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Surface crack fracture toughness and HRR fields of ultra-high strength steel
Materials Science and Engineering A 454–455 (2007) 467–471
Surface crack fracture toughness and HRR fields of ultra-high strength steel Jiquan Zou a,b , Hongyang Jing a , Lianyong Xu a,∗ a
b
School of Material Science & Engineering, Tianjin University, Tianjin 300072, China Electromechanical and Automation School, Tianjin Professional College, Tianjin 300402, China
Received 19 May 2006; received in revised form 8 November 2006; accepted 13 December 2006
Abstract In present, due to continuous increasing of the fracture toughness and continuous reduction of steel plate thickness for ultra-high strength steel used in the aerospace solid rocket motor, it is difficult to measure its fracture toughness of surface crack using the linear elastic fracture mechanics method. In this paper, the tensile test was used to achieve the conditional load when the surface crack initiation occurred. Then, the finite element method (FEM) was conducted to calculate the ductile fracture toughness JIC under the conditional load. Lastly, the surface crack fracture toughness, KIe , can be computed from JIC based on the fracture mechanics method. In addition, the analysis on validity of J-integral showed that KIe converted from JIC was able to evaluate the fracture toughness of ultra-high strength steel. These are important technical parameters for design of aerospace solid rocket motor. © 2006 Elsevier B.V. All rights reserved. Keywords: Solid rocket motor; Ultra-high strength steel; Surface crack; Fracture toughness; FEM; J-dominant zone
1. Introduction In present, the thin-walled structure of ultra-high strength steel was widely used in the aerospace vehicles motor with the development of aerospace technology. However, the brittle fracture accidents often occurred. The studies showed that most of the accidents were resulted from surface crack propagation. Therefore, the surface crack fracture toughness (KIe ) was a critical target in the evaluation for this ultra-high strength steel. 31Si2MnCrMoVE steel is ultra-high strength steel which was specially developed for manufacturing solid rocket motor shell. The content of inclusions decreased and controlling for the chemical constitution became precise in the steel with development of the smelting technology, as well as the harmful elements can be strictly restricted. Therefore, the fracture toughness have been continuously increased, correspondingly, the steel plate thickness can be reduced. However, the increasing of fracture toughness and reduction of plate thickness resulted in the difficulty of measuring its surface crack fracture toughness because the very small ligament sizes of specimens could not satisfy the
∗
Corresponding author. Tel.: +86 022 27402439; fax: +86 022 27407022. E-mail address: [email protected] (LY. Xu).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.12.045
validity criterion. So, the linear elastic fracture mechanics cannot be used to evaluate the fracture toughness of thin ultra-high strength steel, but the ductile fracture toughness (JIC ) of thin thickness ultra-high strength steel can be evaluated using the elasto-plastic fracture mechanics. Thaulow et al. [1] and Zhang and co-workers [2] had studied the fracture resistance and the application of constraint correction in evaluating toughness of high strength steel. Beremin [3] and Toyoda and Minami had already studied the toughness of high strength steel based on the local approach for a long time [4–7]. Dotta and Ruggieri [8] and Croavero and Ruggieri [9] had studied the fracture behavior and the ductile crack extension in a high strength pipeline steel. However, in present, the papers with respect to the fracture toughness of ultra-high strength steel were very rare. In this paper, the surface crack fracture toughness specimens had been fabricated using steel plate of original thickness. The fracture mechanics test and finite element method (FEM) had been conducted to calculate the J-integral in the vicinity of the crack tip under the conditional load in the condition that the conditional fracture toughness (KIQ ) cannot satisfy the validity criterion. The computed J-integral was hoped to be used as the ductile fracture toughness of the material. However, the validity of J-integral must be analyzed before the JIC can be used as the fracture parameter. So, the conservation of Loop J-integration
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J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
Table 1 Chemical compositions % C
Si
Mn
Cr
Ni
Mo
V
P
S
0.29
1.66
0.85
1.10
0.25
0.44
0.091
0.01
0.002
and the J-dominant characteristic of ultra-high strength steel surface crack were studied. 2. Fracture toughness test The test chose the ultra-high strength steel plate (31Si2MnCrMoVE) with 5 mm thickness. The chemical compositions and mechanical properties of 31Si2MnCrMoVE steel were listed in Tables 1 and 2, respectively. This test used the uniaxial tension specimens with surface pre-crack. The configuration of tension specimen was showed in Fig. 1, and the dimension of each specimen was 2L × 2W × B in length, width and thickness, respectively. The number of tension specimens was five, and all specimens were abstracted from the steel plate along the rolling direction of the steel plate. The process of preparing the surface pre-crack was as follows: Firstly, the tension specimens were fabricated using the 31Si2MnCrMoVE steel in the annealed condition. Secondly, a surface indentation in the center of the specimen was formed using a special knife with 1.2 mm width and 0.05–0.10 mm thickness. The indentation pressure was controlled in 0.18–0.22 kN, and the indentation depth was in 0.10–0.20 mm. Table 2 Mechanical properties of 31Si2MnCrMoVE Proof strength, non-proportional extension, RP0.2 (MPa) Tensile strength, Rm (MPa) Percentage elongation after fracture, A
1323.0 1643.0 9.5
Fig. 2. Fracture appearance of surface crack.
Thirdly, the specimens were heated in a salt-bath furnace. The furnace temperature was 930 ◦ C, and the specimens were maintained 10 min under that temperature. Then the specimens were treated oil quenching (the temperature of oil was the roomtemperature). When the furnace temperature reduced to the 300 ◦ C, the specimens were put into the furnace again and were maintained 2 h. After the specimens were cooled in the air, the oxides on the specimen surface were grinded off. Lastly, the surface fatigue pre-crack was created on a 20 kN high frequency fatigue experiment machine using three point bend test. The crack length (2c), and crack depth (a) were controlled so that both a/B and a/c were in the range of 0.45–0.55. The crack depth a can be estimated by the following equation through observing the crack length: a a + =1 B c
(1)
The tension test was on a MTS880 material test machine. An extensometer was used to measure the crack opening displacement. The loading speed was approximately 1.0 mm/min. The load-crack opening displacement curve (P–V curve) was recorded in the test. The test ended until the specimen fractured. Then, a 50× toolmaker’s microscope was used to measure the crack depth and crack length. Fig. 2 showed the appearance of fracture of the surface crack observed. In Fig. 3, the heavy line was the P–V curve, and line OA was the initial tangent line. The rate of slope of the line OD was 15% lower than that of the line OA. The load at the point F which was the crossing point between the P–V curve and OD was defined as the conditional load (PQ ), and the PQ corresponded to the conditional initiation of crack [10]. Pmax was the maximum load in the P–V curve. 3. Test results and analysis 3.1. Conditional fracture toughness
Fig. 1. Surface cracked specimen subjected to tension.
In this paper, the Newman–Raju expression [11] was chosen to compute the stress intensity factor (KI ) of surface crack because of its good precision and small dispersibility. The expression of the conditional fracture toughness (KIQ ) was as
J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
469
Table 3 Results of tests No.
B (mm)
a/B
a/C
Pmax /PQ
KIQ (MPam1/2 )
a/(KIQ /RP0.2 )2
(B − a)/(KIQ /RP0.2 )2
T-1 T-2 T-3 T-4 T-5
5.0 5.0 5.0 5.0 5.0
0.502 0.513 0.519 0.504 0.512
0.515 0.571 0.505 0.502 0.513
1.004 1.034 1.050 1.061 1.031
93.86 87.68 96.86 95.58 93.53
0.36 0.447 0.36 0.375 0.400
0.358 0.436 0.333 0.368 0.381
follows: √ M KIQ = σ πa Φ
(2)
where σ = PQ /BW, Φ2 = 1 + 1.464(a/C)1.65 and a 0.89 a 2 M = 1.13 − 0.09 + −0.54 + C 0.2 + aC B a 4 1.0 a 24 + 0.5 − + 14 1.0 − 0.65 + (a/C) C B
4. Ductile fracture toughness
The surface crack fracture toughness (KIe ) is equal to the conditional fracture toughness (KIQ ) if all the following criterions could be satisfied. Pmax ≤ 1.2 PQ KIQ 2 a ≥ 0.50 σ0.2 KIQ 2 (B − a) ≥ 0.50 σ0.2
ment size were so small that the rather large plastic deformation occurred near the crack front. Therefore, the linear elastic fracture mechanics could not apply with these specimens. Under this condition, the elasto-plastic fracture mechanics must be used to measure the ductile fracture toughness.
(3)
(4)
(5)
3.2. Test results and analysis The results of tests were listed in Table 3. It was found that Eq. (3) could be satisfied, but Eqs. (4) and (5) could not be satisfied. The reason was that the specimen thickness and liga-
Kikuchi and some researchers had studied the stress field near surface crack tip, but these studies constrain to the low strength steels [13–15]. In this section, FEM had been conducted to analyze the stress field near surface crack tip for ultra-high strength steel and calculate the J-integral of the surface crack front of the tension specimens under the conditional load achieved in the experiment in Section 2. The value of J-integral where the surface crack was deepest was the maximal, and it was defined as the J-integral value when the crack initiation occurred. So, the maximum value of J-integral of the surface crack front can be seen as the ductile fracture toughness of the material in the engineering evaluation. The surface crack fracture toughness, KIe , can be computed from JIC based on the following equation:
EJIC (6) KIe = (1 − ν2 ) where E is the Young’s modulus of the material and ν is the Poisson’s ratio. 4.1. Finite element model
Fig. 3. Determination of the conditional load (PQ ).
According to the symmetry quality, the one-quarter of tension specimen was selected to carry out the finite element analysis. The finite elements were 20-node quadratic brick, reduced integration elements; one element had 8 integration points. The mesh of whole model for tension specimen with surface crack was presented in Fig. 4. The mesh contained 15,620 elements and 17,808 nodes. The property of 31Si2MnCrMoVE steel was achieved through the uniaxial tension test, and the true stress–strain curve and nominal stress–strain curve were showed in Fig. 5. The stress–strain curve was the basis of this FEM calculation. The three-dimensional finite element solutions were obtained using general purpose finite element software ABAQUS Version 6.5-1 [12]. In the FEA, the applied load was the conditional load obtained in Section 2. The analysis was 3D elastic–plastic analysis.
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J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
Fig. 7. Distribution of J-integral along the crack front. Table 4 Values of KIe and KIQ Fig. 4. Mesh used in FE-analysis: (a) mesh of 1/4 specimen and region near the crack tip and (b) refined mesh in the crack tip region.
Specimen no.
J-integral (N/mm)
KIe (MPam1/2 )
KIQ (MPam1/2 )
T-1 T-2 T-3 T-4 T-5 Average
52.32 61.60 50.56 45.35 56.94 53.35
109.88 119.23 108.23 102.30 114.63 110.85
93.86 87.68 96.86 95.58 93.53 93.50
value of J-integral was the one where the crack was deepest (Φ = π/2). In this paper, the maximum J-integral value was used as the ductile fracture toughness of the specimen for the tested material. Table 4 gave the JIC values and the corresponding KIe values of each specimen, respectively. Fig. 5. Stress–strain curve of 31Si2MnGrMoVE steel.
4.2. Analysis result Fig. 6 showed the conservation of J-integral for surface crack along eight integrating loop on five cross-sections as the maximum error was 2.30%. Fig. 7 represented the J-integral values of semi-elliptical surface cracks of five specimens on different cross-sections. The cross-section which had the maximum
Fig. 6. Conservation analysis of J-integral.
5. J-dominant analysis If the J-integral was hoped to be used to characterize the intensity of stress–strain field in the elastic–plastic condition, the validity of J-integral must be analyzed when the specimen was loaded with the conditional load. If the HRR fields approximately existed near the crack tip, the condition was called J-dominant. So, the J-integral can be considered effective as a fracture parameter. For a strain hardening material, the strain ε is related to the uniaxial tensile stress σ in the form (Ramberg–Osgood type equation): n ε σ σ = +α (7) ε0 σ0 σ0 where σ 0 is a reference stress (which is related to the yield stress level), ε0 = σ 0 /E the associated reference strain with E as Young’s modulus, α a dimensionless constant and n is the strain hardening exponent. For the 31Si2MnCrMoVE steel, the constant α and the strain hardening exponent n can be fitted from Fig. 4: α = 0.0051, n = 18. Hutchinson [16] and Rice and Rosengren [17] showed that asymptotic field for a crack under monotonically increasing,
J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
471
solution, and the exponent of the stress field was nearly equal to the HRR solution. So, the HRR fields approximately existed near the crack front where the surface crack was deepest. It indicated that the J-integral was effective, and J can be used as a fracture parameter for the specimens in this study. Therefore, the surface crack fracture toughness KIe of 31Si2MnCrMoVE steel which was conversed from JIC can be used in the engineering evaluation for the aerospace industry. 6. Conclusion The study showed the linear elastic fracture mechanics cannot be used to evaluate the fracture toughness of thin ultra-high strength steel used in this paper. So, the FEM was utilized to analyze the stress field of specimens and compute the JIC of each specimen. Through the comparison of FEA results and the HRR field, the validity of J-integral was tested. So, the J can be used as a fracture parameter for the 31Si2MnCrMoVE steel tension specimens in this study. Therefore, the surface crack fracture toughness KIe of 31Si2MnCrMoVE steel can be conversed from JIC .
Fig. 8. θ-Variation of stresses at crack tip.
Acknowledgement The research work of this paper is supported by the National Nature Science Foundation of China, No. 50275107; and Fok Ying Tung Education Foundation No. 81405. References
Fig. 9. Stress fields at the bottom of the surface crack (θ = 0).
mode I loading is given by: 1/n+1 EJ σ˜ ij (θ, n) σij (r, θ) = σ0 ασ02 In r
(8)
where In is integration constant, σ˜ ij (θ, n) the dimensionless functions determined by n and angle θ which is defined by the polar coordinate at the crack tip. Model I dimensionless functions σ˜ ij (θ, n) and In have been tabulated by Shih [18] under the plane strain and plane stress conditions. J is the amplitude of the crack tip fields, which are now referred to as HRR fields, in the same spirit that K scales the elastic near-tip fields. Figs. 8 and 9 gave the HRR field and stress field near the crack front from the results of the FEA where the value of J in Fig. 9 was equal to the value of JIC which was computed in Section 4.2 under the conditional load. In Fig. 8, σ θθ (r,θ) and σ rθ (r,θ) and σ rr (r,θ) varied with θ when r/(J/σ 0 ) was equal to 2 and the figure showed that the FEA results and the HRR field were almost identical. Fig. 9 showed that the comparison of the exponent of the stress field with that of the HRR solution when θ was equal to zero. In the HRR solution, the exponent was −1/(n + 1). In Fig. 9, the crack tip field agreed well with the HRR
[1] C. Thaulow, E. Ostby, B. Nyhus, et al., Eng. Fract. Mech. 71 (November (16–17)) (2004) 2417–2433. [2] M. Hauge, C. Thaulow, ZL. Zhang, in: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering—OMAE, vol. 3, Mater. Eng. (1997) 257–264. [3] F.M. Beremin, Metall. Trans. A 14 (1983) 2277–2287. [4] F. Minami, K. Arimochi, Constraint correction of fracture toughness CTOD for fracture performance evaluation of structural components, ASTM Special Technical Publication, No. 1429, 2002, pp. 48–66. [5] M. Toyoda, M. Ohata, M. Yokota, et al., Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering–OMAE, vol. 3, 2001, pp. 85–98. [6] F. Minami, A. Bruckner-Foit, D. Munz, et al., Int. J. Fract. 54 (1992) 197–210. [7] M. El-Shennawy, F. Minami, M. Toyoda, et al., Eng. Fract. Mech. 63 (June (4)) (1999) 447–479. [8] F. Dotta, C. Ruggieri, Int. J. Pressure Vessels Piping 81 (September (9)) (2004) 761–770. [9] S. Croavero, C. Ruggieri, Eng. Fract. Mech. 72 (June (9)) (2005) 1344–1360. [10] ZY. Cui, Eng. Fract. Mech. 13 (4) (1980) 775–789. [11] J.C. Newman, I.S. Raju, Eng. Fract. Mech. 15 (1) (1981) 85–92. [12] ABAQUS Version 6.5, User’s Manual, Hibbitt. Karlsson & Sorensen, Inc., RI, 2005. [13] M. Kikuchi, Int. J. Fract. 58 (1992) 273–283. [14] M. Kikuchi, Int. J. Press. Vessels Piping 63 (1995) 315–326. [15] M.H. Sharobeam, J.D. Landes, Eng. Fract. Mech. 63 (1999) 131–145. [16] J.W. Hutchinson, J. Mech. Phys. Solids 16 (1968) 13–31. [17] J.R. Rice, G.F. Rosengren, J. Mech. Phys. Solids 16 (1968) 1–12. [18] C.F. Shih, Tables of Hutchinson–Rice–Rosengren Singular Field Quantities. MRL E-147, Materials Research Laboratory, Brown University, 1983.
Surface crack fracture toughness and HRR fields of ultra-high strength steel Jiquan Zou a,b , Hongyang Jing a , Lianyong Xu a,∗ a
b
School of Material Science & Engineering, Tianjin University, Tianjin 300072, China Electromechanical and Automation School, Tianjin Professional College, Tianjin 300402, China
Received 19 May 2006; received in revised form 8 November 2006; accepted 13 December 2006
Abstract In present, due to continuous increasing of the fracture toughness and continuous reduction of steel plate thickness for ultra-high strength steel used in the aerospace solid rocket motor, it is difficult to measure its fracture toughness of surface crack using the linear elastic fracture mechanics method. In this paper, the tensile test was used to achieve the conditional load when the surface crack initiation occurred. Then, the finite element method (FEM) was conducted to calculate the ductile fracture toughness JIC under the conditional load. Lastly, the surface crack fracture toughness, KIe , can be computed from JIC based on the fracture mechanics method. In addition, the analysis on validity of J-integral showed that KIe converted from JIC was able to evaluate the fracture toughness of ultra-high strength steel. These are important technical parameters for design of aerospace solid rocket motor. © 2006 Elsevier B.V. All rights reserved. Keywords: Solid rocket motor; Ultra-high strength steel; Surface crack; Fracture toughness; FEM; J-dominant zone
1. Introduction In present, the thin-walled structure of ultra-high strength steel was widely used in the aerospace vehicles motor with the development of aerospace technology. However, the brittle fracture accidents often occurred. The studies showed that most of the accidents were resulted from surface crack propagation. Therefore, the surface crack fracture toughness (KIe ) was a critical target in the evaluation for this ultra-high strength steel. 31Si2MnCrMoVE steel is ultra-high strength steel which was specially developed for manufacturing solid rocket motor shell. The content of inclusions decreased and controlling for the chemical constitution became precise in the steel with development of the smelting technology, as well as the harmful elements can be strictly restricted. Therefore, the fracture toughness have been continuously increased, correspondingly, the steel plate thickness can be reduced. However, the increasing of fracture toughness and reduction of plate thickness resulted in the difficulty of measuring its surface crack fracture toughness because the very small ligament sizes of specimens could not satisfy the
∗
Corresponding author. Tel.: +86 022 27402439; fax: +86 022 27407022. E-mail address: [email protected] (LY. Xu).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.12.045
validity criterion. So, the linear elastic fracture mechanics cannot be used to evaluate the fracture toughness of thin ultra-high strength steel, but the ductile fracture toughness (JIC ) of thin thickness ultra-high strength steel can be evaluated using the elasto-plastic fracture mechanics. Thaulow et al. [1] and Zhang and co-workers [2] had studied the fracture resistance and the application of constraint correction in evaluating toughness of high strength steel. Beremin [3] and Toyoda and Minami had already studied the toughness of high strength steel based on the local approach for a long time [4–7]. Dotta and Ruggieri [8] and Croavero and Ruggieri [9] had studied the fracture behavior and the ductile crack extension in a high strength pipeline steel. However, in present, the papers with respect to the fracture toughness of ultra-high strength steel were very rare. In this paper, the surface crack fracture toughness specimens had been fabricated using steel plate of original thickness. The fracture mechanics test and finite element method (FEM) had been conducted to calculate the J-integral in the vicinity of the crack tip under the conditional load in the condition that the conditional fracture toughness (KIQ ) cannot satisfy the validity criterion. The computed J-integral was hoped to be used as the ductile fracture toughness of the material. However, the validity of J-integral must be analyzed before the JIC can be used as the fracture parameter. So, the conservation of Loop J-integration
468
J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
Table 1 Chemical compositions % C
Si
Mn
Cr
Ni
Mo
V
P
S
0.29
1.66
0.85
1.10
0.25
0.44
0.091
0.01
0.002
and the J-dominant characteristic of ultra-high strength steel surface crack were studied. 2. Fracture toughness test The test chose the ultra-high strength steel plate (31Si2MnCrMoVE) with 5 mm thickness. The chemical compositions and mechanical properties of 31Si2MnCrMoVE steel were listed in Tables 1 and 2, respectively. This test used the uniaxial tension specimens with surface pre-crack. The configuration of tension specimen was showed in Fig. 1, and the dimension of each specimen was 2L × 2W × B in length, width and thickness, respectively. The number of tension specimens was five, and all specimens were abstracted from the steel plate along the rolling direction of the steel plate. The process of preparing the surface pre-crack was as follows: Firstly, the tension specimens were fabricated using the 31Si2MnCrMoVE steel in the annealed condition. Secondly, a surface indentation in the center of the specimen was formed using a special knife with 1.2 mm width and 0.05–0.10 mm thickness. The indentation pressure was controlled in 0.18–0.22 kN, and the indentation depth was in 0.10–0.20 mm. Table 2 Mechanical properties of 31Si2MnCrMoVE Proof strength, non-proportional extension, RP0.2 (MPa) Tensile strength, Rm (MPa) Percentage elongation after fracture, A
1323.0 1643.0 9.5
Fig. 2. Fracture appearance of surface crack.
Thirdly, the specimens were heated in a salt-bath furnace. The furnace temperature was 930 ◦ C, and the specimens were maintained 10 min under that temperature. Then the specimens were treated oil quenching (the temperature of oil was the roomtemperature). When the furnace temperature reduced to the 300 ◦ C, the specimens were put into the furnace again and were maintained 2 h. After the specimens were cooled in the air, the oxides on the specimen surface were grinded off. Lastly, the surface fatigue pre-crack was created on a 20 kN high frequency fatigue experiment machine using three point bend test. The crack length (2c), and crack depth (a) were controlled so that both a/B and a/c were in the range of 0.45–0.55. The crack depth a can be estimated by the following equation through observing the crack length: a a + =1 B c
(1)
The tension test was on a MTS880 material test machine. An extensometer was used to measure the crack opening displacement. The loading speed was approximately 1.0 mm/min. The load-crack opening displacement curve (P–V curve) was recorded in the test. The test ended until the specimen fractured. Then, a 50× toolmaker’s microscope was used to measure the crack depth and crack length. Fig. 2 showed the appearance of fracture of the surface crack observed. In Fig. 3, the heavy line was the P–V curve, and line OA was the initial tangent line. The rate of slope of the line OD was 15% lower than that of the line OA. The load at the point F which was the crossing point between the P–V curve and OD was defined as the conditional load (PQ ), and the PQ corresponded to the conditional initiation of crack [10]. Pmax was the maximum load in the P–V curve. 3. Test results and analysis 3.1. Conditional fracture toughness
Fig. 1. Surface cracked specimen subjected to tension.
In this paper, the Newman–Raju expression [11] was chosen to compute the stress intensity factor (KI ) of surface crack because of its good precision and small dispersibility. The expression of the conditional fracture toughness (KIQ ) was as
J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
469
Table 3 Results of tests No.
B (mm)
a/B
a/C
Pmax /PQ
KIQ (MPam1/2 )
a/(KIQ /RP0.2 )2
(B − a)/(KIQ /RP0.2 )2
T-1 T-2 T-3 T-4 T-5
5.0 5.0 5.0 5.0 5.0
0.502 0.513 0.519 0.504 0.512
0.515 0.571 0.505 0.502 0.513
1.004 1.034 1.050 1.061 1.031
93.86 87.68 96.86 95.58 93.53
0.36 0.447 0.36 0.375 0.400
0.358 0.436 0.333 0.368 0.381
follows: √ M KIQ = σ πa Φ
(2)
where σ = PQ /BW, Φ2 = 1 + 1.464(a/C)1.65 and a 0.89 a 2 M = 1.13 − 0.09 + −0.54 + C 0.2 + aC B a 4 1.0 a 24 + 0.5 − + 14 1.0 − 0.65 + (a/C) C B
4. Ductile fracture toughness
The surface crack fracture toughness (KIe ) is equal to the conditional fracture toughness (KIQ ) if all the following criterions could be satisfied. Pmax ≤ 1.2 PQ KIQ 2 a ≥ 0.50 σ0.2 KIQ 2 (B − a) ≥ 0.50 σ0.2
ment size were so small that the rather large plastic deformation occurred near the crack front. Therefore, the linear elastic fracture mechanics could not apply with these specimens. Under this condition, the elasto-plastic fracture mechanics must be used to measure the ductile fracture toughness.
(3)
(4)
(5)
3.2. Test results and analysis The results of tests were listed in Table 3. It was found that Eq. (3) could be satisfied, but Eqs. (4) and (5) could not be satisfied. The reason was that the specimen thickness and liga-
Kikuchi and some researchers had studied the stress field near surface crack tip, but these studies constrain to the low strength steels [13–15]. In this section, FEM had been conducted to analyze the stress field near surface crack tip for ultra-high strength steel and calculate the J-integral of the surface crack front of the tension specimens under the conditional load achieved in the experiment in Section 2. The value of J-integral where the surface crack was deepest was the maximal, and it was defined as the J-integral value when the crack initiation occurred. So, the maximum value of J-integral of the surface crack front can be seen as the ductile fracture toughness of the material in the engineering evaluation. The surface crack fracture toughness, KIe , can be computed from JIC based on the following equation:
EJIC (6) KIe = (1 − ν2 ) where E is the Young’s modulus of the material and ν is the Poisson’s ratio. 4.1. Finite element model
Fig. 3. Determination of the conditional load (PQ ).
According to the symmetry quality, the one-quarter of tension specimen was selected to carry out the finite element analysis. The finite elements were 20-node quadratic brick, reduced integration elements; one element had 8 integration points. The mesh of whole model for tension specimen with surface crack was presented in Fig. 4. The mesh contained 15,620 elements and 17,808 nodes. The property of 31Si2MnCrMoVE steel was achieved through the uniaxial tension test, and the true stress–strain curve and nominal stress–strain curve were showed in Fig. 5. The stress–strain curve was the basis of this FEM calculation. The three-dimensional finite element solutions were obtained using general purpose finite element software ABAQUS Version 6.5-1 [12]. In the FEA, the applied load was the conditional load obtained in Section 2. The analysis was 3D elastic–plastic analysis.
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J. Zou et al. / Materials Science and Engineering A 454–455 (2007) 467–471
Fig. 7. Distribution of J-integral along the crack front. Table 4 Values of KIe and KIQ Fig. 4. Mesh used in FE-analysis: (a) mesh of 1/4 specimen and region near the crack tip and (b) refined mesh in the crack tip region.
Specimen no.
J-integral (N/mm)
KIe (MPam1/2 )
KIQ (MPam1/2 )
T-1 T-2 T-3 T-4 T-5 Average
52.32 61.60 50.56 45.35 56.94 53.35
109.88 119.23 108.23 102.30 114.63 110.85
93.86 87.68 96.86 95.58 93.53 93.50
value of J-integral was the one where the crack was deepest (Φ = π/2). In this paper, the maximum J-integral value was used as the ductile fracture toughness of the specimen for the tested material. Table 4 gave the JIC values and the corresponding KIe values of each specimen, respectively. Fig. 5. Stress–strain curve of 31Si2MnGrMoVE steel.
4.2. Analysis result Fig. 6 showed the conservation of J-integral for surface crack along eight integrating loop on five cross-sections as the maximum error was 2.30%. Fig. 7 represented the J-integral values of semi-elliptical surface cracks of five specimens on different cross-sections. The cross-section which had the maximum
Fig. 6. Conservation analysis of J-integral.
5. J-dominant analysis If the J-integral was hoped to be used to characterize the intensity of stress–strain field in the elastic–plastic condition, the validity of J-integral must be analyzed when the specimen was loaded with the conditional load. If the HRR fields approximately existed near the crack tip, the condition was called J-dominant. So, the J-integral can be considered effective as a fracture parameter. For a strain hardening material, the strain ε is related to the uniaxial tensile stress σ in the form (Ramberg–Osgood type equation): n ε σ σ = +α (7) ε0 σ0 σ0 where σ 0 is a reference stress (which is related to the yield stress level), ε0 = σ 0 /E the associated reference strain with E as Young’s modulus, α a dimensionless constant and n is the strain hardening exponent. For the 31Si2MnCrMoVE steel, the constant α and the strain hardening exponent n can be fitted from Fig. 4: α = 0.0051, n = 18. Hutchinson [16] and Rice and Rosengren [17] showed that asymptotic field for a crack under monotonically increasing,
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solution, and the exponent of the stress field was nearly equal to the HRR solution. So, the HRR fields approximately existed near the crack front where the surface crack was deepest. It indicated that the J-integral was effective, and J can be used as a fracture parameter for the specimens in this study. Therefore, the surface crack fracture toughness KIe of 31Si2MnCrMoVE steel which was conversed from JIC can be used in the engineering evaluation for the aerospace industry. 6. Conclusion The study showed the linear elastic fracture mechanics cannot be used to evaluate the fracture toughness of thin ultra-high strength steel used in this paper. So, the FEM was utilized to analyze the stress field of specimens and compute the JIC of each specimen. Through the comparison of FEA results and the HRR field, the validity of J-integral was tested. So, the J can be used as a fracture parameter for the 31Si2MnCrMoVE steel tension specimens in this study. Therefore, the surface crack fracture toughness KIe of 31Si2MnCrMoVE steel can be conversed from JIC .
Fig. 8. θ-Variation of stresses at crack tip.
Acknowledgement The research work of this paper is supported by the National Nature Science Foundation of China, No. 50275107; and Fok Ying Tung Education Foundation No. 81405. References
Fig. 9. Stress fields at the bottom of the surface crack (θ = 0).
mode I loading is given by: 1/n+1 EJ σ˜ ij (θ, n) σij (r, θ) = σ0 ασ02 In r
(8)
where In is integration constant, σ˜ ij (θ, n) the dimensionless functions determined by n and angle θ which is defined by the polar coordinate at the crack tip. Model I dimensionless functions σ˜ ij (θ, n) and In have been tabulated by Shih [18] under the plane strain and plane stress conditions. J is the amplitude of the crack tip fields, which are now referred to as HRR fields, in the same spirit that K scales the elastic near-tip fields. Figs. 8 and 9 gave the HRR field and stress field near the crack front from the results of the FEA where the value of J in Fig. 9 was equal to the value of JIC which was computed in Section 4.2 under the conditional load. In Fig. 8, σ θθ (r,θ) and σ rθ (r,θ) and σ rr (r,θ) varied with θ when r/(J/σ 0 ) was equal to 2 and the figure showed that the FEA results and the HRR field were almost identical. Fig. 9 showed that the comparison of the exponent of the stress field with that of the HRR solution when θ was equal to zero. In the HRR solution, the exponent was −1/(n + 1). In Fig. 9, the crack tip field agreed well with the HRR
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