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Study on uniform parameters characterizing the crack-tip constraint effect of fracture toughness
Engineering Fracture Mechanics 222 (2019) 106706
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Study on uniform parameters characterizing the crack-tip constraint effect of fracture toughness ⁎
Guangwei He, Chen Bao , Lixun Cai
T
⁎
Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Crack-tip constraint Plastic zone Fracture toughness Unified constraint parameter
The plastic zone of crack-tip is connected with the constraint level inherently. Based on this, a unified crack-tip constraint parameter Cp was proposed. And through finite element analysis (FEA), the Cp can be recognized as a characteristic parameter in the description of equivalent stress fields. Moreover, the ductile material SA-508 steel was used to test the effectiveness of Cp as a uniform parameter, three different configurations (CT, SENB, Mini-SENB) of specimens with different crack lengths and thicknesses were employed to distinguish the effects of constraints on ductile fracture toughness. The result showed that clear linear relationship between the Cp and J0.2BL is shown for all specimens with different dimensions, which can be used to determine constraint-dependent fracture toughness for the specimens or flaw components of SA-508 steel, regardless of in-plane or out-of-plane constraint or both.
1. Introduction The constraints of crack-tip have an important influence on the evaluation of J-resistance curves. In current fracture toughness test standards ASTM E1820-15a [1], IS0 12135 [2] and GB/T 21143 [3], extremely high crack-tip constraint specimens, such as single-edge notched bend (SENB) and compact tension (CT) specimens with deep crack and enough thickness, are employed to obtain a lower bound value of material’s fracture toughness [4–6]. However, most of non-standard specimens or real cracked structures have low crack-tip constraint. The direct application of high crack-tip constraint fracture toughness to the safety design of the structures will result in lower allowable load, which may cause a conservative result. Therefore, the constraint effect on the fracture toughness must be corrected so that the fracture toughness determined in laboratory can be accurately applied to real cracked structures. There is a popular belief that the constraint of crack-tip can be divided into in-plane constraint and out-of-plane constraint. The inplane constraint is associated with crack size, geometry of specimen and type of loading. The out-of-plane constraint is associated with thickness. The in-plane and out-of-plane constraint effects on the fracture behavior of material are different in the specimens and structures with different geometries and loading configurations. Therefore, it is important to clarify the crack-tip constraint effect and size-dependence fracture toughness induced by the finite geometry of specimens. During the past decades, different constraint parameters have been proposed, such as T-stress [7], Q [8,9], A2, [10,11], Tz [12–14], h (or b) [15,16], and A [17,18], etc. However, the parameters mentioned previously are not equally sensitive to both in-plane and out-of-plane constraints which exist in the engineering structures simultaneously. For conducting accurate structural integrity assessment incorporating the constraint effects, the unified constraint parameter is desired. The plastic zone around the crack-tip line governs the initiation and propagation
⁎
Corresponding authors. E-mail addresses: [email protected] (G. He), [email protected] (C. Bao), [email protected] (L. Cai).
https://doi.org/10.1016/j.engfracmech.2019.106706 Received 11 June 2019; Received in revised form 30 September 2019; Accepted 30 September 2019 Available online 01 October 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Fracture Mechanics 222 (2019) 106706
G. He, et al.
r, θ SN Uεp V Vp W Wp α σ0 σeq ε0 δij σij ̃ (θ , N ) σeq̃ (θ , N )
Nomenclature a b B Cp E f IN J0.2BL k L N P rp
crack length remaining ligament specimen thickness unified constraint parameter Young’s modulus proportional coefficient an integration constant the intersection of J-R curve and 0.2 mm offset blunt lines constant value for a given specimen characteristic lengths (commonly set to be 1 mm) strain hardening exponent applied load radius of plastic zone
Φ
polar coordinates centered at the crack-tip dimensionless constant 3D plastic strain energy displacement plastic displacement specimen width plastic work done by external load strain hardening coefficient reference stress, and usually equal to yield stress equivalent stress reference strain, and ε0 = σ0/E Kronecker delta dimensionless stress function of crack-tip dimensionless equivalent stress function of cracktip 2D plastic strain energy
of cracks, a unified parameter based on the crack-tip plastic zone may be a promising approach to quantify the crack-tip constraint effects. Based on this idea, Anderson and Dodds [19] proposed a unified parameter which is the normalized area (volume per unit thickness) of a zone surrounded by the contour with σ1/σy = constant (σ1 is the maximum principal stress and σy is the yield strength of the material). Mostafavi et al. [20] modified the A-D method and redefines a uniform parameter φ denoted as the area of plastic region at the onset of fracture (AC) normalized by the reference area of plastic region of a standard plane strain specimen at the onset of fracture (Assy). However, for material with extremely high fracture toughness, the plastic deformation zone will expand to the specimen’s surface, it is impossible to accurately calculate the area of crack-tip plastic zone because the intersection of the plastic zone in the load region and that around the crack-tip region. Yang et al. [21] made an improvement and proposed a modified parameter Ap, which is the area surrounded by the isoline of certain equivalent plastic strain (εp) ahead of the crack-tip normalized by the reference area of the isoline with same εp in a standard specimen with high constraint level. However, the studies by Yang et al. are mainly based on finite element method and lacked of effective experimental verification. In this paper, a new constraint parameter Cp was introduced, which can describe the equivalent stress distributions ahead of crack-tip well. In order to verify its capability of characterize the fracture toughness, specimens with different constraints levels include different crack length, thickness, geometry of specimen and type of loading were used in fracture toughness tests.
2. Theoretical background As mentioned above, the plastic region of crack-tip is closely related to the constraint level for ductile fracture because of the contribution of plasticity to the ductile fracture mechanism of void nucleation, growth and coalescence. Actually, for a cracked specimen subjected to external load, its plastic deformation is mainly concentrated at the crack-tip, the elastic deformation is almost negligible. In other words, for a cracked specimen, the plastic work done Wp of external load is approximately equal to the plastic strain energy Uεp around crack-tip. This idea will guide the following research to propose a unified crack-tip constraint parameter.
Fig. 1. The schematic of the plastic work Wp. 2
Engineering Fracture Mechanics 222 (2019) 106706
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2.1. Expression of plastic work done Wp by external load The study focuses on mode-I cracked body of elastic-plastic materials. In the plot of load versus displacement, the Wp is the area as shown in Fig. 1, and it can be calculated from the following equation as recommended in ASTM E1820-15a [1]:
Wp (i) = Wp (i − 1) + [P(i) + P(i − 1) ][Vp (i) − Vp (i − 1) ]/2
(1)
where Wp(i) and Wp(i-1) are the plastic work done at points i and i-1. P and Vp are the corresponding load and plastic displacement. 2.2. Expression of 3D plastic strain energy Uεp A generally accepted that the HRR crack-tip asymptotic solution for mode-I cracked body made by Hutchinson [22] and Rice and Rosengren [27], is the theoretical foundation of modern elastic-plastic fracture mechanics and it can be given as, 1
σij
N+1 J ⎞ σ~ij (θ , N ) =⎛ σ0 αε σ I r ⎝ 0 0 N ⎠ ⎜
⎟
(2)
where, α is strain hardening coefficient, N is strain hardening exponent. σ 0 is reference stress, and usually equal to yield stress, ε0 = σ0/E. IN is an integration constant and σij (θ , N ) are dimensionless stress function of crack tip. r represents the distance from the crack-tip along the crack surface. Accordingly, the equivalent stress of crack-tip can be expressed as 1
σeq
N+1 J ⎞ σ~eq (θ , N ) =⎛ σ0 ⎝ αε0 σ0 IN r ⎠ ⎜
⎟
(3)
where, σeq (θ , N ) is only related to θ for a certain material. At the boundary of plastic zone, i.e. r = rp, the equivalent stress reaches to yield stress,
σeq = σ0
(4)
combining Eqs. (3) and (4), we can get that − 1 N +1
J ⎞ σ~eq (θ , N ) = ⎛⎜ ⎟ ⎝ αε0 σ0 IN rp ⎠
(5)
then, substituting Eq. (5) into Eq. (3), the equivalent stress can now be expressed by the size of plastic zone rp, 1
rp N + 1 σeq = σ0 ⎛ ⎞ ⎝r⎠
(6)
it should be pointed out that rp is related to θ because σ~eq (θ , N ) in Eq. (5) changes with the variation of θ. But if we suppose a circular shape of the plastic zone around crack-tip by ignoring the difference of rp caused by different θ, the rp can be replaced by an equivalent radius of plastic zone Cp, as shown in Fig. 2. The plastic strain energy Φ in the plastic zone around crack-tip[28] can be expressed as
S Φ = ⎛ N ⎞ JCp ⎝ IN ⎠ ⎜
⎟
(7)
where the dimensionless constant SN is
Fig. 2. The schematic showing of the plastic zone around crack-tip. 3
Engineering Fracture Mechanics 222 (2019) 106706
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SN =
N N+1
∫π
π
σ~eqN + 1 dθ
(8)
combining Eq. (5) and Eq. (8), Eq. (7) can be rewritten as
Φ = 2π
N ασ0 ε0 Cp2 N+1
(9)
It should be noted that above derivations are restricted in the plane condition, but the deformation of an elastic-plastic body is three-dimensional. It is widely accepted to ignore the variation in the thickness direction (usually choose the middle plane of the thickness as the representative) [20,21,23–26] in the research of 3D crack-tip constraint. If the plastic zone size of the plane with different thickness is the same as that of the mid-plane, then we can obtain that
Uεp = f ΦB
(10)
where, f is the proportional coefficient to quantify the difference between the assumed volume of the plastic zone and the real one. A trial-and-error optimization approach was employed to determine the parameter f. This method was based on a large number of finite element calculations, and the value of f was determined by comparing the difference between the assumed plastic zone volume and the actual plastic zone volume. It turned out that f is only related to the initial remaining ligament of the specimen, and doesn’t change in the calculation of Cp, and can be assumed to b/L for all the specimen geometries considered in this work. b is initial remaining ligament. L is characteristic length parameter which can be equal to 1 mm. 2.3. Unified constraint parameter Cp Based on the above derivation, the expressions of Wp and Uεp are obtained, respectively. By establishing the equation Wp = Uεp, the equivalent radius of plastic zone Cp can be obtained as 1/2
Wp ⎛ ⎞ Cp = ⎜ N 2π N + 1 α 0 σ0 ε0 fB ⎟ ⎝ ⎠
(11)
The commercial finite element code, ANSYS 14.5, was used to calculate the equivalent stress distributions ahead of crack-tip. The typical models and meshes for the SENB and CT specimens are shown in Fig. 3. The width W of SENB and CT were 40 mm, 50 mm. The thickness B of SENB and CT were 20 mm, 25 mm. Consider the symmetry of loading and geometry of SENB and CT specimens, only 1/4th of the two specimens were modeled, and the symmetry boundary conditions were applied in the un-cracked ligament. Along the thickness direction (z-axis), the identical planar mesh was repeated from the symmetry plane (mid-plane, z/B = 0) to the free surface (z/B = 0.5). The loading was directly applied as a pressure to upper surface of the load hole for the CT specimen. While for SENB specimen, two contact rollers were defined to simulate the rollers supporting and loading the specimen. The eight-node isoparametric elements (Solid 185) were used to mesh the specimens. Because the crack-tip region contains steep stress gradient, the mesh refinement was made near the crack-tip. The coordinate system in Fig. 3 is the following: the x axis lies in the crack plane and was normal to the straight crack front, the y axis was orthogonal to the crack plane and the z axis lied on the thickness direction from mid-plane (z/B = 0) to free surface (z/B = 0.5). The origin of the coordinate system was located at the crack-tip on the mid-plane. Generally, in fracture mechanics, the material behavior of uniaixal tension is described by the Ramberg-Osgood power-law relation. In this paper, the reference stress σ0 = 400 MPa, the Young’s modulus E = 200 GPa, and the material constant α = 1. Different strain hardening exponent N = 2, 5, 10 and crack length a/W = 0.3, 0.5, 0.7 are considered to cover a wide range of material and specimen geometry. On the other hand, the equivalent stress distributions ahead of crack-tip can be derived from Eq. (11) and Eq. (6)
(a) CT specimen
(b) SENB specimen
Fig. 3. Finite element model of CT specimen and SENB specimen. 4
Engineering Fracture Mechanics 222 (2019) 106706
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σeq
Cp =⎛ ⎞ σ0 ⎝ r ⎠ ⎜
1 N+1
⎟
(12)
The equivalent stress fields of CT and SENB specimens obtained from the finite element calculations were compared with the HRR solution under plane strain and the solution of Cp in Eq. (12) at the same value of applied load J. Fig. 4 shows the equivalent stress distributions ahead of crack-tip of CT specimen, at θ = 0, N = 5, a/W = 0.7, mid-plane (z/B = 0) and bσ0/J = 15. Compared with the result of HRR solution, the results of Eq. (12) are closer to FEA, but there are still differences. By analyzing all calculation results and referring to the definition of Q parameters in J-Q theory, the expression of Eq. (12) was corrected as
σeq
1
N+1 Cp L = ⎡ ⎛ + k⎞ ⎤ ⎢ ⎥ σ0 L r ⎠⎦ ⎣ ⎝
(13)
where k is a constant value for a given specimen, and k = 1 + b/W. The corrected equivalent stress distributions of Eq. (13) are shown in Fig. 5. Similarly, Figs. 6–7 shows parts of the comparison of equivalent stress distribution of CT and SENB specimens under different conditions. It can be seen that Eq. (13) can describe the equivalent stress distributions ahead of crack-tip well in a certain range, either the middle or the free surface of the specimens.
3. Application to fracture constraint analysis 3.1. Material properties The material used in this work is SA-508 steel, which is a typical ferritic low-alloy steel in nuclear pressure vessels. And the more detailed mechanical properties can be referred to reference [29].
3.2. Ductile fracture toughness test of specimens with a wide range of constraints To verify the Cp is a unified measure of constraint, three different configurations (CT, SENB, Mini-SENB) of specimens as shown in Fig. 8, with different crack length and thickness were used to distinguish the effects of constraints on ductile fracture toughness. The machined cracks in each specimen were pre-cracked by fatigue to produce a naturally sharp initial crack. The detailed sizes of all the specimens are shown in Table 1. Fig. 9 shows the fractography of Mini-SENB specimen, it is observed that nearly straight lines are formed along the crack front, which indicates that the crack-tip constraint remains consistent in the direction of thickness. The normalization method [29–32] was employed to estimate the J-resistance curves for all specimens. This method does not require any automatic crack growth measuring equipment. It proposed an individual normalized calibration curve, in which load, displacement, and crack length can be functionally related for each specimen in reference to load versus displacement records only. This calibration curve was then used to determine the instantaneous crack length in conjunction with the load and displacement test data. The test procedure in this paper was completely in accordance with the ASTM E1820-15a [1]. The critical fracture toughness J0.2BL were determined by the intersection of J-resistance curve and 0.2 mm blunting offset line. Fig. 10 shows the great differences of J-resistance curves for Mini-SENB specimens. As expected, different crack lengths and specimen thicknesses affect the crack-tip constraint level of specimens. The J-resistance curves of CT and SENB specimens are shown in Fig. 11. Obviously, the specimen configurations also affect the crack-tip. Furthermore, the J-resistance curves of CT specimens are obviously lower than Mini-SENB and SENB, which means that the constraint levels of CT specimens are higher than Mini-SENB and SENB specimens.
Fig. 4. The equivalent stress distributions along the remaining ligament in CT, with N = 5, a/W = 0.7, mid-plane (z/B = 0) and bσ0/J = 15, obtained from HRR, FEA and Eq. (12). 5
Engineering Fracture Mechanics 222 (2019) 106706
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(a) mid-plane (z/B=0) and bσ 0/J=15
(b) free-plane (z/B=1) and bσ 0/J=27
Fig. 5. The equivalent stress distributions along the remaining ligament in CT, with N = 5 and a/W = 0.7, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 15; (b) free-plane (z/B = 1) and bσ0/J = 27.
(a) mid-plane (z/B=0) and bσ 0/J=6
(b) free-plane (z/B=1) and bσ 0/J=27
Fig. 6. The equivalent stress distributions along the remaining ligament of CT specimen, with N = 10 and a/W = 0.5, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 6; (b) free-plane (z/B = 1) and bσ0/J = 15.
(a) mid-plane (z/B=0) and bσ 0/J=27
(b) free-plane (z/B=1) and bσ 0/J=40
Fig. 7. The equivalent stress distributions along the remaining ligament of SENB specimen, with N = 2 and a/W = 0.3, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 27; (b) free-plane (z/B = 1) and bσ0/J = 40.
3.3. The J-Cp curves of all specimens As shown in Fig. 12, according to the load-displacement curves of each specimen, it is not hard to solve the constraint parameter Cp in Eq. (11). Fig. 13 shows that a clear linear relationship between the Cp and J0.2BL of all specimens is presented. It means that the Cp can well reveals the difference of constraints between different specimens. The larger value of Cp means the loss of the constraint and the corresponding higher fracture toughness, and the lower value of Cp reflects the higher constraints and corresponding lower fracture toughness. Therefore, the parameter Cp can be used as a uniform parameter to simultaneously characterize in-plane
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Fig. 8. Comparation of specimens in three configurations. Table 1 The specimen sizes and test results of fracture toughness. Number
Specimen
W (mm)
B (mm)
a/W
J0.2BL (kJ/m2)
1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14# 15# 16# 17# 18# 19# 20# 21# 22# 23# 24#
Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB SENB SENB SENB SENB SENB SENB CT CT CT CT
4 4 4 4 4 4 4 4 4 4 4 4 4 4 20 20 20 20 20 20 50 50 50 50
2 2 2 2 2 2 2 2 2 2 6 6 6 6 10 5 5 5 20 20 25 25 15 15
0.19 0.20 0.22 0.20 0.15 0.42 0.44 0.44 0.42 0.43 0.45 0.48 0.50 0.44 0.75 0.35 0.55 0.75 0.55 0.75 0.68 0.78 0.58 0.78
306 389 578 448 495 357 404 463 412 267 353 351 369 325 424 296 421 418 441 341 253 156 110 147
constraints, out-of-plane constraints, and both. In addition, the linear relationships in Fig. 13 can be used to determine constraint-dependent fracture toughness for the specimens or flaw components of SA-508 steel. As long as the constraint parameters Cp is directly determined from the load-displacement curve, the conditional fracture toughness of the specimens or flaw components under different constraint levels of SA-508 steel can be determined by the linear relationship of J0.2BL versus Cp.
4. Summary In this paper, combined with the HRR crack-tip asymptotic solution, a unified constraint parameter Cp was proposed based on the plastic work Wp of external load is approximately equal to the plastic strain energy Uεp of crack-tip. Compared with FEA, an approximate solution of the equivalent stress at the crack-tip expressed by the Cp was presented and verified. The ductile material of SA508 steel was employed for fracture toughness testing. Three different configurations (CT, SENB, Mini-SENB) of specimens with different crack lengths and thicknesses were machined to distinguish the effects of constraints. It is found that the parameter Cp can reflect the constraint level of crack-tip uniquely, which will be beneficial to the prediction of material’s fracture toughness and the avoidance of experimental cost. 7
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Fig. 9. Fractography of Mini-SENB specimen.
Fig. 10. J-△a resistance curves for Mini-SENB specimens with different constraints.
Fig. 11. J-△a resistance curves for SENB and CT specimens with different constraints.
Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Acknowledgement This work below is financially supported by the National Natural Science Foundation of China (Grant Nos. 11872320). The authors thank the helpful comments and suggestions from anonymous reviewers. 8
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Experiment or FEA
P-V curve Eq. (2)
Wp Eq. (11)
Cp Fig. 12. The flow chart to get Cp.
Fig. 13. The relationship between critical fracture toughness J0.2BL versus the parameter Cp for all specimens under different constraint levels.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Study on uniform parameters characterizing the crack-tip constraint effect of fracture toughness ⁎
Guangwei He, Chen Bao , Lixun Cai
T
⁎
Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Crack-tip constraint Plastic zone Fracture toughness Unified constraint parameter
The plastic zone of crack-tip is connected with the constraint level inherently. Based on this, a unified crack-tip constraint parameter Cp was proposed. And through finite element analysis (FEA), the Cp can be recognized as a characteristic parameter in the description of equivalent stress fields. Moreover, the ductile material SA-508 steel was used to test the effectiveness of Cp as a uniform parameter, three different configurations (CT, SENB, Mini-SENB) of specimens with different crack lengths and thicknesses were employed to distinguish the effects of constraints on ductile fracture toughness. The result showed that clear linear relationship between the Cp and J0.2BL is shown for all specimens with different dimensions, which can be used to determine constraint-dependent fracture toughness for the specimens or flaw components of SA-508 steel, regardless of in-plane or out-of-plane constraint or both.
1. Introduction The constraints of crack-tip have an important influence on the evaluation of J-resistance curves. In current fracture toughness test standards ASTM E1820-15a [1], IS0 12135 [2] and GB/T 21143 [3], extremely high crack-tip constraint specimens, such as single-edge notched bend (SENB) and compact tension (CT) specimens with deep crack and enough thickness, are employed to obtain a lower bound value of material’s fracture toughness [4–6]. However, most of non-standard specimens or real cracked structures have low crack-tip constraint. The direct application of high crack-tip constraint fracture toughness to the safety design of the structures will result in lower allowable load, which may cause a conservative result. Therefore, the constraint effect on the fracture toughness must be corrected so that the fracture toughness determined in laboratory can be accurately applied to real cracked structures. There is a popular belief that the constraint of crack-tip can be divided into in-plane constraint and out-of-plane constraint. The inplane constraint is associated with crack size, geometry of specimen and type of loading. The out-of-plane constraint is associated with thickness. The in-plane and out-of-plane constraint effects on the fracture behavior of material are different in the specimens and structures with different geometries and loading configurations. Therefore, it is important to clarify the crack-tip constraint effect and size-dependence fracture toughness induced by the finite geometry of specimens. During the past decades, different constraint parameters have been proposed, such as T-stress [7], Q [8,9], A2, [10,11], Tz [12–14], h (or b) [15,16], and A [17,18], etc. However, the parameters mentioned previously are not equally sensitive to both in-plane and out-of-plane constraints which exist in the engineering structures simultaneously. For conducting accurate structural integrity assessment incorporating the constraint effects, the unified constraint parameter is desired. The plastic zone around the crack-tip line governs the initiation and propagation
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Corresponding authors. E-mail addresses: [email protected] (G. He), [email protected] (C. Bao), [email protected] (L. Cai).
https://doi.org/10.1016/j.engfracmech.2019.106706 Received 11 June 2019; Received in revised form 30 September 2019; Accepted 30 September 2019 Available online 01 October 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Fracture Mechanics 222 (2019) 106706
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r, θ SN Uεp V Vp W Wp α σ0 σeq ε0 δij σij ̃ (θ , N ) σeq̃ (θ , N )
Nomenclature a b B Cp E f IN J0.2BL k L N P rp
crack length remaining ligament specimen thickness unified constraint parameter Young’s modulus proportional coefficient an integration constant the intersection of J-R curve and 0.2 mm offset blunt lines constant value for a given specimen characteristic lengths (commonly set to be 1 mm) strain hardening exponent applied load radius of plastic zone
Φ
polar coordinates centered at the crack-tip dimensionless constant 3D plastic strain energy displacement plastic displacement specimen width plastic work done by external load strain hardening coefficient reference stress, and usually equal to yield stress equivalent stress reference strain, and ε0 = σ0/E Kronecker delta dimensionless stress function of crack-tip dimensionless equivalent stress function of cracktip 2D plastic strain energy
of cracks, a unified parameter based on the crack-tip plastic zone may be a promising approach to quantify the crack-tip constraint effects. Based on this idea, Anderson and Dodds [19] proposed a unified parameter which is the normalized area (volume per unit thickness) of a zone surrounded by the contour with σ1/σy = constant (σ1 is the maximum principal stress and σy is the yield strength of the material). Mostafavi et al. [20] modified the A-D method and redefines a uniform parameter φ denoted as the area of plastic region at the onset of fracture (AC) normalized by the reference area of plastic region of a standard plane strain specimen at the onset of fracture (Assy). However, for material with extremely high fracture toughness, the plastic deformation zone will expand to the specimen’s surface, it is impossible to accurately calculate the area of crack-tip plastic zone because the intersection of the plastic zone in the load region and that around the crack-tip region. Yang et al. [21] made an improvement and proposed a modified parameter Ap, which is the area surrounded by the isoline of certain equivalent plastic strain (εp) ahead of the crack-tip normalized by the reference area of the isoline with same εp in a standard specimen with high constraint level. However, the studies by Yang et al. are mainly based on finite element method and lacked of effective experimental verification. In this paper, a new constraint parameter Cp was introduced, which can describe the equivalent stress distributions ahead of crack-tip well. In order to verify its capability of characterize the fracture toughness, specimens with different constraints levels include different crack length, thickness, geometry of specimen and type of loading were used in fracture toughness tests.
2. Theoretical background As mentioned above, the plastic region of crack-tip is closely related to the constraint level for ductile fracture because of the contribution of plasticity to the ductile fracture mechanism of void nucleation, growth and coalescence. Actually, for a cracked specimen subjected to external load, its plastic deformation is mainly concentrated at the crack-tip, the elastic deformation is almost negligible. In other words, for a cracked specimen, the plastic work done Wp of external load is approximately equal to the plastic strain energy Uεp around crack-tip. This idea will guide the following research to propose a unified crack-tip constraint parameter.
Fig. 1. The schematic of the plastic work Wp. 2
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2.1. Expression of plastic work done Wp by external load The study focuses on mode-I cracked body of elastic-plastic materials. In the plot of load versus displacement, the Wp is the area as shown in Fig. 1, and it can be calculated from the following equation as recommended in ASTM E1820-15a [1]:
Wp (i) = Wp (i − 1) + [P(i) + P(i − 1) ][Vp (i) − Vp (i − 1) ]/2
(1)
where Wp(i) and Wp(i-1) are the plastic work done at points i and i-1. P and Vp are the corresponding load and plastic displacement. 2.2. Expression of 3D plastic strain energy Uεp A generally accepted that the HRR crack-tip asymptotic solution for mode-I cracked body made by Hutchinson [22] and Rice and Rosengren [27], is the theoretical foundation of modern elastic-plastic fracture mechanics and it can be given as, 1
σij
N+1 J ⎞ σ~ij (θ , N ) =⎛ σ0 αε σ I r ⎝ 0 0 N ⎠ ⎜
⎟
(2)
where, α is strain hardening coefficient, N is strain hardening exponent. σ 0 is reference stress, and usually equal to yield stress, ε0 = σ0/E. IN is an integration constant and σij (θ , N ) are dimensionless stress function of crack tip. r represents the distance from the crack-tip along the crack surface. Accordingly, the equivalent stress of crack-tip can be expressed as 1
σeq
N+1 J ⎞ σ~eq (θ , N ) =⎛ σ0 ⎝ αε0 σ0 IN r ⎠ ⎜
⎟
(3)
where, σeq (θ , N ) is only related to θ for a certain material. At the boundary of plastic zone, i.e. r = rp, the equivalent stress reaches to yield stress,
σeq = σ0
(4)
combining Eqs. (3) and (4), we can get that − 1 N +1
J ⎞ σ~eq (θ , N ) = ⎛⎜ ⎟ ⎝ αε0 σ0 IN rp ⎠
(5)
then, substituting Eq. (5) into Eq. (3), the equivalent stress can now be expressed by the size of plastic zone rp, 1
rp N + 1 σeq = σ0 ⎛ ⎞ ⎝r⎠
(6)
it should be pointed out that rp is related to θ because σ~eq (θ , N ) in Eq. (5) changes with the variation of θ. But if we suppose a circular shape of the plastic zone around crack-tip by ignoring the difference of rp caused by different θ, the rp can be replaced by an equivalent radius of plastic zone Cp, as shown in Fig. 2. The plastic strain energy Φ in the plastic zone around crack-tip[28] can be expressed as
S Φ = ⎛ N ⎞ JCp ⎝ IN ⎠ ⎜
⎟
(7)
where the dimensionless constant SN is
Fig. 2. The schematic showing of the plastic zone around crack-tip. 3
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SN =
N N+1
∫π
π
σ~eqN + 1 dθ
(8)
combining Eq. (5) and Eq. (8), Eq. (7) can be rewritten as
Φ = 2π
N ασ0 ε0 Cp2 N+1
(9)
It should be noted that above derivations are restricted in the plane condition, but the deformation of an elastic-plastic body is three-dimensional. It is widely accepted to ignore the variation in the thickness direction (usually choose the middle plane of the thickness as the representative) [20,21,23–26] in the research of 3D crack-tip constraint. If the plastic zone size of the plane with different thickness is the same as that of the mid-plane, then we can obtain that
Uεp = f ΦB
(10)
where, f is the proportional coefficient to quantify the difference between the assumed volume of the plastic zone and the real one. A trial-and-error optimization approach was employed to determine the parameter f. This method was based on a large number of finite element calculations, and the value of f was determined by comparing the difference between the assumed plastic zone volume and the actual plastic zone volume. It turned out that f is only related to the initial remaining ligament of the specimen, and doesn’t change in the calculation of Cp, and can be assumed to b/L for all the specimen geometries considered in this work. b is initial remaining ligament. L is characteristic length parameter which can be equal to 1 mm. 2.3. Unified constraint parameter Cp Based on the above derivation, the expressions of Wp and Uεp are obtained, respectively. By establishing the equation Wp = Uεp, the equivalent radius of plastic zone Cp can be obtained as 1/2
Wp ⎛ ⎞ Cp = ⎜ N 2π N + 1 α 0 σ0 ε0 fB ⎟ ⎝ ⎠
(11)
The commercial finite element code, ANSYS 14.5, was used to calculate the equivalent stress distributions ahead of crack-tip. The typical models and meshes for the SENB and CT specimens are shown in Fig. 3. The width W of SENB and CT were 40 mm, 50 mm. The thickness B of SENB and CT were 20 mm, 25 mm. Consider the symmetry of loading and geometry of SENB and CT specimens, only 1/4th of the two specimens were modeled, and the symmetry boundary conditions were applied in the un-cracked ligament. Along the thickness direction (z-axis), the identical planar mesh was repeated from the symmetry plane (mid-plane, z/B = 0) to the free surface (z/B = 0.5). The loading was directly applied as a pressure to upper surface of the load hole for the CT specimen. While for SENB specimen, two contact rollers were defined to simulate the rollers supporting and loading the specimen. The eight-node isoparametric elements (Solid 185) were used to mesh the specimens. Because the crack-tip region contains steep stress gradient, the mesh refinement was made near the crack-tip. The coordinate system in Fig. 3 is the following: the x axis lies in the crack plane and was normal to the straight crack front, the y axis was orthogonal to the crack plane and the z axis lied on the thickness direction from mid-plane (z/B = 0) to free surface (z/B = 0.5). The origin of the coordinate system was located at the crack-tip on the mid-plane. Generally, in fracture mechanics, the material behavior of uniaixal tension is described by the Ramberg-Osgood power-law relation. In this paper, the reference stress σ0 = 400 MPa, the Young’s modulus E = 200 GPa, and the material constant α = 1. Different strain hardening exponent N = 2, 5, 10 and crack length a/W = 0.3, 0.5, 0.7 are considered to cover a wide range of material and specimen geometry. On the other hand, the equivalent stress distributions ahead of crack-tip can be derived from Eq. (11) and Eq. (6)
(a) CT specimen
(b) SENB specimen
Fig. 3. Finite element model of CT specimen and SENB specimen. 4
Engineering Fracture Mechanics 222 (2019) 106706
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σeq
Cp =⎛ ⎞ σ0 ⎝ r ⎠ ⎜
1 N+1
⎟
(12)
The equivalent stress fields of CT and SENB specimens obtained from the finite element calculations were compared with the HRR solution under plane strain and the solution of Cp in Eq. (12) at the same value of applied load J. Fig. 4 shows the equivalent stress distributions ahead of crack-tip of CT specimen, at θ = 0, N = 5, a/W = 0.7, mid-plane (z/B = 0) and bσ0/J = 15. Compared with the result of HRR solution, the results of Eq. (12) are closer to FEA, but there are still differences. By analyzing all calculation results and referring to the definition of Q parameters in J-Q theory, the expression of Eq. (12) was corrected as
σeq
1
N+1 Cp L = ⎡ ⎛ + k⎞ ⎤ ⎢ ⎥ σ0 L r ⎠⎦ ⎣ ⎝
(13)
where k is a constant value for a given specimen, and k = 1 + b/W. The corrected equivalent stress distributions of Eq. (13) are shown in Fig. 5. Similarly, Figs. 6–7 shows parts of the comparison of equivalent stress distribution of CT and SENB specimens under different conditions. It can be seen that Eq. (13) can describe the equivalent stress distributions ahead of crack-tip well in a certain range, either the middle or the free surface of the specimens.
3. Application to fracture constraint analysis 3.1. Material properties The material used in this work is SA-508 steel, which is a typical ferritic low-alloy steel in nuclear pressure vessels. And the more detailed mechanical properties can be referred to reference [29].
3.2. Ductile fracture toughness test of specimens with a wide range of constraints To verify the Cp is a unified measure of constraint, three different configurations (CT, SENB, Mini-SENB) of specimens as shown in Fig. 8, with different crack length and thickness were used to distinguish the effects of constraints on ductile fracture toughness. The machined cracks in each specimen were pre-cracked by fatigue to produce a naturally sharp initial crack. The detailed sizes of all the specimens are shown in Table 1. Fig. 9 shows the fractography of Mini-SENB specimen, it is observed that nearly straight lines are formed along the crack front, which indicates that the crack-tip constraint remains consistent in the direction of thickness. The normalization method [29–32] was employed to estimate the J-resistance curves for all specimens. This method does not require any automatic crack growth measuring equipment. It proposed an individual normalized calibration curve, in which load, displacement, and crack length can be functionally related for each specimen in reference to load versus displacement records only. This calibration curve was then used to determine the instantaneous crack length in conjunction with the load and displacement test data. The test procedure in this paper was completely in accordance with the ASTM E1820-15a [1]. The critical fracture toughness J0.2BL were determined by the intersection of J-resistance curve and 0.2 mm blunting offset line. Fig. 10 shows the great differences of J-resistance curves for Mini-SENB specimens. As expected, different crack lengths and specimen thicknesses affect the crack-tip constraint level of specimens. The J-resistance curves of CT and SENB specimens are shown in Fig. 11. Obviously, the specimen configurations also affect the crack-tip. Furthermore, the J-resistance curves of CT specimens are obviously lower than Mini-SENB and SENB, which means that the constraint levels of CT specimens are higher than Mini-SENB and SENB specimens.
Fig. 4. The equivalent stress distributions along the remaining ligament in CT, with N = 5, a/W = 0.7, mid-plane (z/B = 0) and bσ0/J = 15, obtained from HRR, FEA and Eq. (12). 5
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(a) mid-plane (z/B=0) and bσ 0/J=15
(b) free-plane (z/B=1) and bσ 0/J=27
Fig. 5. The equivalent stress distributions along the remaining ligament in CT, with N = 5 and a/W = 0.7, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 15; (b) free-plane (z/B = 1) and bσ0/J = 27.
(a) mid-plane (z/B=0) and bσ 0/J=6
(b) free-plane (z/B=1) and bσ 0/J=27
Fig. 6. The equivalent stress distributions along the remaining ligament of CT specimen, with N = 10 and a/W = 0.5, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 6; (b) free-plane (z/B = 1) and bσ0/J = 15.
(a) mid-plane (z/B=0) and bσ 0/J=27
(b) free-plane (z/B=1) and bσ 0/J=40
Fig. 7. The equivalent stress distributions along the remaining ligament of SENB specimen, with N = 2 and a/W = 0.3, obtained from HRR, FEA and Eq. (13). (a) mid-plane (z/B = 0) and bσ0/J = 27; (b) free-plane (z/B = 1) and bσ0/J = 40.
3.3. The J-Cp curves of all specimens As shown in Fig. 12, according to the load-displacement curves of each specimen, it is not hard to solve the constraint parameter Cp in Eq. (11). Fig. 13 shows that a clear linear relationship between the Cp and J0.2BL of all specimens is presented. It means that the Cp can well reveals the difference of constraints between different specimens. The larger value of Cp means the loss of the constraint and the corresponding higher fracture toughness, and the lower value of Cp reflects the higher constraints and corresponding lower fracture toughness. Therefore, the parameter Cp can be used as a uniform parameter to simultaneously characterize in-plane
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Fig. 8. Comparation of specimens in three configurations. Table 1 The specimen sizes and test results of fracture toughness. Number
Specimen
W (mm)
B (mm)
a/W
J0.2BL (kJ/m2)
1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14# 15# 16# 17# 18# 19# 20# 21# 22# 23# 24#
Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB Mini-SENB SENB SENB SENB SENB SENB SENB CT CT CT CT
4 4 4 4 4 4 4 4 4 4 4 4 4 4 20 20 20 20 20 20 50 50 50 50
2 2 2 2 2 2 2 2 2 2 6 6 6 6 10 5 5 5 20 20 25 25 15 15
0.19 0.20 0.22 0.20 0.15 0.42 0.44 0.44 0.42 0.43 0.45 0.48 0.50 0.44 0.75 0.35 0.55 0.75 0.55 0.75 0.68 0.78 0.58 0.78
306 389 578 448 495 357 404 463 412 267 353 351 369 325 424 296 421 418 441 341 253 156 110 147
constraints, out-of-plane constraints, and both. In addition, the linear relationships in Fig. 13 can be used to determine constraint-dependent fracture toughness for the specimens or flaw components of SA-508 steel. As long as the constraint parameters Cp is directly determined from the load-displacement curve, the conditional fracture toughness of the specimens or flaw components under different constraint levels of SA-508 steel can be determined by the linear relationship of J0.2BL versus Cp.
4. Summary In this paper, combined with the HRR crack-tip asymptotic solution, a unified constraint parameter Cp was proposed based on the plastic work Wp of external load is approximately equal to the plastic strain energy Uεp of crack-tip. Compared with FEA, an approximate solution of the equivalent stress at the crack-tip expressed by the Cp was presented and verified. The ductile material of SA508 steel was employed for fracture toughness testing. Three different configurations (CT, SENB, Mini-SENB) of specimens with different crack lengths and thicknesses were machined to distinguish the effects of constraints. It is found that the parameter Cp can reflect the constraint level of crack-tip uniquely, which will be beneficial to the prediction of material’s fracture toughness and the avoidance of experimental cost. 7
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Fig. 9. Fractography of Mini-SENB specimen.
Fig. 10. J-△a resistance curves for Mini-SENB specimens with different constraints.
Fig. 11. J-△a resistance curves for SENB and CT specimens with different constraints.
Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Acknowledgement This work below is financially supported by the National Natural Science Foundation of China (Grant Nos. 11872320). The authors thank the helpful comments and suggestions from anonymous reviewers. 8
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Experiment or FEA
P-V curve Eq. (2)
Wp Eq. (11)
Cp Fig. 12. The flow chart to get Cp.
Fig. 13. The relationship between critical fracture toughness J0.2BL versus the parameter Cp for all specimens under different constraint levels.
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