ. These parameters are also used to construct the 3D SIF solutions from existing 2D SIF solutions. The 3D SIF solution can potentially serve as a driving force for an assessment in actual structures that has similar crack geometries and loading configurations. Furthermore, the 3D SIF solution is useful in an ongoing study that investigating the impacts of crack front curvature and depth of side groove on the constraint parameter (e.g. Tstress) along the crack front of the SENT specimen [24]. The rest of the paper is structured as follows. Section 2 presents a brief review of the existing 2D SIF solution for the SENT specimen. Section 3 describes the characterization of the curved crack front employed in this study. Section 4 describes the configurations of the FE models, material properties and computational procedures involved in the present study. The SIFs for specimens containing curved crack fronts and side grooves are evaluated and compared with the existing 2D SIF in Section 5, followed by conclusions in Section 6.
2. Review of 2D SIF Solutions for SENT Specimens The 2D SIF (K2D) solution for SENT Specimens can be expressed as
a K 2 D F 2 D n a W
(1)
where the remote nominal tensile stress σn = P/BW with P being the applied tensile force, and F2D is the normalized SIF associated with K2D. Recently, Zhu [12, 13] proposed the following analytical F2D solution for SENT specimens with H/W = 10
a F 2D W
2 3 a a a 1.9873 0.7422 11.188 45.82 W W W 4 5 6 a a a 100.49 120.03 51.448 W W W
W 1
2a a 1 W W
3 2
(2)
Eq. (2) is more representative than other F2D solutions reported in the literature [9-11] because it has a wider applicable range of a/W, i.e. 0 < a/W < 1.
3. Characteristics of Curved Crack Front Previous studies [25, 26] indicated that curved crack fronts in specimens made of homogeneous materials are typically bowed (i.e. crack length at the mid-plane greater than that at the free surface) and symmetric about the mid-plane. Therefore, such a curved crack front was assumed in the present study, and characterized by the following power-law function suggested by Nikishkov et al. [25]: p 2 z W a z aave 1 p 1 p B aave a z B /2 W 1 B aave B 0 a z dz
(a) (b) (c)
(3)
where z (–B/2 ≤ z ≤ B/2) is the coordinate in the specimen thickness direction; az is the crack length at the location of coordinate z; aave and az = ±B/2 denote the average crack length and crack lengths at the free surfaces of the specimen, respectively, and λ and p (p > 1) are parameters characterizing the curvature of the crack front. Note that λ = 0 corresponds to a straight crack front. By examining the fatigue pre-crack fronts of a series of plane-sided CT, SENB and SENT
specimens, Nikishkov et al. [25], and Yan and Zhou [26] showed that the parameter p in Eq. (3) can be adequately set to a fixed value of 3.0 for different curved crack fronts.
Detailed
evaluation process of λ based on the nine-point crack length measurement can be found in Ref. [17]
4. Numerical Analysis Linear-elastic analyses of 198 PS and 60 SG SE(T) models were performed to evaluate the stress intensity factors. Young's modulus (E) and Poisson's ratio (ν) were set to be 207 GPa and 0.3, respectively to simulate the typical structural steel. All the specimens included in this study have a width W = 20 mm and a daylight (H) of 10W. Six aave/W ratios (i.e. aave/W = 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7) were included in the analyses. For the PS models, three B/W ratios (B/W = 0.5, 1 and 2) and eleven λ values (λ= 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 and 0.1) were considered. A typical FE model with aave/W = 0.5, λ = 0.1 and B/W = 1 is schematically shown in Fig. 2(a) together with the fixation and loading conditions. The analyses matrix for the SG models consists of one B/W ratio (B/W = 1) and ten side groove depths (i.e. dsg/B = 3%, 4%, 5%, 6%, 7%, 7.5%, 8%, 9%, 10%, and 12.5%) with the corresponding thickness reduction ratios χ = BN/B = 0.94, 0.92, 0.9, 0.88, 0.86, 0.85, 0.84, 0.82, 0.8 and 0.75, respectively. The side groove is modeled as a U-shape notch with a fixed root radius rsg = 0.5 mm as recommended in ASTM E1820 [21]. Schematics of U-notched side grooves are shown in Figs. 2(d). Stationary cracks were assumed in all the analyses. Only a quarter of the specimen with appropriate constraints imposed on the remaining ligament was modelled due to symmetry. The FEA code ADINA® [27] was employed to analyze all the models. A typical quartersymmetric 3D model has 10 layers over the half net thickness (BN/2). For the SG model, the side groove ((B – BN)/2) was divided into 12 layers. The thickness of each layer was arranged such
that the corresponding mesh density increases from the mid-plane to the free surface (or root of the side groove) to capture the high stress gradients at these locations (see Figs. 2(b) and 2(d)). The 8-node 3D brick elements with 2×2×2 integration points were adopted in the analyses. A spider-web mesh around the crack tip was established with 45 concentric semicircles (i.e. rings) surrounding the crack tip (see Fig. 2(c)). The in-plane dimension of the elements in the 45th ring is about 2,000 times that of the element in the first ring (i.e. closest to the crack tip). The total number of elements is approximately 12,000 in a typical PS specimen, and 28,000 in a typical SG specimen. Uniform displacements (u) were applied on two lateral surfaces that are considered as the clamped surfaces each with a length of 2W (see Fig. 2(a)). The applied tension P was calculated as the total reactions of the nodes on the clamped surface. All the analyses results were obtained from one loading step that corresponds to an applied displacement u = 2 m. The SIFs along crack front were calculated from the J-integral evaluated in the FEA. The values of J in each layer along the thickness direction, i.e. the local J values, denoted by Jloc0, Jloc2,…Jloc10, were calculated using the virtual crack extension method [27]. Note that the local J value at the midplane equals Jloc0. Let zi denote the distance between the end of the ith layer and mid-plane (i.e. z = z0 = 0) as shown in Fig. 3. The weighted average J value over the entire crack front, Jave, is then calculated as follows based on the trapezoidal rule:
J ave
z1 z0 J loc 0 J loc1 z2 z1 J loc1 J loc 2 1 2 2 B z10 z9 z9 z8 J loc8 J loc 9 z10 z9 J loc 9 2 2 2 2
(4)
Note that the local J value at the specimen free surface, i.e. Jloc10, cannot be accurately evaluated from FEA. The two outermost semicircular rings surrounding the crack tip were used to define
the virtual shifts. Within the realm of linear elastic fracture mechanics, the local and average stress intensity factors in 3D analyses, K3Dloc and K3Dave respectively, can be calculated from the following equations:
3D J loc E K loc 1 2 K 3 D J ave E ave 1 2
(a) (5)
(b)
By analogy to Eq. (1), the corresponding 3D normalized SIFs, F3Dloc and F3Dave, are defined as 3D 3 D aave B z K loc F , , , , loc W W B n aave 3D F 3 D aave , B , , K ave ave W W n aave
(a) (6)
(b)
5. Results and Discussions 5.1 Stress intensity factor for plane-sided specimens containing curved crack fronts The values of F3Dloc and F3Dave corresponding to various aave/W, B/W and λ are estimated and compared with the F2D obtained from Eq. (2). For better comparisons, define the parameters
and ψ: 3D Floc 2D F 3D Fave F 2D
(7)
The difference (%) between F3Dloc (or F3Dave) and F2D can be calculated as 1 100% and
1 100% . The subscripts "CR" and "SG" denote the configurations of the specimens (i.e. with curved crack fronts or side grooves) for which and ψ are applicable. Figures 4 shows ψCR
values plotted against λ and aave/W for specimens with B/W = 0.5, 1 and 2. Figures 5 through 7 show variation of CR with 2|z|/B and λ for specimens with various aave/W and B/W ratios. Figures (4a) through (4c) suggest that for given aave/W and B/W artios, F3Dave as well as ψCR decreases as λ increases. On the other hand, for given B/W artios and λ value, F3Dave generally increases as aave/W decreases. The impact of λ on F3Dave decreases as B/W increases. The maximum difference between F3Dave and F2D is about 12%, 7% and 4% for specimens with B/W = 0.5, 1 and 2. Figures 5 through 7 suggest that for λ ≤ 0.01, as 2|z|/B increases from 0 to 0.8, the value of CR remains constant at around 1.00 and then decreases to 0.9 – 0.95 as 2|z|/B further increases to 0.98. For λ = 0.01 to 0.02, the distribution of CR along 2|z|/B is relatively uniform. On the other hand, for λ ≥ 0.03, CR generally increases as 2|z|/B increases. Higher values of λ lead to smaller CR at the mid thickness and larger CR at the specimen free surface. Similar observations regarding the F3Dloc – aave/W and F3Dave – 2|z|/B relationships are reported in the literature for CT [28-30] and SENB [31] specimens containing curved crack fronts. Huang and Zhou [17] proposed the following crack front straightness criteria for SENT specimens:
aave aave B Min 0.06 0.1 W , 0.03 , for W 0.5, 0.2 W 0.7 a a B Min 0.03 0.2 ave , 0.08 , for 1, 0.2 ave 0.7 W W W a a a B Min 0.06 0.5 ave , 0.3 ave , for 2, 0.2 ave 0.7 W W W W
(8)
It is reported that Eq. (8) ensures the differences in Jave (corresponding to loading level of 1.0 to 1.3 times of the limit load) and the elastic compliance between the specimens with curved and straight crack fronts to be within 5%. For λ values satisfying Eq. (8), Figures 4 through 7
suggest that the difference between F3Dave and F2D is also within 5%, and difference between F3Dloc and F2D is within 10% corresponding to 2|z|/B ≤ 0.9. Therefore, the 2D SIF obtained from Eq. (2) is considered adequate for SENT specimens with crack front curvature characterized by Eq. (8).
5.2 Stress intensity factor for side-grooved specimens containing straight crack fronts Figures 8 illustrates the ψSG values plotted against χ for specimens with B/W = 1 and aave/W = 0.2 to 0.7 while Figures 9(a) through 9(f) show the distributions of CR along the specimen thickness. Figure 8 suggests that ψSG decreasing from 1.17 – 1.23 to 1.0 as χ increases from 0.75 to 1.0. Figure 8 also suggests that aave/W has little impact on ψSG. It is observed from Figs. 9(a) through 9(f) that the value of SG increases as 2|z|/BN increases. Lower χ values lead to higher
SG values at both the specimen mid thickness and free surface. Similar observations are reported in the literature [31]. It is interesting to observe that along the crack front, the distribution of SG corresponding to 0.75 ≤ χ ≤ 0.94 is somewhat similar to that of CR corresponding to λ ≥ 0.03.
5.3 Three-dimensional stress intensity factor solution For given aave/W and B/W ratios, the following polynomial equations were proposed to express
CR as a function of (2|z|/BN) and λ, and SG as a functions of χ based on the values of CR and SG shown in Figs. (5) – (7) and (9): i a z 6 2 z ave B CR , , , i W W B i 0 BN 2 M ij j i j 0
(9)
i a 6 2 z z ave B SG , , , i W W BN i 0 BN 2 j i N ij j 0
(10)
Similarly, regression is performed on the values of ψCR shown in Fig. 4:
aave B 2 , , Qi i W W i 0
CR
(11)
where the fitting coefficients Mij, Nij and Qi are listed in Tables 1, 2 and 3, respectively. The fitting errors of Mij and Nij are in general within 1%. For specimens with aave/W = 0.2 and of λ = 0.1, the fitting errors of Mij increase up to 3% at the position very close to the free surface, i.e. 2|z|/B = 0.976. Using Qi leads to a very accurate prediction of ψCR such that the corresponding fitting errors are less than 0.1% for all the cases considered in the present study. On the other hand, as indicated in Fig. 8, ψSG is relatively insensitive to aave/W. The following equation for ψSG as a power-law function of χ is proposed based on the ψSG values presented in Fig. 8:
1 SG q q 0.6265
for 0.2
aave B 0.7, 1 W W
The maximum fitting error of Eq. (12) is about 2%.
(12)
Note that using Eq. (12) is more
advantageous than using q = 0.5, which is commonly adopted in the literature [3, 9-13, 21, 22] and yields a maximum fitting error of 6%. Note also that in a previous study by the present authors [18], q = 0.58 was proposed based on the modification with respect to F3Dave whereas Eq. (12) was developed based on the modification with respect to F2D. To investigate the adequacy of Eqs. (9) and (11) for specimens containing curved crack fronts characterized by the shape parameter p other than 3.0, additional FEA for specimens with aave/W
= 0.2 to 0.7, B/W = 1, and λ = 0.08 were performed. The value of p was set to 2.5 for the additional analyses. The rationale for selecting p = 2.5, as opposed to, say, p = 3.5, in the analysis is that for a given λ, decreaseing p leads to the central portion of the crack front, which has the largest contribution to K3Dave, to become more curved [17, 26]. The prediction errors (%) associated with the value of CR and ψCR predicted from Eqs. (9) and (11) for these specimens, e and eψ, are shown in Table 4. The results indicate that decreasing p from 3.0 to 2.5 but maintaining aave/W and λ has small impacts on ψCR (values of eψ are within 0.7% for p = 3.0 and 2.5). The maximum e increases from 1% to about 5% as p decreases from 3.0 to 2.5. Note that the maximum e for p = 2.5 occurs at the 2|z|/B = 0.9755. Within the region of 0 ≤ 2|z|/B ≤ 0.95, the values of e for p = 2.5 are less than 3.5%. Given the above, Eqs. (9) and (11) are also considered applicable for curved crack fronts with p = 2.5.
5.4 Stress intensity factor for side-grooved specimens containing curved crack fronts The above discussions are with respect to PS specimens containing curved crack fronts and SG specimens containing straight crack fronts. Additional FEA for SG specimens containing curved crack fronts was performed to investigate the SIFs for SG specimens containing curved crack fronts. The geometry of the analyzed specimens is characterized byaave/W = 0.2 to 0.7, B/W = 1, χ = 0.85, p = 3 and λ = 0.05. Note that the net thickness of the specimen, BN, was used to evaluate az and aave in Eq. (3) for the specimens. For simplicity, the side groove was modelled as a sharp V-notch with an opening angle of 45° as schematically shown in Fig. 10. Let * and ψ* denote the 3D modification factors for the local and average SIFs, respectively, for SG specimens containing curved crack fronts. Figures 11(a) through 11(f) show values of * as a function of 2|z|/BN. For comparasions, CR and SG estimated by Eqs. (9) and (10) were also
plotted in the figures. It is observed that Eq. (9) can accurately predict * for 2|z|/BN ≤ 0.5 but underestimates * when 2|z|/B further increases.
On the contrary, Eq. (10) in general
overestimates * and results in good predictions only when 2|z|/BN = 0.9 ~ 0.95. Figure 11 suggests that for a specimen with given λ and χ, CR and SG serve as the lower and upper bounds, respectively, of *. Figure 12 illustrates the ψ* values, as well as ψCR and ψSG estimated by Eqs. (11) and (12), plotted against aave/W. The figure suggests that values of ψ* are between the corresponding ψCR and ψSG. Two empirical expressions are developed to estimate * and ψ*, respectively as follows:
and
* CR 1 SG m 2 z BN
(13)
* CR
(14)
1
SG
where m and θ are non-negative dimensionless factors that depend on λ and χ. In the limit when ω = θ = 0 (i.e. z = 0 and m ≠ 0, or 2|z|/BN < 1 and m = ∞), * and ψ* coincide with CR and ψCR, respectively. Similarly, ω = θ = 1 (i.e. z ≠ 0 and m = 0, or 2|z|/BN = 1) yields * = SG and ψ* = ψSG. The potential dependence of m and θ on λ and χ will be investigated in a future study. Based on the numerical results, m = 3 and θ = 0.3 are proposed for specimens with χ = 0.85 and λ = 0.05. Equations (13) and (14) with the proposed m and θ are also plotted in Figs. (11) and (12) for comparison. The prediction errors of these equations are within 5% (for 2|z|/BN ≤ 0.95) and 1%, respectively. Finally, the full-field 3D SIF solution are presented as Eqs. (15a) and (15b)
3D 2D aave B z K loc W , W , B , , F n aave K 3 D aave , B , , F 2 D a ave n ave W W
(a) (15)
(b)
where
and
CR SG *
for 0 and 1
CR SG *
for 0 and 1
for 0 and 1
(16)
for 0 and 1
for 0 and 1
(17)
for 0 and 1
6. Summary and Conclusions Three-dimensional linear-elastic finite element analyses of clamped single-edge notched tension (SENT) specimens were carried out to evaluate both the local and average stress intensity factors along the crack front (F3Dloc and F3Dave). Plane-sided (PS) specimens containing both straight and curved crack fronts and side-grooved (SG) specimens containing straight crack fronts are considered including six average crack lengths (i.e. aave/W = 0.2 to 0.7). For PS specimens, symmetric bowed crack fronts characterized by a power-law expression were considered in the analysis. Three specimen thicknesses (B/W = 0.5, 1 and 2) and eleven crack front curvatures (λ = 0 to 0.1) were included in this study. For SG specimens, one relative thickness (B/W = 1) and ten depths of side groove (χ = 0.94 to 0.75) are considered. Key observations and findings are summarized in the following. 1) Two parameters, and ψ, are introduced to relate F2D and F3Dloc, and F2D and F3Dave, respectively. For plane-sided specimens containing curved crack fronts, F3Dave as well as ψCR
decreases as λ increases for given aave/W and B/W artios. For given B/W artios and λ value, F3Dave generally increases as aave/W decreases. The impact of λ on F3Dave decreases as B/W increases. The maximum difference between F3Dave and F2D is about 12%, 7% and 4% for specimens with B/W = 0.5, 1 and 2. For λ ≤ 0.01, as 2|z|/B increases from 0 to 0.8, the value of
CR remains constant at around 1.00 and then decreases to 0.9 – 0.95 as 2|z|/B further increases to 0.98. For λ = 0.01 to 0.02, the distribution of CR along 2|z|/B is relatively uniform. On the other hand, for λ ≥ 0.03, F3Dloc or CR generally increases as 2|z|/B increases.
Equation (2) is
considered adequate for SENT specimens with crack front straightness satisfying Eq. (8). 2) For side-grooved specimens containing straight fronts, F3Dave as well as ψSG decreases as χ increases from 0.75 to 1.0. The ψSG – χ relationship is not sensitive to aave/W for specimens with B/W = 1. Value of SG increases as 2|z|/BN increases, which is similar to the distribution of CR along specimen thickness corresponding to λ ≥ 0.03. 3) Based on the numerical results, empirical equations were developed to evaluate CR, SG, ψCR and ψSG, respectively. The errors of predictions of these equations are less than 2% in most cases.
Additional FEA for specimens containing curved crack fronts with p = 2.5 were
performed to validate the proposed CR and ψCR. The validation suggests that they are also applicable for curved crack fronts with p = 2.5. 4) The modification factors output from additional FEA of side-grooved specimens containing curved crack fronts, * and ψ*, were also compared with the proposed 3D modification factors. It is found that the proposed CR agrees well with * at the regions 2|z|/BN ≤ 0.5. The proposed
SG agrees well with * when 2|z|/BN = 0.9 ~ 0.95. The values of *(ψ*) basically lie between the corresponding CR(ψCR) and SG(ψSG).
A general expression of *(ψ*) is proposed as a
combination of CR(ψCR) and SG(ψSG). The empirical 3D SIF solution can potentially serve as a
driving force for an assessment in actual structures that has similar crack geometries and loading configurations. Future investigations will be focused on the dependences of m and θ on λ and χ as well as SIFs for SG specimens containing concave crack fronts, i.e. λ < 0.
Acknowledgments The study was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada. Constructive comments by the anonymous reviewer are much appreciated.
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[5] Wu, X. R., Newman, J. C., Zhao, W., Swain, M. H., Ding, C. F., and Phillips, E. P. (1998). Small crack growth and fatigue life predictions for high-strength aluminium alloys: Part I: Experimental and fracture mechanics analysis. Fatigue and Fracture of Engineering Materials and Structures, 21(11), 1289-1306. [6] Newman, J. C., Wu, X. R., Swain, M. H., Zhao, W., Phillips, E. P., and Ding, C. F. (2000). Small-crack growth and fatigue life predictions for high-strength aluminium alloys. Part II: crack closure and fatigue analyses. Fatigue and Fracture of Engineering Materials and Structures, 23(1), 59-72. [7] Newman, J. C., Reuter, W. G., and Aveline, C. R. (2000). Stress and fracture analyses of semi-elliptical surface cracks. ASTM STP 1360: Fatigue and Fracture Mechanics, P.C. Harris and K.L. Jerina (Eds.), West Conshohocken, 30, 403–423. [8] Jin, Z., and Wang, X. (2013). Weight functions for the determination of stress intensity factor and T‐ stress for semi‐ elliptical cracks in finite thickness plate. Fatigue and Fracture of Engineering Materials and Structures, 36(10), 1051-1066. [9] Cravero, S., and Ruggieri, C. (2007). Estimation procedure of J-resistance curves for SE(T) fracture specimens using unloading compliance. Engineering Fracture Mechanics, 74(17), 2735-2757. [10] Shen, G., Gianetto, J. A., Tyson, W. R. (2008). Report 2008-18(TR): Development of procedure for low-constraint toughness testing using a single-specimen technique. Ottawa, Canada: CANMET-MTL. [11] John, R., and Rigling, B. (1998). Effect of height to width ratio on K and CMOD solutions for a single edge cracked geometry with clamped ends. Engineering Fracture Mechanics, 60(2), 147-156.
[12] Zhu, X. K. (2016). Corrected stress intensity factor solution for a British standard single edge notched tension (SENT) specimen. Fatigue and Fracture of Engineering Materials and Structures, 39(1), 120-131. [13] Zhu, X. K. (2017). Full-range stress intensity factor solutions for clamped SENT specimens. International Journal of Pressure Vessels and Piping, 149, 1-13. [14] Huang, Y., and Zhou, W. (2014). Investigation of plastic eta factors for clamped SE(T) specimens based on three-dimensional finite element analyses. Engineering Fracture Mechanics, 132, 120-135. [15] Huang, Y., and Zhou, W. (2014). J-CTOD relationship for clamped SE(T) specimens based on three-dimensional finite element analyses. Engineering Fracture Mechanics. 131, 643-655. [16] Huang, Y., and Zhou, W. (2015). Numerical investigation of compliance equations used in the R‐ curve testing for clamped SE(T) specimens. Fatigue and Fracture of Engineering Materials and Structures, 38(10), 1137-1154. [17] Huang, Y., and Zhou, W. (2015). Effects of crack front curvature on J–R curve testing using clamped SE(T) specimens of homogeneous materials. International Journal of Pressure Vessels and Piping, 134, 112-127. [18] Huang, Y. and Zhou, W.. (2016). Effective thickness of side-grooved clamped SE(T) specimens
for
J-R
curve
testing.
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σn a (z)
Uncracked ligament
A
A Fatigue crack growth H
A W
A
B
Machined notch a
σn
-B/2
0
B/2
z
(a) Surface-notched specimen a (z)
a (z)
Side groove Uncracked ligament Ductile crack growth
Fatigue pre-crack front W
A
A Fatigue crack growth
Machined notch
aave
A W
A Fatigue crack growth
aave
Machined notch
BN/2 -B/2
0
B/2
(b) Through-thickness-notched plane-sided specimen
Figure 1
z
-B/2
0
(c) Through-thickness-notched side-grooved specimen
Crosss sections of typical SENT specimens.
B/2
z
Clamped surfaces
u
see (b)
see (c)
x
aave
y
(a)
Configuration of a typical FE model with aave/W = 0.5, λ = 0.1 and B/W = 1
W
z
W
az = 0
az = -B/2
dsg
B/2 rsg
(c) Mesh around the sharp crack tip
(b) Front view of the FE model
Figure 2
(d) Configuration of the U-shape side groove
Configuration and meshing of a typical finite element model.
Jloc9
J
Jloc8 Jloc1
Jloc0
Jloc2
Jave
Jloc7
Jloc3 Jloc4
z0
z1
z2
z3
z4
Jloc5
z5
Jloc6
z6 z7 z8 z9 z10 = B/2
Figure 3
Schematic of the calculation of the weighted average J along the crack front.
z
1.04
aave/W increases
1.02 1.04
1.00
1.04
1.04
1.02
1.02
1.00
1.00
0.98
0.98
1.02 1.00
0.98
0.98 CR
0.96
0.96
0.94
0.94 0.92
0.92 0.90
0.90
0.88 0.88 0.86 0.00
0.86 0.00 0.02
CR a ave /W = 0.2 a/W = 0.2 a ave /W = 0.3 a/W = 0.3 a ave /W = 0.4 a/W = 0.4 a ave /W = 0.5 a/W = 0.4 a ave /W = 0.6 a/W = 0.6 a ave /W = 0.7 a/W = 0.7
0.02 0.04
0.04 0.08 0.06 0.10 0.08 0.06
0.10
0.96
CR
0.96
0.94
0.94
0.92
0.92
0.90
0.90
0.88
0.88
0.86 0.00
0.02
0.04
0.06
0.08
0.10
0.86 0.00
0.02
(a) B/W = 0.5
Figure 4
(b) B/W = 1
Variation of ψCR with λ for PS specimens with various aave/W and B/W ratios.
0.04
0.06
(c) B/W = 2
0.08
0.10
1.9
1.9 2.0 1.8 1.9
1.7
1.8
1.6
1.7
1.5
1.6
CR
1.4 1.5
1.3 1.4 1.3 1.2 1.2 1.1
(a) aave/W = 0.2
1.9
(b) aave/W = 0.3
1.8
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
CR
1.1
1.0
1.0
0.9
(c) aave/W = 0.4
1.8
1.7
1.7
1.6
1.6
1.5
1.5
1.4
CR
1.3
1.4
1.3
1.2
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.9
0.8
λ increases
0.8
0.7 0.0 0.0
0.2
0.2
0.4
0.6
0.4
0.8
0.6
1.0
0.8
0.7
0.7 0.0
1.0
0.2
2z /B
0.6
0.8
1.0
0.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.7
1.7
1.6
1.6
1.6
1.5
1.5
1.5
1.4
1.4
1.4
CR
1.3 1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8
0.8
0.8
0.2
0.4
0.6
2z /B Figure 5
0.8
1.0
0.6
0.8
1.0
0.7
0.7 0.0
1.0
1.3
1.2
0.7
0.8
(f) aave/W = 0.7
1.8
1.7
CR
0.6
1.9
1.8
1.3
0.4
2z /B
1.9
1.8
0.2
2z /B
1.9
CR
0.4
0.0
0.2
0.4
2z /B
0.6
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 0.5 and various aave/W ratios.
1.5
(a) aave/W = 0.2
1.9 1.4
(b) aave/W = 0.3
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
1.8 1.7 1.3 1.6
CR
1.5
1.5
2.0
1.5 1.2 1.4 1.3 1.1 1.2
CR
1.1 1.0
(c) aave/W = 0.4
1.4
1.4
1.3
1.3
1.2
CR
1.2
1.1
1.1
1.0
1.0
0.9
0.9
1.0 0.9 0.9
λ 0.8 increases 1.0
0.8
0.8
0.0
0.0
0.2
0.2
0.4
0.6
0.4
0.6
0.8
0.8
0.8
1.0
0.0
0.2
2z /B
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6 1.4
1.3
1.3
1.3
1.2
1.2
1.2
CR
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8
0.2
0.4
0.6
2z /B Figure 6
0.8
1.0
0.8
1.0
0.6
0.8
1.0
(f) aave/W = 0.7
1.4
0.8
0.6
1.5
1.4
CR
0.4
2z /B
1.5
0.0
0.2
2z /B
1.5
CR
0.4
0.8
0.0
0.2
0.4
0.6
2z /B
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 1 and various aave/W ratios.
2.0
1.30
1.9
(a) aave/W = 0.2
1.25
1.8 1.7
1.20
1.6
1.15
1.5 1.4
1.10
CR
1.3
1.05
1.2
1.00
1.1
CR
0.9
0.90
0.0
0.2
λ increases
0.4
1.20
1.20
1.15
1.15
1.10
1.10
CR
1.05
1.00
0.95
0.95 0.90 0.8
1.0
0.85
0.85
0.80
0.80
0.80 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
2z /B
CR
0.4
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.20
1.20
1.15
1.15
1.15
1.10
1.10
1.10
CR
1.05 1.00
1.00
0.95
0.95
0.95
0.90
0.90
0.90
0.85
0.85
0.85
0.80
0.2
0.4
0.6
2z /B Figure 7
0.8
1.0
1.0
0.6
0.8
1.0
1.05
1.00
0.80
0.8
(f) aave/W = 0.7
1.25
1.20
CR
0.6
1.30
1.25
1.05
0.4
2z /B
1.30
0.0
0.2
2z /B
1.30 1.25
1.05
1.00
0.6
(c) aave/W = 0.4
1.25
0.90
0.8
0.85
(b) aave/W = 0.3
1.25
1.0
0.95
1.30
1.30
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
0.80
0.0
0.2
0.4
0.6
2z /B
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 2 and various aave/W ratios.
1.00 0.98 0.96
1.25
0.94 0.92
1.20
0.90 0.88
1.15
a ave /W = 0.2 a/W = 0.2 a ave /W = 0.3 a/W = 0.3 a ave /W = 0.4 a/W = 0.4 a ave /W = 0.5 a/W = 0.4 a ave /W = 0.6 a/W = 0.6 a ave /W = 0.7 a/W = 0.7
0.86
SG 0.001.10 0.02
0.04
0.06
0.08
0.10
aave/W increases
1.05
1.00
Eq. (11) 0.95 1.00
0.95
0.90
0.85
0.80
0.75
Figure 8
Variation of ψSG with χ for SG specimens with B/W = 1 and various aave/W ratios.
2.0 1.9
1.8
1.8
1.7
1.7
1.6
1.6
SG
2.0
2.0
2.0 1.9
1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2
(a) aave/W = 0.2 χ =1 χ = 0.94 χ = 0.92 χ = 0.90 χ = 0.88 χ = 0.86 χ = 0.85 χ = 0.84 χ = 0.82 χ = 0.80 χ = 0.75
(b) aave/W = 0.3
1.9
χ decreases
SG
1.1
1.1
1.0
1.0
(c) aave/W = 0.4
1.9
1.8
1.8
1.7
1.7
1.6
1.6
1.5
SG
1.4
1.5
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.9
0.9
0.8
0.8 0.0 0.0
0.2
0.2
0.4
0.6
0.8
0.4
0.6
1.0
0.8
0.8
0.8 1.0
0.0
0.2
2 z / BN
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.8
1.8
1.7
1.7
1.7
1.6
1.6
1.6
1.4
1.5
SG
1.4
1.3
1.2
1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8 0.2
0.4
0.6
2 z / BN Figure 9
0.8
1.0
0.6
0.8
1.0
1.4
1.3
0.0
1.0
1.5
1.3
0.8
0.8
(f) aave/W = 0.7
1.9
1.8
SG
0.6
2.0
1.9
1.5
0.4
2 z / BN
2.0
1.9
0.2
2 z / BN
2.0
SG
0.4
0.8 0.0
0.2
0.4
2 z / BN
0.6
0.8
1.0
0.0
0.2
0.4
2 z / BN
Variation of SG with 2|z|/BN and χ for SG specimens with B/W = 1 and various aave/W ratios.
22.5◦
Figure 10
Configuration of the sharp V-shape side groove.
1.7
1.7
1.7
1.7 1.6
(a) aave/W = 0.2
(b) aave/W = 0.3
* obtained from FEA FEA
1.6 1.5
(c) aave/W = 0.4
1.6
1.6
1.5
1.5
1.4
1.4
Eq. (9) 1.5 1.4
Eq. (10)
Eq. (13) with m = 3
1.4 1.3
1.3
1.3 1.2
1.2
1.2
1.2 1.1
1.1
1.1
1.1 1.0
1.0
1.0
1.0 0.9 0.0 0.9 0.0
0.9
0.9 0.2
0.4
0.6
0.8
0.2
0.4
2 z / BN0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.5
1.5
1.4
1.4
1.4
1.3 1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9 0.0
0.2
0.4
0.6
2 z / BN Figure 11
0.8
1.0
1.0
0.6
0.8
1.0
1.3
1.2
0.9
0.8
(f) aave/W = 0.7
1.6
1.5
0.6
1.7
1.6
1.3
0.4
2 z / BN
1.7
1.6
0.2
2 z / BN
1.0
1.7
1.3
0.9 0.0
0.2
0.4
2 z / BN
0.6
0.8
1.0
0.0
0.2
0.4
2 z / BN
Variation of with 2|z|/BN for SG specimens with B/W = 1, χ = 0.85, λ = 0.05 and various aave/W ratios.
1.12 1.12 1.10 1.08 1.08 1.06
1.06 1.04
1.02 1.04 1.00 1.02 0.98 * obtained from FEA ψ FEA Eq. (11) Eq. (12) Eq. (14) with θ = 0.3
0.96 1.00 0.94 0.98 0.92
0.96 0.90 0.1
0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
0.8 0.8
aave/W
Figure 12
Variation of ψ with aave/W for SG specimens with B/W = 1, χ = 0.85, λ = 0.05 and various ratios.
Table 1 Coeffieients Mij in Eq. (9). aave/W
B/W
j 2
0.5
1 0
1
2
0.3
0.5
6
5
4 3 -710.34 1512.26 485.33 1268.63 1885.77 6445.75 5605.03 3536.43 -16.82 50.24 -57.98 31.98
2
1
0
-69.87
4.87
-4.72
915.43
-89.47
-1.12
-8.35
0.81
1.04
41.91
-2.08
-39.87
-0.16
0.68
1.02
28.28
-0.80
0
2865.74 1715.24 -435.15 1013.59 3191.89 844.47 2878.21 407.42 2505.80 1577.96 -14.26 42.52 -49.00 27.00 -7.03
2
-638.80 1830.49
1
384.77
0
-12.69
2
1683.20
2 0.2
i
1
1 0
1133.47 -291.08 2073.04
1307.82 1139.26 37.74 -43.43
-717.96
184.42
-18.12
0.35
23.94
-6.22
0.61
1.00
6938.20 1097.78 -108.03 5624.06 4023.10 1911.23 6535.96 929.38 -90.85 5681.83 3586.79 -17.46 52.08 -60.07 33.12 -8.66 0.84
-4.96 -1.35 1.04
1
2
2
871.82
1
793.58
0
-14.74
2
-23.66
1
379.16
0
-13.99
0.4
1
2
32.44 -12.99 1291.90 1124.84 41.64 -47.95
528.01
-52.19
-2.40
380.36
-37.17
-0.47
-7.24
0.70
1.03
-3.92
4.52
-0.55
-1.03
-707.75
182.28
-17.86
0.00
26.41
-6.86
0.67
1.01
485.49
-46.83
-5.17
974.25
-95.23
-1.48
-8.90
0.86
1.04
0
3036.01 2442.92 1764.64 1995.76 6839.22 5939.98 3756.09 -17.97 53.59 -61.78 34.05
2
1901.63
1
765.71
0
-15.11
2
438.49
1
361.19
0
-14.87
2 0.5
3409.96 2808.60 1958.09 2693.80 2349.67 1473.35 43.86 -50.48 27.77
1
698.48
6977.96 1043.19 -102.30 5893.69 3931.71 2596.16 366.35 -35.79 2265.59 1418.67 44.90 -51.63 28.38 -7.40 0.72 1579.72 1346.26 1228.66 1070.62 44.19 -50.81
-2.61 -0.66 1.03
-884.74
234.04
-22.88
-1.18
-671.92
173.28
-16.96
-0.23
27.95
-7.26
0.71
1.01
continued
aave/W
B/W
j
0
6 5 4 3 2 2891.99 1867.97 -485.06 1002.22 3356.88 1975.59 6759.97 961.79 5875.28 3709.64 -17.42 51.83 -59.65 32.82 -8.57
2
2374.90
1
762.66
0
-15.51
2
764.82
1
347.31
0
-15.66
2 0.5
0.5
1
2
1
2 0.6
0.5
i
1 0
1
0
49.24
-5.70
-93.96
-1.56
0.83
1.03
8550.18 1263.61 -123.49 7280.10 4787.59 2589.49 365.90 -35.74 2258.22 1415.40 46.03 -52.90 29.07 -7.58 0.74 2704.77 2320.69 1506.61 1180.17 -644.51 1028.87 46.45 -53.33 29.30
-3.14 -0.76 1.02
396.16
-38.64
-1.69
166.37
-16.27
-0.36
-7.62
0.74
1.01
119.80
-5.68
-88.07
-1.73
0.78
1.02
6903.99 4564.25 2288.81 8133.71 1201.28 1867.16 6362.52 902.18 5540.24 3484.74 -16.55 49.12 -56.41 30.98 -8.08
1
2
2
2393.39
1
772.42
0
-15.96
2
1022.60
1
337.85
0
-16.61
0.7
1
2
3590.39 3089.56 1995.18 1147.97 -626.30 1000.78 49.20 -56.41 30.96
1.01 -1.68
161.84
-15.82
-0.56
-8.04
0.78
1.00
157.45
-6.01
-84.96
-1.90
0.77
1.02
2
2333.32
1
796.09
0
-16.70
2
1282.92
1 0
333.57 -17.97
8175.96 1187.28 -115.19 7047.35 4535.30 2714.67 384.91 -37.60 2362.77 1485.38 49.47 -56.78 31.18 -8.12 0.79 4490.15 3868.63 2493.29 -988.59 1134.39 -618.26 53.25 -61.02 33.48
-0.94
-50.97
0
1
-3.17
523.32
9054.13 6004.05 2977.08 10691.8 1583.09 1811.36 6155.68 870.92 5366.77 3366.79 -16.41 48.62 -55.77 30.60 -7.97
2 0.5
8494.96 1244.02 -121.15 7279.49 4733.24 2628.03 371.97 -36.34 2289.67 1437.21 47.34 -54.37 29.87 -7.79 0.76
-3.45 -1.13 1.01
653.53
-63.65
-1.88
159.90 -8.69
-15.62 0.84
-0.75 1.00
Table 2 Coeffieients Nij in Eq. (10). aave/W
j
0
4 2 -402.11 1143.00 686.40 -174.50 1275.85 650.10 2089.49 287.20 1862.20 1129.43 -199.05 573.91 -647.50 352.08 -89.15
2
-449.94 1297.32
2 0.2
0.3
0.4
1
1464.39
795.23
-204.07
0 2
-503.72 1464.28
1
2 1 0 0.6
5
2470.24 346.48 2176.20 1348.01 -249.78 733.69 -839.67 461.72 -118.83
1
0
0.5
6
i 3
2
748.59
1662.46
906.86
-233.78
2907.06 411.89 2545.29 1593.80 -315.86 937.14 596.64 -154.72 1079.85 867.98
-528.33 1537.50
1745.28
951.27
-245.11
3163.39 448.24 2768.27 1733.90 -365.75 1087.71 692.68 -179.82 1254.14 942.16
-488.16 1411.66
-
865.16
-222.17
1
0
16.82
1.64
-27.81
-3.21
8.71
2.59
19.74
1.64
-33.64
-3.27
11.62
2.66
22.66
1.61
-40.06
-3.24
15.13
2.66
23.73
1.52
-43.54
-3.09
17.56
2.60
21.44
1.38
1594.23 1 0
2 0.7
1 0
3017.22 425.39 2649.42 1648.90 -369.64 1096.27 694.79 -180.17 1260.83 905.35
-421.90 1205.97
1349.27
726.46
-185.35
2639.34 368.27 2333.57 1434.09 -335.04 986.84 619.07 -159.95 1128.67 804.21
-41.24
-2.85
17.56
2.48
17.80
1.19
-35.57
-2.50
15.54
2.31
Table 3 Coeffieients Qi in Eq. (11).
2
i 1
0
0.2
0.5 1 2
-5.15 -2.43 -0.87
-0.55 -0.15 -0.01
1.03 1.02 1.01
0.3
0.5 1 2
-5.01 -2.64 -1.15
-0.58 -0.19 -0.05
1.03 1.02 1.01
0.4
0.5 1 2
-4.97 -2.72 -1.28
-0.60 -0.23 -0.09
1.03 1.02 1.01
0.5
0.5 1 2
-5.31 -3.15 -1.75
-0.58 -0.22 -0.09
1.02 1.01 1.00
0.6
0.5 1 2
-5.14 -3.09 -1.71
-0.63 -0.29 -0.16
1.01 1.00 1.00
0.7
0.5 1 2
-5.28 -3.29 -1.89
-0.66 -0.34 -0.22
1.00 1.00 1.00
aave/W
B/W
1
Table 4. Prediction errors (%) associated with the value of CR and ψCR predicted from Eqs. (9) and (11) for PS specimens with aave/W = 0.2 to 0.7, B/W = 1, and λ = 0.08 aave/W 2|z|/B
e
e
0 0.2447 0.4341 0.5808 0.6944 0.7823 0.8504 0.9031 0.9439 0.9755
0.2 p = 2.5 p=3
0.3 p = 2.5 p=3
0.4 p = 2.5 p=3
0.5 p = 2.5 p=3
0.6 p = 2.5 p=3
0.7 p = 2.5
p=3
0.21
0.01
0.32
0.01
0.38
0.01
0.43
0.02
0.49
0.01
0.69
0.01
1.04 0.49 0.55 2.42 0.62 1.11 1.51 0.43 2.10 3.11
0.00 0.02 0.16 0.42 0.46 0.18 0.52 0.31 0.80 0.51
0.96 0.34 0.78 2.87 0.80 1.22 1.57 0.22 2.77 3.97
0.01 0.02 0.20 0.47 0.55 0.22 0.57 0.32 0.92 0.49
0.93 0.25 0.93 3.14 0.90 1.28 1.60 0.10 3.14 4.47
0.01 0.02 0.22 0.52 0.59 0.24 0.61 0.35 0.97 0.52
0.91 0.18 1.07 3.36 1.02 1.35 1.64 0.00 3.41 4.85
0.02 0.01 0.24 0.54 0.62 0.25 0.64 0.34 0.99 0.53
0.90 0.11 1.24 3.58 1.17 1.45 1.64 0.13 3.58 5.13
0.01 0.02 0.23 0.55 0.62 0.23 0.64 0.32 0.94 0.54
0.96 0.11 1.27 3.38 2.25 2.22 2.16 0.24 3.24 4.90
0.02 0.02 0.24 0.55 0.63 0.22 0.64 0.29 0.88 0.54
1
Highlights
Evaluate the SIF for SENT specimen containing curved crack front and side grooves
Validates the 2D SIF for specimens with curved crack fronts and side grooves.
Develop the modification factors to construct the 3D SIF solution for SENT specimen
S0167-8442(17)30047-2 http://dx.doi.org/10.1016/j.tafmec.2017.07.011 TAFMEC 1916
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
25 January 2017 22 April 2017 12 July 2017
Please cite this article as: Y. Huang, W. Zhou, Stress Intensity Factor for Clamped SENT Specimen Containing Non-Straight Crack Front and Side Grooves, Theoretical and Applied Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.tafmec.2017.07.011
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Stress Intensity Factor for Clamped SENT Specimen Containing NonStraight Crack Front and Side Grooves Yifan Huanga, b, * and Wenxing Zhouc a
Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By
Drive, Ottawa, Ontario K1S 5B6, Canada b
School of Engineering Science, Simon Fraser University, 8888 University Drive, Burnaby,
British Columbia V5A 1S6, Canada c
Department of Civil and Environmental Engineering, The University of Western Ontario, 1151
Richmond Street, London, Ontario N6A 5B9, Canada *: Corresponding author, Email: [email protected]
Abstract:
Three-dimensional (3D) finite element analyses (FEA) of clamped single-edge
notched tension (SENT) specimens are performed to evaluate the stress intensity factor (SIF). Both the average and local SIF values over the crack front are evaluated. Plane-sided (PS) specimens containing both straight and curved crack fronts and side-grooved (SG) specimens containing straight crack fronts are considered including six average crack lengths (i.e. aave/W = 0.2 to 0.7). PS specimens with three thickness-to-width ratios (i.e. B/W = 0.5, 1 and 2) are analyzed. The curved crack front is assumed to be bowed symmetrically and characterized by a power-law expression with a wide range of curvatures. For SG specimens, one relative thickness (B/W = 1) and ten depths of side groove are considered. The 3D SIF are compared with the present two-dimensional (2D) SIF. It is observed that the 2D SIF are sufficient as the difference between the 2D and 3D SIFs is in general within 10% if the crack front straightness requirements reported in the literature is satisfied.
An empirical 3D SIF solution is developed that can
potentially serve as a driving force for an assessment in actual structures that has similar crack geometries and loading configurations.
Keywords:
Stress intensity factor (SIF); Three-dimensional finite element analysis (3D
FEA); Single-edge notched tension (SENT); Crack front curvature; Side groove
Nomenclature a
crack length
aave
average crack length over the specimen thickness
az = 0, az = ±B/2
crack lengths at the mid-plane and surfaces of the specimen
B
gross thickness of the specimen
BN
net thickness of the specimen
CT
compact tension
CTOD
crack tip opening displacement
dsg
depth of the side groove on each side
E
elastic (Young’s) modulus
2D
F
normalized SIFs associated with K2D
F3Dloc, F3Dave
normalized SIFs associated with K3Dloc and K3Dave
H
daylight length (distance between grips)
J
J-integral, nonlinear energy release rate
Jave
average J evaluated over the crack front
Jloc(i)
the local J value at the ith layer
K2D
SIF in two-dimensional analyses
K3Dloc
local SIF in three-dimensional analyses
K3Dave
average SIF over the crack front in three-dimensional analyses
Mij, Nij
fitting coefficients of Eqs. (9) and (10)
m
non-dimensional factor in Eq. (13)
P
applied load
p
shape parameter used in Eq. (3)
PS
plane-sided
Qi
fitting coefficients of Eq. (11)
q
fitting coefficients of Eq. (12)
SENB
single-edge notched bend
SENT
single-edge notched tension
SG
side-grooved
SIF
stress intensity factor
u
applied displacement on clamped surface
W
the width of the specimen
non-dimensional factor in Eq. (14)
parameters used to characterize the crack front curvature
n
remote nominal tensile stress
Poisson’s ratio
CR, SG, *
modification factors to construct K3Dloc solutions
CR, SG, *
modification factors to construct K3Dave solutions
non-dimensional factor in Eq. (13)
ratio between the net and gross specimen thicknesses
1. Introduction The single-edge notched tension (SENT) specimen [1-6] is one of the commonly used specimens in fatigue and fracture toughness tests of metallic materials. Based on the notch configuration, the SENT specimens can be classified into two subsets. The first group of SENT specimens contain a surface notch (i.e. partly-through-thickness notch) and are generally used to evaluate the fatigue life of the materials [5, 6]. The notch is usually elliptical and requires the calculation of the stress intensity factor (SIF) along the three-dimensional (3D) crack front (e.g. [7, 8]).
The second group of SENT specimens are through-thickness notched and the
corresponding SIF can be determined from the simplified two-dimensional (2D) analyses [9-13]. Figure 1 schematically shows the configuration of a typical SENT specimen including the width (W), thickness (B), crack length (a) and the daylight distance (distance between the clamped surfaces, H) of the specimen. Cross sections with surface notch and through-thickness notch are also shown in the figure. The through-thickness-notched SENT specimens usually need to be fatigue pre-cracked to simulate natural cracks before the subsequent ductile tearing test. In the recent years, the use of the through-thickness-notched SENT specimen with clamped ends in the
experimental evaluation of ductile fracture toughness, such as the J-integral resistance (J-R) or crack tip opening displacement resistance (CTOD-R) curve, has gained significant research interests [14-19] especially in the oil and gas pipeline industry. Because the crack-tip stress and strain fields of such specimens are similar to those of the full-scale pipe containing surface cracks under longitudinal tension and/or internal pressure [10, 20], the fracture toughness determined from the SENT specimen can lead to less-conservative design and assessment of pipelines with respect to cracks compared with the typical standard single-edge notched bend (SENB) and compact tension (CT) specimens [21, 22]. The present paper is focused on the through-thickness notched SENT specimens with clamped ends, which will be referred to as the SENT specimen hereafter for simplicity. Testing on plane-sided (PS) through-thickness-notched specimens generally leads to a curved crack front and shear failure of the trailing edge [23] (see Fig. (1b)). These phenomena are caused by the difference in the states of stress along the crack front. At the region near the center of the PS specimen, the stress state is close to the plane-strain condition with higher SIF, which promotes a relatively fast crack growth. On the other hand, the stress state near the side surface is close to the plane-stress condition with lower SIF, therefore causing relatively slow crack growth and the shear lips near the free surface [23]. The SIFs over the crack front in this case are similar to the ones in surface-notched specimens. The side-grooved (SG) specimens are used in the J(CTOD)-R curve testing to achieve relatively straight crack fronts. As shown in Fig. 1(c), one groove is machined into each lateral side of the specimen to remove the low-SIF region. The net thickness between the side grooves (BN) can be calculated as BN = B – 2dsg with dsg being the depth of each side groove. The present authors [18] have previously investigated the impact of the BN/B ratio on the evaluated SIF and proposed a modification factor to account for this effect.
Numerical and analytical studies concerning the calculation of SIF for SENT specimens are reported in the literature [9-13, 24]. For example, Cravero and Ruggieri [9] and Shen et al. [10] respectively proposed the SIF solutions for SENT specimens with the crack length (a) over specimen width (W) ratio in the range of 0.1 ≤ a/W ≤ 0.7 and 0.05 ≤ a/W ≤ 0.95 based on 2D plane-strain finite element analyses (FEA). By using 2D plane-stress FEA, John and Rigling [11] developed a 6th-order polynomial SIF solution for SENT specimens with 0.1 ≤ a/W ≤ 0.9. Zhu [12, 13] proposed an analytical SIF solution for SENT specimens with 0 < a/W < 1. Jin and Wang [24] conducted 3D FEA to evaluate the local SIF distribution along the straight crack front for PS SENT specimens with a/W = 0.2, 0.4, 0.6 and 0.8 and the thickness-over-width ratio (B/W) equal to 0.1, 0.2, 0.5, 1, 2 and 4. It is noted that all of the existing SIF solutions [9-13] are based on the results from 2D analytical or numerical methods and 3D FEA of plane-sided SENT specimens with straight crack fronts [24]. There is a lack of a systematic study that validates the applicabilities of these solutions to the SENT specimens with curved crack fronts or/and side grooves. Such a study is valuable in that it provides guidance in the standardization of the testing procedure for the SENT specimen. Note that there are currently no crack front straightness criteria specified in any SENT test standards.
The objective of the present study was to carry out a systematic
investigation of the SIF for SENT specimens containing curved crack fronts or side grooves. We carried out 3D FEA of both PS and SG SENT specimens with wide ranges of the average crack length (aave) over specimen width ratio (aave/W), B/W, crack front curvature and side-groove depth to evaluate the local SIF distributed along the crack front. The focus of the present study is the clamped SENT specimen with a daylight-over-width ratio H/W = 10 because the crack-tip stress fields of such specimens correspond closely to those of the full-scale pipes containing
circumferential cracks [10]. The impacts of the crack front curvature and side-groove depth on the 3D SIF are studied. Both the local and average 3D SIF are compared with the 2D SIF and characterized by two parameters and ψ. These parameters are also used to construct the 3D SIF solutions from existing 2D SIF solutions. The 3D SIF solution can potentially serve as a driving force for an assessment in actual structures that has similar crack geometries and loading configurations. Furthermore, the 3D SIF solution is useful in an ongoing study that investigating the impacts of crack front curvature and depth of side groove on the constraint parameter (e.g. Tstress) along the crack front of the SENT specimen [24]. The rest of the paper is structured as follows. Section 2 presents a brief review of the existing 2D SIF solution for the SENT specimen. Section 3 describes the characterization of the curved crack front employed in this study. Section 4 describes the configurations of the FE models, material properties and computational procedures involved in the present study. The SIFs for specimens containing curved crack fronts and side grooves are evaluated and compared with the existing 2D SIF in Section 5, followed by conclusions in Section 6.
2. Review of 2D SIF Solutions for SENT Specimens The 2D SIF (K2D) solution for SENT Specimens can be expressed as
a K 2 D F 2 D n a W
(1)
where the remote nominal tensile stress σn = P/BW with P being the applied tensile force, and F2D is the normalized SIF associated with K2D. Recently, Zhu [12, 13] proposed the following analytical F2D solution for SENT specimens with H/W = 10
a F 2D W
2 3 a a a 1.9873 0.7422 11.188 45.82 W W W 4 5 6 a a a 100.49 120.03 51.448 W W W
W 1
2a a 1 W W
3 2
(2)
Eq. (2) is more representative than other F2D solutions reported in the literature [9-11] because it has a wider applicable range of a/W, i.e. 0 < a/W < 1.
3. Characteristics of Curved Crack Front Previous studies [25, 26] indicated that curved crack fronts in specimens made of homogeneous materials are typically bowed (i.e. crack length at the mid-plane greater than that at the free surface) and symmetric about the mid-plane. Therefore, such a curved crack front was assumed in the present study, and characterized by the following power-law function suggested by Nikishkov et al. [25]: p 2 z W a z aave 1 p 1 p B aave a z B /2 W 1 B aave B 0 a z dz
(a) (b) (c)
(3)
where z (–B/2 ≤ z ≤ B/2) is the coordinate in the specimen thickness direction; az is the crack length at the location of coordinate z; aave and az = ±B/2 denote the average crack length and crack lengths at the free surfaces of the specimen, respectively, and λ and p (p > 1) are parameters characterizing the curvature of the crack front. Note that λ = 0 corresponds to a straight crack front. By examining the fatigue pre-crack fronts of a series of plane-sided CT, SENB and SENT
specimens, Nikishkov et al. [25], and Yan and Zhou [26] showed that the parameter p in Eq. (3) can be adequately set to a fixed value of 3.0 for different curved crack fronts.
Detailed
evaluation process of λ based on the nine-point crack length measurement can be found in Ref. [17]
4. Numerical Analysis Linear-elastic analyses of 198 PS and 60 SG SE(T) models were performed to evaluate the stress intensity factors. Young's modulus (E) and Poisson's ratio (ν) were set to be 207 GPa and 0.3, respectively to simulate the typical structural steel. All the specimens included in this study have a width W = 20 mm and a daylight (H) of 10W. Six aave/W ratios (i.e. aave/W = 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7) were included in the analyses. For the PS models, three B/W ratios (B/W = 0.5, 1 and 2) and eleven λ values (λ= 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 and 0.1) were considered. A typical FE model with aave/W = 0.5, λ = 0.1 and B/W = 1 is schematically shown in Fig. 2(a) together with the fixation and loading conditions. The analyses matrix for the SG models consists of one B/W ratio (B/W = 1) and ten side groove depths (i.e. dsg/B = 3%, 4%, 5%, 6%, 7%, 7.5%, 8%, 9%, 10%, and 12.5%) with the corresponding thickness reduction ratios χ = BN/B = 0.94, 0.92, 0.9, 0.88, 0.86, 0.85, 0.84, 0.82, 0.8 and 0.75, respectively. The side groove is modeled as a U-shape notch with a fixed root radius rsg = 0.5 mm as recommended in ASTM E1820 [21]. Schematics of U-notched side grooves are shown in Figs. 2(d). Stationary cracks were assumed in all the analyses. Only a quarter of the specimen with appropriate constraints imposed on the remaining ligament was modelled due to symmetry. The FEA code ADINA® [27] was employed to analyze all the models. A typical quartersymmetric 3D model has 10 layers over the half net thickness (BN/2). For the SG model, the side groove ((B – BN)/2) was divided into 12 layers. The thickness of each layer was arranged such
that the corresponding mesh density increases from the mid-plane to the free surface (or root of the side groove) to capture the high stress gradients at these locations (see Figs. 2(b) and 2(d)). The 8-node 3D brick elements with 2×2×2 integration points were adopted in the analyses. A spider-web mesh around the crack tip was established with 45 concentric semicircles (i.e. rings) surrounding the crack tip (see Fig. 2(c)). The in-plane dimension of the elements in the 45th ring is about 2,000 times that of the element in the first ring (i.e. closest to the crack tip). The total number of elements is approximately 12,000 in a typical PS specimen, and 28,000 in a typical SG specimen. Uniform displacements (u) were applied on two lateral surfaces that are considered as the clamped surfaces each with a length of 2W (see Fig. 2(a)). The applied tension P was calculated as the total reactions of the nodes on the clamped surface. All the analyses results were obtained from one loading step that corresponds to an applied displacement u = 2 m. The SIFs along crack front were calculated from the J-integral evaluated in the FEA. The values of J in each layer along the thickness direction, i.e. the local J values, denoted by Jloc0, Jloc2,…Jloc10, were calculated using the virtual crack extension method [27]. Note that the local J value at the midplane equals Jloc0. Let zi denote the distance between the end of the ith layer and mid-plane (i.e. z = z0 = 0) as shown in Fig. 3. The weighted average J value over the entire crack front, Jave, is then calculated as follows based on the trapezoidal rule:
J ave
z1 z0 J loc 0 J loc1 z2 z1 J loc1 J loc 2 1 2 2 B z10 z9 z9 z8 J loc8 J loc 9 z10 z9 J loc 9 2 2 2 2
(4)
Note that the local J value at the specimen free surface, i.e. Jloc10, cannot be accurately evaluated from FEA. The two outermost semicircular rings surrounding the crack tip were used to define
the virtual shifts. Within the realm of linear elastic fracture mechanics, the local and average stress intensity factors in 3D analyses, K3Dloc and K3Dave respectively, can be calculated from the following equations:
3D J loc E K loc 1 2 K 3 D J ave E ave 1 2
(a) (5)
(b)
By analogy to Eq. (1), the corresponding 3D normalized SIFs, F3Dloc and F3Dave, are defined as 3D 3 D aave B z K loc F , , , , loc W W B n aave 3D F 3 D aave , B , , K ave ave W W n aave
(a) (6)
(b)
5. Results and Discussions 5.1 Stress intensity factor for plane-sided specimens containing curved crack fronts The values of F3Dloc and F3Dave corresponding to various aave/W, B/W and λ are estimated and compared with the F2D obtained from Eq. (2). For better comparisons, define the parameters
and ψ: 3D Floc 2D F 3D Fave F 2D
(7)
The difference (%) between F3Dloc (or F3Dave) and F2D can be calculated as 1 100% and
1 100% . The subscripts "CR" and "SG" denote the configurations of the specimens (i.e. with curved crack fronts or side grooves) for which and ψ are applicable. Figures 4 shows ψCR
values plotted against λ and aave/W for specimens with B/W = 0.5, 1 and 2. Figures 5 through 7 show variation of CR with 2|z|/B and λ for specimens with various aave/W and B/W ratios. Figures (4a) through (4c) suggest that for given aave/W and B/W artios, F3Dave as well as ψCR decreases as λ increases. On the other hand, for given B/W artios and λ value, F3Dave generally increases as aave/W decreases. The impact of λ on F3Dave decreases as B/W increases. The maximum difference between F3Dave and F2D is about 12%, 7% and 4% for specimens with B/W = 0.5, 1 and 2. Figures 5 through 7 suggest that for λ ≤ 0.01, as 2|z|/B increases from 0 to 0.8, the value of CR remains constant at around 1.00 and then decreases to 0.9 – 0.95 as 2|z|/B further increases to 0.98. For λ = 0.01 to 0.02, the distribution of CR along 2|z|/B is relatively uniform. On the other hand, for λ ≥ 0.03, CR generally increases as 2|z|/B increases. Higher values of λ lead to smaller CR at the mid thickness and larger CR at the specimen free surface. Similar observations regarding the F3Dloc – aave/W and F3Dave – 2|z|/B relationships are reported in the literature for CT [28-30] and SENB [31] specimens containing curved crack fronts. Huang and Zhou [17] proposed the following crack front straightness criteria for SENT specimens:
aave aave B Min 0.06 0.1 W , 0.03 , for W 0.5, 0.2 W 0.7 a a B Min 0.03 0.2 ave , 0.08 , for 1, 0.2 ave 0.7 W W W a a a B Min 0.06 0.5 ave , 0.3 ave , for 2, 0.2 ave 0.7 W W W W
(8)
It is reported that Eq. (8) ensures the differences in Jave (corresponding to loading level of 1.0 to 1.3 times of the limit load) and the elastic compliance between the specimens with curved and straight crack fronts to be within 5%. For λ values satisfying Eq. (8), Figures 4 through 7
suggest that the difference between F3Dave and F2D is also within 5%, and difference between F3Dloc and F2D is within 10% corresponding to 2|z|/B ≤ 0.9. Therefore, the 2D SIF obtained from Eq. (2) is considered adequate for SENT specimens with crack front curvature characterized by Eq. (8).
5.2 Stress intensity factor for side-grooved specimens containing straight crack fronts Figures 8 illustrates the ψSG values plotted against χ for specimens with B/W = 1 and aave/W = 0.2 to 0.7 while Figures 9(a) through 9(f) show the distributions of CR along the specimen thickness. Figure 8 suggests that ψSG decreasing from 1.17 – 1.23 to 1.0 as χ increases from 0.75 to 1.0. Figure 8 also suggests that aave/W has little impact on ψSG. It is observed from Figs. 9(a) through 9(f) that the value of SG increases as 2|z|/BN increases. Lower χ values lead to higher
SG values at both the specimen mid thickness and free surface. Similar observations are reported in the literature [31]. It is interesting to observe that along the crack front, the distribution of SG corresponding to 0.75 ≤ χ ≤ 0.94 is somewhat similar to that of CR corresponding to λ ≥ 0.03.
5.3 Three-dimensional stress intensity factor solution For given aave/W and B/W ratios, the following polynomial equations were proposed to express
CR as a function of (2|z|/BN) and λ, and SG as a functions of χ based on the values of CR and SG shown in Figs. (5) – (7) and (9): i a z 6 2 z ave B CR , , , i W W B i 0 BN 2 M ij j i j 0
(9)
i a 6 2 z z ave B SG , , , i W W BN i 0 BN 2 j i N ij j 0
(10)
Similarly, regression is performed on the values of ψCR shown in Fig. 4:
aave B 2 , , Qi i W W i 0
CR
(11)
where the fitting coefficients Mij, Nij and Qi are listed in Tables 1, 2 and 3, respectively. The fitting errors of Mij and Nij are in general within 1%. For specimens with aave/W = 0.2 and of λ = 0.1, the fitting errors of Mij increase up to 3% at the position very close to the free surface, i.e. 2|z|/B = 0.976. Using Qi leads to a very accurate prediction of ψCR such that the corresponding fitting errors are less than 0.1% for all the cases considered in the present study. On the other hand, as indicated in Fig. 8, ψSG is relatively insensitive to aave/W. The following equation for ψSG as a power-law function of χ is proposed based on the ψSG values presented in Fig. 8:
1 SG q q 0.6265
for 0.2
aave B 0.7, 1 W W
The maximum fitting error of Eq. (12) is about 2%.
(12)
Note that using Eq. (12) is more
advantageous than using q = 0.5, which is commonly adopted in the literature [3, 9-13, 21, 22] and yields a maximum fitting error of 6%. Note also that in a previous study by the present authors [18], q = 0.58 was proposed based on the modification with respect to F3Dave whereas Eq. (12) was developed based on the modification with respect to F2D. To investigate the adequacy of Eqs. (9) and (11) for specimens containing curved crack fronts characterized by the shape parameter p other than 3.0, additional FEA for specimens with aave/W
= 0.2 to 0.7, B/W = 1, and λ = 0.08 were performed. The value of p was set to 2.5 for the additional analyses. The rationale for selecting p = 2.5, as opposed to, say, p = 3.5, in the analysis is that for a given λ, decreaseing p leads to the central portion of the crack front, which has the largest contribution to K3Dave, to become more curved [17, 26]. The prediction errors (%) associated with the value of CR and ψCR predicted from Eqs. (9) and (11) for these specimens, e and eψ, are shown in Table 4. The results indicate that decreasing p from 3.0 to 2.5 but maintaining aave/W and λ has small impacts on ψCR (values of eψ are within 0.7% for p = 3.0 and 2.5). The maximum e increases from 1% to about 5% as p decreases from 3.0 to 2.5. Note that the maximum e for p = 2.5 occurs at the 2|z|/B = 0.9755. Within the region of 0 ≤ 2|z|/B ≤ 0.95, the values of e for p = 2.5 are less than 3.5%. Given the above, Eqs. (9) and (11) are also considered applicable for curved crack fronts with p = 2.5.
5.4 Stress intensity factor for side-grooved specimens containing curved crack fronts The above discussions are with respect to PS specimens containing curved crack fronts and SG specimens containing straight crack fronts. Additional FEA for SG specimens containing curved crack fronts was performed to investigate the SIFs for SG specimens containing curved crack fronts. The geometry of the analyzed specimens is characterized byaave/W = 0.2 to 0.7, B/W = 1, χ = 0.85, p = 3 and λ = 0.05. Note that the net thickness of the specimen, BN, was used to evaluate az and aave in Eq. (3) for the specimens. For simplicity, the side groove was modelled as a sharp V-notch with an opening angle of 45° as schematically shown in Fig. 10. Let * and ψ* denote the 3D modification factors for the local and average SIFs, respectively, for SG specimens containing curved crack fronts. Figures 11(a) through 11(f) show values of * as a function of 2|z|/BN. For comparasions, CR and SG estimated by Eqs. (9) and (10) were also
plotted in the figures. It is observed that Eq. (9) can accurately predict * for 2|z|/BN ≤ 0.5 but underestimates * when 2|z|/B further increases.
On the contrary, Eq. (10) in general
overestimates * and results in good predictions only when 2|z|/BN = 0.9 ~ 0.95. Figure 11 suggests that for a specimen with given λ and χ, CR and SG serve as the lower and upper bounds, respectively, of *. Figure 12 illustrates the ψ* values, as well as ψCR and ψSG estimated by Eqs. (11) and (12), plotted against aave/W. The figure suggests that values of ψ* are between the corresponding ψCR and ψSG. Two empirical expressions are developed to estimate * and ψ*, respectively as follows:
and
* CR 1 SG m 2 z BN
(13)
* CR
(14)
1
SG
where m and θ are non-negative dimensionless factors that depend on λ and χ. In the limit when ω = θ = 0 (i.e. z = 0 and m ≠ 0, or 2|z|/BN < 1 and m = ∞), * and ψ* coincide with CR and ψCR, respectively. Similarly, ω = θ = 1 (i.e. z ≠ 0 and m = 0, or 2|z|/BN = 1) yields * = SG and ψ* = ψSG. The potential dependence of m and θ on λ and χ will be investigated in a future study. Based on the numerical results, m = 3 and θ = 0.3 are proposed for specimens with χ = 0.85 and λ = 0.05. Equations (13) and (14) with the proposed m and θ are also plotted in Figs. (11) and (12) for comparison. The prediction errors of these equations are within 5% (for 2|z|/BN ≤ 0.95) and 1%, respectively. Finally, the full-field 3D SIF solution are presented as Eqs. (15a) and (15b)
3D 2D aave B z K loc W , W , B , , F n aave K 3 D aave , B , , F 2 D a ave n ave W W
(a) (15)
(b)
where
and
CR SG *
for 0 and 1
CR SG *
for 0 and 1
for 0 and 1
(16)
for 0 and 1
for 0 and 1
(17)
for 0 and 1
6. Summary and Conclusions Three-dimensional linear-elastic finite element analyses of clamped single-edge notched tension (SENT) specimens were carried out to evaluate both the local and average stress intensity factors along the crack front (F3Dloc and F3Dave). Plane-sided (PS) specimens containing both straight and curved crack fronts and side-grooved (SG) specimens containing straight crack fronts are considered including six average crack lengths (i.e. aave/W = 0.2 to 0.7). For PS specimens, symmetric bowed crack fronts characterized by a power-law expression were considered in the analysis. Three specimen thicknesses (B/W = 0.5, 1 and 2) and eleven crack front curvatures (λ = 0 to 0.1) were included in this study. For SG specimens, one relative thickness (B/W = 1) and ten depths of side groove (χ = 0.94 to 0.75) are considered. Key observations and findings are summarized in the following. 1) Two parameters, and ψ, are introduced to relate F2D and F3Dloc, and F2D and F3Dave, respectively. For plane-sided specimens containing curved crack fronts, F3Dave as well as ψCR
decreases as λ increases for given aave/W and B/W artios. For given B/W artios and λ value, F3Dave generally increases as aave/W decreases. The impact of λ on F3Dave decreases as B/W increases. The maximum difference between F3Dave and F2D is about 12%, 7% and 4% for specimens with B/W = 0.5, 1 and 2. For λ ≤ 0.01, as 2|z|/B increases from 0 to 0.8, the value of
CR remains constant at around 1.00 and then decreases to 0.9 – 0.95 as 2|z|/B further increases to 0.98. For λ = 0.01 to 0.02, the distribution of CR along 2|z|/B is relatively uniform. On the other hand, for λ ≥ 0.03, F3Dloc or CR generally increases as 2|z|/B increases.
Equation (2) is
considered adequate for SENT specimens with crack front straightness satisfying Eq. (8). 2) For side-grooved specimens containing straight fronts, F3Dave as well as ψSG decreases as χ increases from 0.75 to 1.0. The ψSG – χ relationship is not sensitive to aave/W for specimens with B/W = 1. Value of SG increases as 2|z|/BN increases, which is similar to the distribution of CR along specimen thickness corresponding to λ ≥ 0.03. 3) Based on the numerical results, empirical equations were developed to evaluate CR, SG, ψCR and ψSG, respectively. The errors of predictions of these equations are less than 2% in most cases.
Additional FEA for specimens containing curved crack fronts with p = 2.5 were
performed to validate the proposed CR and ψCR. The validation suggests that they are also applicable for curved crack fronts with p = 2.5. 4) The modification factors output from additional FEA of side-grooved specimens containing curved crack fronts, * and ψ*, were also compared with the proposed 3D modification factors. It is found that the proposed CR agrees well with * at the regions 2|z|/BN ≤ 0.5. The proposed
SG agrees well with * when 2|z|/BN = 0.9 ~ 0.95. The values of *(ψ*) basically lie between the corresponding CR(ψCR) and SG(ψSG).
A general expression of *(ψ*) is proposed as a
combination of CR(ψCR) and SG(ψSG). The empirical 3D SIF solution can potentially serve as a
driving force for an assessment in actual structures that has similar crack geometries and loading configurations. Future investigations will be focused on the dependences of m and θ on λ and χ as well as SIFs for SG specimens containing concave crack fronts, i.e. λ < 0.
Acknowledgments The study was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada. Constructive comments by the anonymous reviewer are much appreciated.
References [1] Srawley, J. E., Jones, M. H., and Gross, B. (1964). Experimental determination of the dependence of crack extension force on crack length for a single-edge-notch tension specimen (NO. NASA-TN-D-2396). National Aeronautics and Space Administration Cleveland OH Lewis Research Center. [2] Hollstein, T., Schmitt, W., and Blauel, J. G. (1983). Numerical analysis of ductile fracture experiments using single-edge notched tension specimens. Journal of Testing and Evaluation, 11(3), 174-181. [3] BS 8571. Method of Test for Determination of Fracture Toughness in Metallic Materials Using Single Edge Notched Tension (SENT) Specimens, British Standards Institution, London, 2014. [4] Det Norske Veritas, DNV-RP-F108. Fracture Control for Pipeline Installation Methods Introducing Cyclic Plastic Strain, DNV Recommended Practice, Norway, 2006
[5] Wu, X. R., Newman, J. C., Zhao, W., Swain, M. H., Ding, C. F., and Phillips, E. P. (1998). Small crack growth and fatigue life predictions for high-strength aluminium alloys: Part I: Experimental and fracture mechanics analysis. Fatigue and Fracture of Engineering Materials and Structures, 21(11), 1289-1306. [6] Newman, J. C., Wu, X. R., Swain, M. H., Zhao, W., Phillips, E. P., and Ding, C. F. (2000). Small-crack growth and fatigue life predictions for high-strength aluminium alloys. Part II: crack closure and fatigue analyses. Fatigue and Fracture of Engineering Materials and Structures, 23(1), 59-72. [7] Newman, J. C., Reuter, W. G., and Aveline, C. R. (2000). Stress and fracture analyses of semi-elliptical surface cracks. ASTM STP 1360: Fatigue and Fracture Mechanics, P.C. Harris and K.L. Jerina (Eds.), West Conshohocken, 30, 403–423. [8] Jin, Z., and Wang, X. (2013). Weight functions for the determination of stress intensity factor and T‐ stress for semi‐ elliptical cracks in finite thickness plate. Fatigue and Fracture of Engineering Materials and Structures, 36(10), 1051-1066. [9] Cravero, S., and Ruggieri, C. (2007). Estimation procedure of J-resistance curves for SE(T) fracture specimens using unloading compliance. Engineering Fracture Mechanics, 74(17), 2735-2757. [10] Shen, G., Gianetto, J. A., Tyson, W. R. (2008). Report 2008-18(TR): Development of procedure for low-constraint toughness testing using a single-specimen technique. Ottawa, Canada: CANMET-MTL. [11] John, R., and Rigling, B. (1998). Effect of height to width ratio on K and CMOD solutions for a single edge cracked geometry with clamped ends. Engineering Fracture Mechanics, 60(2), 147-156.
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J-R
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σn a (z)
Uncracked ligament
A
A Fatigue crack growth H
A W
A
B
Machined notch a
σn
-B/2
0
B/2
z
(a) Surface-notched specimen a (z)
a (z)
Side groove Uncracked ligament Ductile crack growth
Fatigue pre-crack front W
A
A Fatigue crack growth
Machined notch
aave
A W
A Fatigue crack growth
aave
Machined notch
BN/2 -B/2
0
B/2
(b) Through-thickness-notched plane-sided specimen
Figure 1
z
-B/2
0
(c) Through-thickness-notched side-grooved specimen
Crosss sections of typical SENT specimens.
B/2
z
Clamped surfaces
u
see (b)
see (c)
x
aave
y
(a)
Configuration of a typical FE model with aave/W = 0.5, λ = 0.1 and B/W = 1
W
z
W
az = 0
az = -B/2
dsg
B/2 rsg
(c) Mesh around the sharp crack tip
(b) Front view of the FE model
Figure 2
(d) Configuration of the U-shape side groove
Configuration and meshing of a typical finite element model.
Jloc9
J
Jloc8 Jloc1
Jloc0
Jloc2
Jave
Jloc7
Jloc3 Jloc4
z0
z1
z2
z3
z4
Jloc5
z5
Jloc6
z6 z7 z8 z9 z10 = B/2
Figure 3
Schematic of the calculation of the weighted average J along the crack front.
z
1.04
aave/W increases
1.02 1.04
1.00
1.04
1.04
1.02
1.02
1.00
1.00
0.98
0.98
1.02 1.00
0.98
0.98 CR
0.96
0.96
0.94
0.94 0.92
0.92 0.90
0.90
0.88 0.88 0.86 0.00
0.86 0.00 0.02
CR a ave /W = 0.2 a/W = 0.2 a ave /W = 0.3 a/W = 0.3 a ave /W = 0.4 a/W = 0.4 a ave /W = 0.5 a/W = 0.4 a ave /W = 0.6 a/W = 0.6 a ave /W = 0.7 a/W = 0.7
0.02 0.04
0.04 0.08 0.06 0.10 0.08 0.06
0.10
0.96
CR
0.96
0.94
0.94
0.92
0.92
0.90
0.90
0.88
0.88
0.86 0.00
0.02
0.04
0.06
0.08
0.10
0.86 0.00
0.02
(a) B/W = 0.5
Figure 4
(b) B/W = 1
Variation of ψCR with λ for PS specimens with various aave/W and B/W ratios.
0.04
0.06
(c) B/W = 2
0.08
0.10
1.9
1.9 2.0 1.8 1.9
1.7
1.8
1.6
1.7
1.5
1.6
CR
1.4 1.5
1.3 1.4 1.3 1.2 1.2 1.1
(a) aave/W = 0.2
1.9
(b) aave/W = 0.3
1.8
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
CR
1.1
1.0
1.0
0.9
(c) aave/W = 0.4
1.8
1.7
1.7
1.6
1.6
1.5
1.5
1.4
CR
1.3
1.4
1.3
1.2
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.9
0.8
λ increases
0.8
0.7 0.0 0.0
0.2
0.2
0.4
0.6
0.4
0.8
0.6
1.0
0.8
0.7
0.7 0.0
1.0
0.2
2z /B
0.6
0.8
1.0
0.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.7
1.7
1.6
1.6
1.6
1.5
1.5
1.5
1.4
1.4
1.4
CR
1.3 1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8
0.8
0.8
0.2
0.4
0.6
2z /B Figure 5
0.8
1.0
0.6
0.8
1.0
0.7
0.7 0.0
1.0
1.3
1.2
0.7
0.8
(f) aave/W = 0.7
1.8
1.7
CR
0.6
1.9
1.8
1.3
0.4
2z /B
1.9
1.8
0.2
2z /B
1.9
CR
0.4
0.0
0.2
0.4
2z /B
0.6
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 0.5 and various aave/W ratios.
1.5
(a) aave/W = 0.2
1.9 1.4
(b) aave/W = 0.3
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
1.8 1.7 1.3 1.6
CR
1.5
1.5
2.0
1.5 1.2 1.4 1.3 1.1 1.2
CR
1.1 1.0
(c) aave/W = 0.4
1.4
1.4
1.3
1.3
1.2
CR
1.2
1.1
1.1
1.0
1.0
0.9
0.9
1.0 0.9 0.9
λ 0.8 increases 1.0
0.8
0.8
0.0
0.0
0.2
0.2
0.4
0.6
0.4
0.6
0.8
0.8
0.8
1.0
0.0
0.2
2z /B
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6 1.4
1.3
1.3
1.3
1.2
1.2
1.2
CR
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8
0.2
0.4
0.6
2z /B Figure 6
0.8
1.0
0.8
1.0
0.6
0.8
1.0
(f) aave/W = 0.7
1.4
0.8
0.6
1.5
1.4
CR
0.4
2z /B
1.5
0.0
0.2
2z /B
1.5
CR
0.4
0.8
0.0
0.2
0.4
0.6
2z /B
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 1 and various aave/W ratios.
2.0
1.30
1.9
(a) aave/W = 0.2
1.25
1.8 1.7
1.20
1.6
1.15
1.5 1.4
1.10
CR
1.3
1.05
1.2
1.00
1.1
CR
0.9
0.90
0.0
0.2
λ increases
0.4
1.20
1.20
1.15
1.15
1.10
1.10
CR
1.05
1.00
0.95
0.95 0.90 0.8
1.0
0.85
0.85
0.80
0.80
0.80 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
2z /B
CR
0.4
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.20
1.20
1.15
1.15
1.15
1.10
1.10
1.10
CR
1.05 1.00
1.00
0.95
0.95
0.95
0.90
0.90
0.90
0.85
0.85
0.85
0.80
0.2
0.4
0.6
2z /B Figure 7
0.8
1.0
1.0
0.6
0.8
1.0
1.05
1.00
0.80
0.8
(f) aave/W = 0.7
1.25
1.20
CR
0.6
1.30
1.25
1.05
0.4
2z /B
1.30
0.0
0.2
2z /B
1.30 1.25
1.05
1.00
0.6
(c) aave/W = 0.4
1.25
0.90
0.8
0.85
(b) aave/W = 0.3
1.25
1.0
0.95
1.30
1.30
λ=0 λ = 0.01 λ = 0.02 λ = 0.03 λ = 0.04 λ = 0.05 λ = 0.06 λ = 0.07 λ = 0.08 λ = 0.09 λ = 0.10
0.80
0.0
0.2
0.4
0.6
2z /B
0.8
1.0
0.0
0.2
0.4
2z /B
Variation of CR with 2|z|/B and λ for PS specimens with B/W = 2 and various aave/W ratios.
1.00 0.98 0.96
1.25
0.94 0.92
1.20
0.90 0.88
1.15
a ave /W = 0.2 a/W = 0.2 a ave /W = 0.3 a/W = 0.3 a ave /W = 0.4 a/W = 0.4 a ave /W = 0.5 a/W = 0.4 a ave /W = 0.6 a/W = 0.6 a ave /W = 0.7 a/W = 0.7
0.86
SG 0.001.10 0.02
0.04
0.06
0.08
0.10
aave/W increases
1.05
1.00
Eq. (11) 0.95 1.00
0.95
0.90
0.85
0.80
0.75
Figure 8
Variation of ψSG with χ for SG specimens with B/W = 1 and various aave/W ratios.
2.0 1.9
1.8
1.8
1.7
1.7
1.6
1.6
SG
2.0
2.0
2.0 1.9
1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2
(a) aave/W = 0.2 χ =1 χ = 0.94 χ = 0.92 χ = 0.90 χ = 0.88 χ = 0.86 χ = 0.85 χ = 0.84 χ = 0.82 χ = 0.80 χ = 0.75
(b) aave/W = 0.3
1.9
χ decreases
SG
1.1
1.1
1.0
1.0
(c) aave/W = 0.4
1.9
1.8
1.8
1.7
1.7
1.6
1.6
1.5
SG
1.4
1.5
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.9
0.9
0.8
0.8 0.0 0.0
0.2
0.2
0.4
0.6
0.8
0.4
0.6
1.0
0.8
0.8
0.8 1.0
0.0
0.2
2 z / BN
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.8
1.8
1.7
1.7
1.7
1.6
1.6
1.6
1.4
1.5
SG
1.4
1.3
1.2
1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9
0.9
0.9
0.8 0.2
0.4
0.6
2 z / BN Figure 9
0.8
1.0
0.6
0.8
1.0
1.4
1.3
0.0
1.0
1.5
1.3
0.8
0.8
(f) aave/W = 0.7
1.9
1.8
SG
0.6
2.0
1.9
1.5
0.4
2 z / BN
2.0
1.9
0.2
2 z / BN
2.0
SG
0.4
0.8 0.0
0.2
0.4
2 z / BN
0.6
0.8
1.0
0.0
0.2
0.4
2 z / BN
Variation of SG with 2|z|/BN and χ for SG specimens with B/W = 1 and various aave/W ratios.
22.5◦
Figure 10
Configuration of the sharp V-shape side groove.
1.7
1.7
1.7
1.7 1.6
(a) aave/W = 0.2
(b) aave/W = 0.3
* obtained from FEA FEA
1.6 1.5
(c) aave/W = 0.4
1.6
1.6
1.5
1.5
1.4
1.4
Eq. (9) 1.5 1.4
Eq. (10)
Eq. (13) with m = 3
1.4 1.3
1.3
1.3 1.2
1.2
1.2
1.2 1.1
1.1
1.1
1.1 1.0
1.0
1.0
1.0 0.9 0.0 0.9 0.0
0.9
0.9 0.2
0.4
0.6
0.8
0.2
0.4
2 z / BN0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
0.0
1.0
(d) aave/W = 0.5
(e) aave/W = 0.6
1.5
1.5
1.4
1.4
1.4
1.3 1.2
1.2
1.1
1.1
1.1
1.0
1.0
1.0
0.9 0.0
0.2
0.4
0.6
2 z / BN Figure 11
0.8
1.0
1.0
0.6
0.8
1.0
1.3
1.2
0.9
0.8
(f) aave/W = 0.7
1.6
1.5
0.6
1.7
1.6
1.3
0.4
2 z / BN
1.7
1.6
0.2
2 z / BN
1.0
1.7
1.3
0.9 0.0
0.2
0.4
2 z / BN
0.6
0.8
1.0
0.0
0.2
0.4
2 z / BN
Variation of with 2|z|/BN for SG specimens with B/W = 1, χ = 0.85, λ = 0.05 and various aave/W ratios.
1.12 1.12 1.10 1.08 1.08 1.06
1.06 1.04
1.02 1.04 1.00 1.02 0.98 * obtained from FEA ψ FEA Eq. (11) Eq. (12) Eq. (14) with θ = 0.3
0.96 1.00 0.94 0.98 0.92
0.96 0.90 0.1
0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
0.8 0.8
aave/W
Figure 12
Variation of ψ with aave/W for SG specimens with B/W = 1, χ = 0.85, λ = 0.05 and various ratios.
Table 1 Coeffieients Mij in Eq. (9). aave/W
B/W
j 2
0.5
1 0
1
2
0.3
0.5
6
5
4 3 -710.34 1512.26 485.33 1268.63 1885.77 6445.75 5605.03 3536.43 -16.82 50.24 -57.98 31.98
2
1
0
-69.87
4.87
-4.72
915.43
-89.47
-1.12
-8.35
0.81
1.04
41.91
-2.08
-39.87
-0.16
0.68
1.02
28.28
-0.80
0
2865.74 1715.24 -435.15 1013.59 3191.89 844.47 2878.21 407.42 2505.80 1577.96 -14.26 42.52 -49.00 27.00 -7.03
2
-638.80 1830.49
1
384.77
0
-12.69
2
1683.20
2 0.2
i
1
1 0
1133.47 -291.08 2073.04
1307.82 1139.26 37.74 -43.43
-717.96
184.42
-18.12
0.35
23.94
-6.22
0.61
1.00
6938.20 1097.78 -108.03 5624.06 4023.10 1911.23 6535.96 929.38 -90.85 5681.83 3586.79 -17.46 52.08 -60.07 33.12 -8.66 0.84
-4.96 -1.35 1.04
1
2
2
871.82
1
793.58
0
-14.74
2
-23.66
1
379.16
0
-13.99
0.4
1
2
32.44 -12.99 1291.90 1124.84 41.64 -47.95
528.01
-52.19
-2.40
380.36
-37.17
-0.47
-7.24
0.70
1.03
-3.92
4.52
-0.55
-1.03
-707.75
182.28
-17.86
0.00
26.41
-6.86
0.67
1.01
485.49
-46.83
-5.17
974.25
-95.23
-1.48
-8.90
0.86
1.04
0
3036.01 2442.92 1764.64 1995.76 6839.22 5939.98 3756.09 -17.97 53.59 -61.78 34.05
2
1901.63
1
765.71
0
-15.11
2
438.49
1
361.19
0
-14.87
2 0.5
3409.96 2808.60 1958.09 2693.80 2349.67 1473.35 43.86 -50.48 27.77
1
698.48
6977.96 1043.19 -102.30 5893.69 3931.71 2596.16 366.35 -35.79 2265.59 1418.67 44.90 -51.63 28.38 -7.40 0.72 1579.72 1346.26 1228.66 1070.62 44.19 -50.81
-2.61 -0.66 1.03
-884.74
234.04
-22.88
-1.18
-671.92
173.28
-16.96
-0.23
27.95
-7.26
0.71
1.01
continued
aave/W
B/W
j
0
6 5 4 3 2 2891.99 1867.97 -485.06 1002.22 3356.88 1975.59 6759.97 961.79 5875.28 3709.64 -17.42 51.83 -59.65 32.82 -8.57
2
2374.90
1
762.66
0
-15.51
2
764.82
1
347.31
0
-15.66
2 0.5
0.5
1
2
1
2 0.6
0.5
i
1 0
1
0
49.24
-5.70
-93.96
-1.56
0.83
1.03
8550.18 1263.61 -123.49 7280.10 4787.59 2589.49 365.90 -35.74 2258.22 1415.40 46.03 -52.90 29.07 -7.58 0.74 2704.77 2320.69 1506.61 1180.17 -644.51 1028.87 46.45 -53.33 29.30
-3.14 -0.76 1.02
396.16
-38.64
-1.69
166.37
-16.27
-0.36
-7.62
0.74
1.01
119.80
-5.68
-88.07
-1.73
0.78
1.02
6903.99 4564.25 2288.81 8133.71 1201.28 1867.16 6362.52 902.18 5540.24 3484.74 -16.55 49.12 -56.41 30.98 -8.08
1
2
2
2393.39
1
772.42
0
-15.96
2
1022.60
1
337.85
0
-16.61
0.7
1
2
3590.39 3089.56 1995.18 1147.97 -626.30 1000.78 49.20 -56.41 30.96
1.01 -1.68
161.84
-15.82
-0.56
-8.04
0.78
1.00
157.45
-6.01
-84.96
-1.90
0.77
1.02
2
2333.32
1
796.09
0
-16.70
2
1282.92
1 0
333.57 -17.97
8175.96 1187.28 -115.19 7047.35 4535.30 2714.67 384.91 -37.60 2362.77 1485.38 49.47 -56.78 31.18 -8.12 0.79 4490.15 3868.63 2493.29 -988.59 1134.39 -618.26 53.25 -61.02 33.48
-0.94
-50.97
0
1
-3.17
523.32
9054.13 6004.05 2977.08 10691.8 1583.09 1811.36 6155.68 870.92 5366.77 3366.79 -16.41 48.62 -55.77 30.60 -7.97
2 0.5
8494.96 1244.02 -121.15 7279.49 4733.24 2628.03 371.97 -36.34 2289.67 1437.21 47.34 -54.37 29.87 -7.79 0.76
-3.45 -1.13 1.01
653.53
-63.65
-1.88
159.90 -8.69
-15.62 0.84
-0.75 1.00
Table 2 Coeffieients Nij in Eq. (10). aave/W
j
0
4 2 -402.11 1143.00 686.40 -174.50 1275.85 650.10 2089.49 287.20 1862.20 1129.43 -199.05 573.91 -647.50 352.08 -89.15
2
-449.94 1297.32
2 0.2
0.3
0.4
1
1464.39
795.23
-204.07
0 2
-503.72 1464.28
1
2 1 0 0.6
5
2470.24 346.48 2176.20 1348.01 -249.78 733.69 -839.67 461.72 -118.83
1
0
0.5
6
i 3
2
748.59
1662.46
906.86
-233.78
2907.06 411.89 2545.29 1593.80 -315.86 937.14 596.64 -154.72 1079.85 867.98
-528.33 1537.50
1745.28
951.27
-245.11
3163.39 448.24 2768.27 1733.90 -365.75 1087.71 692.68 -179.82 1254.14 942.16
-488.16 1411.66
-
865.16
-222.17
1
0
16.82
1.64
-27.81
-3.21
8.71
2.59
19.74
1.64
-33.64
-3.27
11.62
2.66
22.66
1.61
-40.06
-3.24
15.13
2.66
23.73
1.52
-43.54
-3.09
17.56
2.60
21.44
1.38
1594.23 1 0
2 0.7
1 0
3017.22 425.39 2649.42 1648.90 -369.64 1096.27 694.79 -180.17 1260.83 905.35
-421.90 1205.97
1349.27
726.46
-185.35
2639.34 368.27 2333.57 1434.09 -335.04 986.84 619.07 -159.95 1128.67 804.21
-41.24
-2.85
17.56
2.48
17.80
1.19
-35.57
-2.50
15.54
2.31
Table 3 Coeffieients Qi in Eq. (11).
2
i 1
0
0.2
0.5 1 2
-5.15 -2.43 -0.87
-0.55 -0.15 -0.01
1.03 1.02 1.01
0.3
0.5 1 2
-5.01 -2.64 -1.15
-0.58 -0.19 -0.05
1.03 1.02 1.01
0.4
0.5 1 2
-4.97 -2.72 -1.28
-0.60 -0.23 -0.09
1.03 1.02 1.01
0.5
0.5 1 2
-5.31 -3.15 -1.75
-0.58 -0.22 -0.09
1.02 1.01 1.00
0.6
0.5 1 2
-5.14 -3.09 -1.71
-0.63 -0.29 -0.16
1.01 1.00 1.00
0.7
0.5 1 2
-5.28 -3.29 -1.89
-0.66 -0.34 -0.22
1.00 1.00 1.00
aave/W
B/W
1
Table 4. Prediction errors (%) associated with the value of CR and ψCR predicted from Eqs. (9) and (11) for PS specimens with aave/W = 0.2 to 0.7, B/W = 1, and λ = 0.08 aave/W 2|z|/B
e
e
0 0.2447 0.4341 0.5808 0.6944 0.7823 0.8504 0.9031 0.9439 0.9755
0.2 p = 2.5 p=3
0.3 p = 2.5 p=3
0.4 p = 2.5 p=3
0.5 p = 2.5 p=3
0.6 p = 2.5 p=3
0.7 p = 2.5
p=3
0.21
0.01
0.32
0.01
0.38
0.01
0.43
0.02
0.49
0.01
0.69
0.01
1.04 0.49 0.55 2.42 0.62 1.11 1.51 0.43 2.10 3.11
0.00 0.02 0.16 0.42 0.46 0.18 0.52 0.31 0.80 0.51
0.96 0.34 0.78 2.87 0.80 1.22 1.57 0.22 2.77 3.97
0.01 0.02 0.20 0.47 0.55 0.22 0.57 0.32 0.92 0.49
0.93 0.25 0.93 3.14 0.90 1.28 1.60 0.10 3.14 4.47
0.01 0.02 0.22 0.52 0.59 0.24 0.61 0.35 0.97 0.52
0.91 0.18 1.07 3.36 1.02 1.35 1.64 0.00 3.41 4.85
0.02 0.01 0.24 0.54 0.62 0.25 0.64 0.34 0.99 0.53
0.90 0.11 1.24 3.58 1.17 1.45 1.64 0.13 3.58 5.13
0.01 0.02 0.23 0.55 0.62 0.23 0.64 0.32 0.94 0.54
0.96 0.11 1.27 3.38 2.25 2.22 2.16 0.24 3.24 4.90
0.02 0.02 0.24 0.55 0.63 0.22 0.64 0.29 0.88 0.54
1
Highlights
Evaluate the SIF for SENT specimen containing curved crack front and side grooves
Validates the 2D SIF for specimens with curved crack fronts and side grooves.
Develop the modification factors to construct the 3D SIF solution for SENT specimen