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Pressure vessel steels crack driving force assessment using different models
Journal of Constructional Steel Research 72 (2012) 29–34
Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
Pressure vessel steels crack driving force assessment using different models Goran Vukelic ⁎, Josip Brnic Department of Engineering Mechanics, Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia
a r t i c l e
i n f o
Article history: Received 22 July 2011 Accepted 30 September 2011 Available online 22 October 2011 Keywords: Pressure vessel steel Crack Crack driving force SENB specimen DCT specimen
a b s t r a c t Fracture behavior of two pressure vessel steels, 20MnMoNi55 and 50CrMo4, was numerically investigated in this work. The research was conducted using numerical models of single-edge notched bend (SENB) and disk compact type (DCT) specimens. J-integral, an important fracture mechanics parameter, was chosen as a criterion for the fracture behavior comparison of two mentioned steels. J-integral values were determined using newly developed numerical algorithm coupled with finite element (FE) analysis. Numerically obtained J-integral values are presented as a measure of crack driving force versus crack growth size (Δa) for a range of initial crack sizes (a/W = 0.25, 0.375, 0.5, 0.625). Fracture behavior of two steels was investigated using numerical models of pressure vessels containing also inner axial crack of different sizes (a/t = 0.25, 0.375, 0.5, 0.625). Although J-integral values cannot be transferred from specimens to real structures, results obtained on pressure vessels have proved useful in engineering assessment of fracture behavior of such structures. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Cracks appearing in structures due to imperfections in design, manufacturing, service or maintaining, can exert an influence on their reliability and safety. Therefore, it is important to determine how existing cracks can threat integrity of structures and their components. Fracture mechanics provides theories and parameters useful for such tasks. One of the fracture mechanics parameters is J-integral, suitable for quantifying crack driving force when material ahead of crack tip exhibits elastic-plastic behavior. When dealing with a crack growing from initial length a, J-integral values should be obtained for a number of crack extensions (Δa). Such results can be presented in J-Δa sets of values, i.e. variation of crack driving force during crack extension. Although most of the researches concentrate on determining Jintegral values using standardized specimens and procedures [1], sometimes it is useful to conduct investigation on real structures and their components. This is particularly important during the process of the design optimization. Nowadays, developed numerical methods and finite element (FE) software offer a powerful tool for conducting such investigations. Still, numerically obtained results need to be verified by some kind of experiment. In this paper, fracture behavior of pressure vessel materials was numerically investigated on the FE models of single-edge notched bend (SENB) specimen, disk compact type (DCT) specimen and pressure vessel. For the purpose of determining J-integral, a numerical
⁎ Corresponding author. Tel.: + 385 51 651 560; fax: + 385 51 651 490. E-mail address: [email protected] (G. Vukelic). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.09.015
algorithm was developed that uses FE analysis results as input. Resulting J values are presented as a measure of crack driving force versus crack growth size (Δa) for a range of initial crack lengths, a. Some of the previous works on similar matter include a problem solution study regarding stress intensity factors (SIF) of semi-elliptical cracks located in the stress concentration areas of a pressure vessel [2]. The problem was numerically solved using a global–local FE analysis. The stress field at the crack area varies along the axial, the circumferential, as well as, through-the-thickness directions. Further, the stress triaxiality is an important parameter in explaining the geometry dependence of crack resistance curves [3]. By comparing the so-called stress triaxiality across the ligament of a specimen and a cracked component, it is possible to assess whether the cracked component exhibits similar fracture behavior to the specimen. The influence of crack depth on fracture behavior was considered in a case of 20MnMoNi55 steel [4]. For this purpose crack resistance curves were obtained from specimens pre-cracked to various a/t. An optimal pressure vessels design procedure requires knowledge of their failure behavior in case of existing imperfections or cracks [5]. 2. Theoretical background J-integral is a path-independent integral that can be drawn around the tip of a crack and viewed as both an energy release rate parameter and a stress intensity parameter. It was presented by Rice [6] in a two-dimensional form and with reference to Fig. 1 it can be written as: ∂u J ¼ ∫ Wdy−Ti i ds ; ∂x Γ
ð1Þ
30
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34 Table 1 Chemical composition of considered materials (wt.%). Material
C
Mn
Si
S
Mo
Cr
Ni
P
Rest
20MnMoNi55 50CrMo4
0.2 0.487
1.25 0.735
0.3 0.257
0.05 0.028
0.5 0.185
0.17 0.999
0.6 –
0.01 0.018
96.92 97.29
Table 2 Mechanical properties of considered materials.
Fig. 1. J-integral arbitrary contour path surrounding the crack tip.
where W is strain energy density, Ti = σijnj are components of the traction vector, ui are the displacement vector components and ds is an incremental length along the contour Γ. When applied to FE models, J-integral can be written as [7]:
Material
σYS [MPa]
σTS [MPa]
20MnMoNi55 50CrMo4
490 1090.2
620 1146.9
mentioned materials are given in Table 1 [9, 10], while their mechanical properties (σYS—yield strength, σTS—tensile strength) are given in Table 2. Stress–strain curves for considered materials, later used for determining elastic-plastic behavior of materials, are shown in Fig. 3.
4. Determination of crack driving force using single-edge notched bend (SENB) and disk compact type (DCT) specimen model
! # ( " ∂uy ∂y 1 ∂ux ∂ux ∂uy ∂ux þ σxy þ þ σyy J¼∫ σ 2 xx ∂x ∂y ∂x ∂x ∂y ∂η Γ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #s 2 ) ∂u ∂u ∂x 2 ∂y y x dη: þ σxy n1 þσyy n2 þ − σxx n1 þσxy n2 ∂x ∂x ∂η ∂η
ð2Þ Expression (2) is previously briefly mentioned in an attempt to calculate stress intensity factor using the concept of J-integral for crack propagation in adhesive bonded joints [8], but the application which is to predict crack driving force values for various materials using numerical models of SENB and DCT specimens and pressure vessels was not found. Based on expression (2), a numerical algorithm was developed in Matlab using stress analysis results from integration points of finite elements surrounding crack tip. Evaluating J-integral values in these points and summing them along a path that encloses crack tip, the total value of J has been calculated, Fig. 2. In each example, three different paths around the crack tip have been defined and their average value was taken as final since J-integral may differ in the vicinity and away from the crack tip. 3. Materials under consideration 20MnMoNi55 and 50CrMo4 steels, usually used in manufacturing of pressure vessels, were considered in this work. Compositions of the
First, J values were determined by numerical simulation of experimental single specimen test method defined by ASTM E1820 [1]. Single specimen test method follows elastic unloading compliance technique that uses measured crack mouth opening displacement to estimate the crack size. Collected values can be presented in terms of variation of crack driving force during crack extension. Some of the previous work on similar matter includes establishment of local ductile fracture criterion used in modeling of crack growth and J–R curves simulation [11]. Additionally, using cohesive elements numerical simulation of experimental techniques for J determination was conducted [12]. An attempt to numerically evaluate J integral in cracked sheets had been previously made [13]. Nevertheless, all mentioned researches are void of extensive data that describe and compare fracture behavior of different pressure vessel materials on standardized specimens and real structures. Two-dimensional FE models of SENB and DCT specimens were defined in Ansys, Fig. 4, ensuring plain-strain conditions. Material behavior was considered to be multilinear isotropic hardening type and data extracted from stress–strain curves, Fig. 3, were used to model material behavior. FE mesh consists of 8-node isoparamateric quadrilateral elements. Particular care was taken when meshing area around the crack tip because mesh needs to be fine enough to properly capture stresses and deformations. The specimen was loaded quasi-statically to simulate compliance procedure of single specimen test method. Several initial crack sizes were modeled, a/W = 0.25,
Integ. points
Crack tip
Fig. 2. Single J-integral path Γ surrounding crack tip through finite element integration points.
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
31
σ [MPa]
a
ε [%]
b
σ [MPa]
Fig. 5. 20MnMoNi55 steel: comparison of predicted J values for crack extension Δa using SENB and DCT specimen.
ε [%] Fig. 3. Stress–strain curve for: a) 20MnMoNi55 [9]. b) 50CrMo4 [10].
0.375, 0.5, 0.625, 0.75. A gradual release of node constraints was used to simulate crack propagation. For every crack extension, FE stress analysis results were recorded in the integration points of the elements around the crack tip and these values were used in
Fig. 6. 50CrMo4 steel: comparison of predicted J values for crack extension Δa using SENB and DCT specimen.
b
a
Crack tip
W
a
0.1W
0.2W 2.25W
2.25W
c
d W 0.125
a 0.55 W
W
1.35 W
Crack tip
Fig. 4. a) SENB specimen geometry, b) SENB specimen FE model, c) DCT specimen geometry, d) DCT specimen FE model.
32
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
Fig. 7. Comparison of numerically predicted and experimental resistance curves for SENB specimen made of 20MnMoNi55 steel.
the numerical algorithm that calculates J values. Final J values are shown as a measure of crack driving force versus crack growth size (Δa), Figs. 5 and 6. Having compared obtained J-Δa data for considered steels, equivalent in terms of specimen type and crack size, it can be noted that 20MnMoNi55 steel has higher resulting values of J-integral than 50CrMo4 steel, Figs. 5 and 6. Also, it is evident that higher a/W ratios, i.e. the crack size, correspond with lower J-integral values of materials and vice versa. Besides, J-integral values obtained using FE model of DCT specimen, give somewhat conservative results when comparing with ones obtained using the FE model of SENB specimen. To verify the developed algorithm, numerically predicted J values are compared with available experimental results for SENB specimen made of 20MnMoNi55 steel containing three different initial crack sizes [4], Fig. 7. Good compatibility between experimental values of J for a growing crack in SENB specimen made of 20MnMoNi55 steel and numerically predicted values is noticeable. Such results gave confidence in further application of the J-integral evaluating method and using predicted J values for 50CrMo4 for which experimental results were not available.
a
b
A-A
Fig. 8. Pressure vessel. a) Geometry and dimensions of pressure vessel. b) A detail of the vessel cross section with the crack dimensions.
c/2
det. A
a l
crack front
b
c Fig. 9. FE mesh of: a) pressure vessel sub-model. b) crack. c) crack front from below.
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
33
Table 4 Pressure vessel with crack—comparison of pressure vessel circumferential, εc, and axial strains, εa, obtained numerically and experimentally. Pressure [MPa] 1.6
Circumferential strain, εc [·10− 2]
Axial strain, εa [·10− 2]
Numerically
Experimentally
Numerically
Experimentally
5.115
5.368
1.203
1.263
Fig. 10. Pressure vessel with positioned strain gages and connected to water pump.
5. Comparison of J values for different materials on real structure After validating numerical procedure and developed algorithm on SENB specimens, the same technique was used for determining J values of real structure with a crack. The pressure vessel was modeled according to geometry defined in Fig. 8, using FE software Ansys [14]. It is a standard geometry used in fire extinguishers, manufactured with length L = 385 mm, inner radius Ri = 86 mm and wall thickness t = 2 mm. An initial FE model containing full geometry was used only to obtain displacement values at boundaries sufficiently away from the crack whose length is 2l. At these boundaries, values of stresses and strains are practically identical in both uncracked and cracked body. Using this principle, displacement values were recorded at distance of c = 6l, which has proven to be enough for crack problems [15]. These displacement values were then set on the edges of the sub-model sized c × c. The use of sub-modeling is encouraged because of significant computer memory and process time saving. Further into the analysis, sub-models with different crack sizes were used. Several crack sizes, a, were modeled on the inner edge of pressure vessel wall having thickness t, with ratios a/t = 0.25, 0.375, 0.5, 0.625. The same materials were taken into consideration again (20MnMoNi55 and 50CrMo4). Sub-models of pressure vessel were meshed with 20-node solid elements, Fig. 9. Because of the symmetry, only a quarter of the submodel is modeled with particular care in meshing the crack front. For every example, stress analysis results at constant pressure were recorded in the integration points. The crack growth was again simulated using node releasing technique. With this in mind, it is necessary to ensure that the size of the finite elements correspond to desired measure of crack extension, Δa.
5.1. Tensometric verification of FE model In order to verify suitability of the FE model, tensometric measurements on real life-size pressure vessel made of 50CrMo4 were conducted simultaneously. Measurements were made using HBM-DMCplus apparatus with rectangular strain gage rosettes placed on the pressure vessel
Fig. 11. 20MnMoNi55 steel: J values for crack extension at p = 1.6 MPa.
outer wall. The vessel was pressurized with the manual water pump, Fig. 10. First, a vessel without a crack was experimentally investigated in order to determine strain distribution. Resulting strains at working pressure of p = 1.6 MPa are compared to values obtained numerically and calculated analytically in Table 3. Results show good agreement which verifies suitability of the FE model and encourages the use of its displacement values for the sub-model. Tensometric measurements were also made on a pressure vessel with axial crack. A crack with a length of 2l = 100 mm, opening of d = 2 mm and a depth of a = 1 mm, Fig. 3, was machined in the pressure vessel inner wall with a precise lathe grinder. Strains were recorded on the pressure vessel outer wall with a strain gage positioned along the path of the crack. Resulting strains are presented in Table 4 along with numerical data and they also show good agreement giving confidence in further use of cracked FE sub-model. Experimental values are obtained for pressure p = 1.6 MPa. 5.2. Real structure—results and discussion Stress analysis results, in the integration points of selected elements along the path of the J-integral, are used to calculate the contribution of each element to the total value of J-integral. The path passes through two of the integration points in each element. Three paths were defined and their average value was taken as final. This procedure is repeated for crack extensions and resulting values of J as a measure of crack driving force are plotted against crack extension Δa for pressure vessels made of two considered materials at working pressure p = 1.6 MPa, Figs. 11 and 12. To additionally verify the suitability of the procedure and numerical models, resulting values are compared with those obtained by commercial FE software Ansys that uses Shih formulation [16] for
Table 3 The pressure vessel without crack—comparison of pressure vessel circumferential, εc, and axial strains, εa, obtained analytically, numerically and experimentally. Pressure [MPa] 1.6
Circumferential strain, εc [·10− 4]
Axial strain, εa [·10− 4]
Analytically
Numerically
Experimentally
Analytically
Numerically
Experimentally
2.85
2.752
2.888
0.67
0.648
0.679
34
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
Fig. 13. J values for crack size a/t = 0.5 in correlation to pressure variation. Fig. 12. 50CrMo4 steel: J values for crack extension at p = 1.6 MPa.
J-integral calculation. This similar approach relies on energy domain integral methodology [17] and gives a bit more conservative results, but good compatibility with results obtained from proposed method is still evident. An advantage of the numerical algorithm based on expression (2) is that it is relatively easy to implement and can be used as a supplement to user-developed FE programs. When comparing J-integral values for the same material obtained by using specimen or pressure vessel model, Figs. 5 and 6 with Figs. 11 and 12, a similar trend of J values distribution can be noticed. In comparison with J values obtained on specimen models, 20MnMoNi55 steel has higher resulting values of J-integral than 50CrMo4 steel when comparing their J values on cracked pressure vessel model. Here also higher a/W ratios correspond with lower J-integral values of materials and vice versa. Although it is obvious that J values obtained on specimen models cannot be transferred to real structures, J values calculated on pressure vessels prove useful in the design or inspection process of such vessels. Further, numerical research was conducted for pressure vessel containing inner axial crack of a/t = 0.5 for different pressures. Resulting J-integral in correlation to pressure variation values are presented in Fig. 13. It can be noticed that higher J values correspond with higher pressure when dealing with constant crack size. J values behavior is similar to previous work of other authors when dealing with stationary cracks in pressurized cylinders [18]. 6. Conclusion A numerical algorithm for J-integral calculation based on FE stress analysis results was developed in this work. Obtained results give an insight into the values of crack driving force for 20MnMoNi55 and 50CrMo4 steels using modeled SENB and DCT specimens and pressure vessels, both containing a range of crack sizes. Comparing J values obtained by SENB specimen models with available experimental results, good compatibility is visible, Fig. 7. This gives confidence in using developed algorithm for evaluating J values of 50CrMo4 steel for which experimental results were not available to authors. With this suggested method, extensive experimental procedures can be reduced when having numerical results as a starting point in the investigation of J values for new materials. Such results can be of great help in the process of material selection during the design of structures. This procedure is also used for predicting J values of mentioned materials on models of cracked pressure vessels. The results obtained by developed algorithm correspond to those obtained by commercial FE software.
Although J values obtained on specimen models cannot be transferred into real structures, the results calculated on pressure vessels have proved themselves useful in the designing process of the pressure vessels with dimension similar to those used in this work. Also, they are valuable for inspecting bodies in their process of assessing the condition of pressure vessels in use. The developed algorithm gives an optional mean of J-integral determination, useful as a supplement to in-house built FE programs without that capability.
References [1] ASTM. Standard test method for measurement of fracture toughness, E1820-01. Baltimore: ASTM; 2005. [2] Diamantoudis ATh, Labeas GN. Stress intensity of semi-elliptical surface cracks in pressure vessels by global–local finite element methodology. Eng Fract Mech 2005;72:1299–312. [3] Pavankumar TV, Chattopadhyay J, Dutta BK, Kushwaha HS. Transferability of specimen J–R curve to straight pipes with throughwall circumferential flaws. Int J Press Vess Pip 2002;79:127–34. [4] Narasaiah N, Tarafder S, Sivaprasad S. Effect of crack depth on fracture toughness of 20MnMoNi55 pressure vessel steel. Mater Sci Eng, A 2010;527:2408–11. [5] Eisele U, Schiedermaier J. Application of ductile fracture assessment methods for the assessment of pressure vessels from high strength steels (HSS). Int J Press Vess Pip 2004;81:879–87. [6] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J App Mech 1968;35:379–86. [7] Mohammadi S. Extended finite element method. Singapore: Blackwell Publishing; 2008. p. 56–8. [8] Premchand VP, Sajikumar KS. Fracture analysis in adhesive bonded joints with centre crack. Proc of the 10th National Conference on Technological Trends, NCTT 2009, College of Engineering Trivandrum, Trivandrum, 1; 2009. p. 45–50. [9] Krieg R, Devos J, Caroli C, Solomos G, Ennis PJ, Kalkhof D. On the prediction of the reactor vessel integrity under severe accident loadings (RPVSA). Nucl Eng Des 2001;209(1–3):117–25. [10] Brnic J, Canadija M, Turkalj G, Lanc D. 50CrMo4 steel—determination of mechanical properties at lowered and elevated temperatures, creep behavior and fracture toughness calculation. J Eng Mater Technol 2010;132:021004-1–6. [11] Margolin BZ, Kostylev VI. Analysis of biaxial loading effect on fracture toughness of reactor pressure vessel steels. Int J Press Vess Pip 1998;75(8):589–601. [12] Kozak V, Dlouhy I. J–R curve prediction using cohesive model and its sensitivity to a material curve. Trans of SMiRT, 19; 2007. Toronto. [13] Taroco E, de Las Casas EB, Cimini CA. Fracture mechanics via sensitivity analysis and numerical evaluation of J integral in cracked sheets. J Constr Steel Res 1994;28(2):153–65. [14] Ansys basic analysis guide. SAS IP, Inc; 2005. [15] Diamantakos ID, Labeas GN, Pantelakis SpG, Kermanidis ThB. A model to assess the fatigue behavior of ageing aircraft fuselage. Fatigue Fract Eng Mater Struct 2001;24:677–86. [16] Shih CF, Moran B, Nakamura T. Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fract 1986;30(2):79–102. [17] Anderson TL. Fracture mechanics—fundamentals and application. Boca Raton: CRC Press; 1995. p. 577–85. [18] Brocks W, Noack HD. Elastic-plastic J analysis for an inner surface flaw in a pressure vessel. Exp Mech 1988;28(2):205–9.
Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
Pressure vessel steels crack driving force assessment using different models Goran Vukelic ⁎, Josip Brnic Department of Engineering Mechanics, Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia
a r t i c l e
i n f o
Article history: Received 22 July 2011 Accepted 30 September 2011 Available online 22 October 2011 Keywords: Pressure vessel steel Crack Crack driving force SENB specimen DCT specimen
a b s t r a c t Fracture behavior of two pressure vessel steels, 20MnMoNi55 and 50CrMo4, was numerically investigated in this work. The research was conducted using numerical models of single-edge notched bend (SENB) and disk compact type (DCT) specimens. J-integral, an important fracture mechanics parameter, was chosen as a criterion for the fracture behavior comparison of two mentioned steels. J-integral values were determined using newly developed numerical algorithm coupled with finite element (FE) analysis. Numerically obtained J-integral values are presented as a measure of crack driving force versus crack growth size (Δa) for a range of initial crack sizes (a/W = 0.25, 0.375, 0.5, 0.625). Fracture behavior of two steels was investigated using numerical models of pressure vessels containing also inner axial crack of different sizes (a/t = 0.25, 0.375, 0.5, 0.625). Although J-integral values cannot be transferred from specimens to real structures, results obtained on pressure vessels have proved useful in engineering assessment of fracture behavior of such structures. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Cracks appearing in structures due to imperfections in design, manufacturing, service or maintaining, can exert an influence on their reliability and safety. Therefore, it is important to determine how existing cracks can threat integrity of structures and their components. Fracture mechanics provides theories and parameters useful for such tasks. One of the fracture mechanics parameters is J-integral, suitable for quantifying crack driving force when material ahead of crack tip exhibits elastic-plastic behavior. When dealing with a crack growing from initial length a, J-integral values should be obtained for a number of crack extensions (Δa). Such results can be presented in J-Δa sets of values, i.e. variation of crack driving force during crack extension. Although most of the researches concentrate on determining Jintegral values using standardized specimens and procedures [1], sometimes it is useful to conduct investigation on real structures and their components. This is particularly important during the process of the design optimization. Nowadays, developed numerical methods and finite element (FE) software offer a powerful tool for conducting such investigations. Still, numerically obtained results need to be verified by some kind of experiment. In this paper, fracture behavior of pressure vessel materials was numerically investigated on the FE models of single-edge notched bend (SENB) specimen, disk compact type (DCT) specimen and pressure vessel. For the purpose of determining J-integral, a numerical
⁎ Corresponding author. Tel.: + 385 51 651 560; fax: + 385 51 651 490. E-mail address: [email protected] (G. Vukelic). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.09.015
algorithm was developed that uses FE analysis results as input. Resulting J values are presented as a measure of crack driving force versus crack growth size (Δa) for a range of initial crack lengths, a. Some of the previous works on similar matter include a problem solution study regarding stress intensity factors (SIF) of semi-elliptical cracks located in the stress concentration areas of a pressure vessel [2]. The problem was numerically solved using a global–local FE analysis. The stress field at the crack area varies along the axial, the circumferential, as well as, through-the-thickness directions. Further, the stress triaxiality is an important parameter in explaining the geometry dependence of crack resistance curves [3]. By comparing the so-called stress triaxiality across the ligament of a specimen and a cracked component, it is possible to assess whether the cracked component exhibits similar fracture behavior to the specimen. The influence of crack depth on fracture behavior was considered in a case of 20MnMoNi55 steel [4]. For this purpose crack resistance curves were obtained from specimens pre-cracked to various a/t. An optimal pressure vessels design procedure requires knowledge of their failure behavior in case of existing imperfections or cracks [5]. 2. Theoretical background J-integral is a path-independent integral that can be drawn around the tip of a crack and viewed as both an energy release rate parameter and a stress intensity parameter. It was presented by Rice [6] in a two-dimensional form and with reference to Fig. 1 it can be written as: ∂u J ¼ ∫ Wdy−Ti i ds ; ∂x Γ
ð1Þ
30
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34 Table 1 Chemical composition of considered materials (wt.%). Material
C
Mn
Si
S
Mo
Cr
Ni
P
Rest
20MnMoNi55 50CrMo4
0.2 0.487
1.25 0.735
0.3 0.257
0.05 0.028
0.5 0.185
0.17 0.999
0.6 –
0.01 0.018
96.92 97.29
Table 2 Mechanical properties of considered materials.
Fig. 1. J-integral arbitrary contour path surrounding the crack tip.
where W is strain energy density, Ti = σijnj are components of the traction vector, ui are the displacement vector components and ds is an incremental length along the contour Γ. When applied to FE models, J-integral can be written as [7]:
Material
σYS [MPa]
σTS [MPa]
20MnMoNi55 50CrMo4
490 1090.2
620 1146.9
mentioned materials are given in Table 1 [9, 10], while their mechanical properties (σYS—yield strength, σTS—tensile strength) are given in Table 2. Stress–strain curves for considered materials, later used for determining elastic-plastic behavior of materials, are shown in Fig. 3.
4. Determination of crack driving force using single-edge notched bend (SENB) and disk compact type (DCT) specimen model
! # ( " ∂uy ∂y 1 ∂ux ∂ux ∂uy ∂ux þ σxy þ þ σyy J¼∫ σ 2 xx ∂x ∂y ∂x ∂x ∂y ∂η Γ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #s 2 ) ∂u ∂u ∂x 2 ∂y y x dη: þ σxy n1 þσyy n2 þ − σxx n1 þσxy n2 ∂x ∂x ∂η ∂η
ð2Þ Expression (2) is previously briefly mentioned in an attempt to calculate stress intensity factor using the concept of J-integral for crack propagation in adhesive bonded joints [8], but the application which is to predict crack driving force values for various materials using numerical models of SENB and DCT specimens and pressure vessels was not found. Based on expression (2), a numerical algorithm was developed in Matlab using stress analysis results from integration points of finite elements surrounding crack tip. Evaluating J-integral values in these points and summing them along a path that encloses crack tip, the total value of J has been calculated, Fig. 2. In each example, three different paths around the crack tip have been defined and their average value was taken as final since J-integral may differ in the vicinity and away from the crack tip. 3. Materials under consideration 20MnMoNi55 and 50CrMo4 steels, usually used in manufacturing of pressure vessels, were considered in this work. Compositions of the
First, J values were determined by numerical simulation of experimental single specimen test method defined by ASTM E1820 [1]. Single specimen test method follows elastic unloading compliance technique that uses measured crack mouth opening displacement to estimate the crack size. Collected values can be presented in terms of variation of crack driving force during crack extension. Some of the previous work on similar matter includes establishment of local ductile fracture criterion used in modeling of crack growth and J–R curves simulation [11]. Additionally, using cohesive elements numerical simulation of experimental techniques for J determination was conducted [12]. An attempt to numerically evaluate J integral in cracked sheets had been previously made [13]. Nevertheless, all mentioned researches are void of extensive data that describe and compare fracture behavior of different pressure vessel materials on standardized specimens and real structures. Two-dimensional FE models of SENB and DCT specimens were defined in Ansys, Fig. 4, ensuring plain-strain conditions. Material behavior was considered to be multilinear isotropic hardening type and data extracted from stress–strain curves, Fig. 3, were used to model material behavior. FE mesh consists of 8-node isoparamateric quadrilateral elements. Particular care was taken when meshing area around the crack tip because mesh needs to be fine enough to properly capture stresses and deformations. The specimen was loaded quasi-statically to simulate compliance procedure of single specimen test method. Several initial crack sizes were modeled, a/W = 0.25,
Integ. points
Crack tip
Fig. 2. Single J-integral path Γ surrounding crack tip through finite element integration points.
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
31
σ [MPa]
a
ε [%]
b
σ [MPa]
Fig. 5. 20MnMoNi55 steel: comparison of predicted J values for crack extension Δa using SENB and DCT specimen.
ε [%] Fig. 3. Stress–strain curve for: a) 20MnMoNi55 [9]. b) 50CrMo4 [10].
0.375, 0.5, 0.625, 0.75. A gradual release of node constraints was used to simulate crack propagation. For every crack extension, FE stress analysis results were recorded in the integration points of the elements around the crack tip and these values were used in
Fig. 6. 50CrMo4 steel: comparison of predicted J values for crack extension Δa using SENB and DCT specimen.
b
a
Crack tip
W
a
0.1W
0.2W 2.25W
2.25W
c
d W 0.125
a 0.55 W
W
1.35 W
Crack tip
Fig. 4. a) SENB specimen geometry, b) SENB specimen FE model, c) DCT specimen geometry, d) DCT specimen FE model.
32
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
Fig. 7. Comparison of numerically predicted and experimental resistance curves for SENB specimen made of 20MnMoNi55 steel.
the numerical algorithm that calculates J values. Final J values are shown as a measure of crack driving force versus crack growth size (Δa), Figs. 5 and 6. Having compared obtained J-Δa data for considered steels, equivalent in terms of specimen type and crack size, it can be noted that 20MnMoNi55 steel has higher resulting values of J-integral than 50CrMo4 steel, Figs. 5 and 6. Also, it is evident that higher a/W ratios, i.e. the crack size, correspond with lower J-integral values of materials and vice versa. Besides, J-integral values obtained using FE model of DCT specimen, give somewhat conservative results when comparing with ones obtained using the FE model of SENB specimen. To verify the developed algorithm, numerically predicted J values are compared with available experimental results for SENB specimen made of 20MnMoNi55 steel containing three different initial crack sizes [4], Fig. 7. Good compatibility between experimental values of J for a growing crack in SENB specimen made of 20MnMoNi55 steel and numerically predicted values is noticeable. Such results gave confidence in further application of the J-integral evaluating method and using predicted J values for 50CrMo4 for which experimental results were not available.
a
b
A-A
Fig. 8. Pressure vessel. a) Geometry and dimensions of pressure vessel. b) A detail of the vessel cross section with the crack dimensions.
c/2
det. A
a l
crack front
b
c Fig. 9. FE mesh of: a) pressure vessel sub-model. b) crack. c) crack front from below.
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
33
Table 4 Pressure vessel with crack—comparison of pressure vessel circumferential, εc, and axial strains, εa, obtained numerically and experimentally. Pressure [MPa] 1.6
Circumferential strain, εc [·10− 2]
Axial strain, εa [·10− 2]
Numerically
Experimentally
Numerically
Experimentally
5.115
5.368
1.203
1.263
Fig. 10. Pressure vessel with positioned strain gages and connected to water pump.
5. Comparison of J values for different materials on real structure After validating numerical procedure and developed algorithm on SENB specimens, the same technique was used for determining J values of real structure with a crack. The pressure vessel was modeled according to geometry defined in Fig. 8, using FE software Ansys [14]. It is a standard geometry used in fire extinguishers, manufactured with length L = 385 mm, inner radius Ri = 86 mm and wall thickness t = 2 mm. An initial FE model containing full geometry was used only to obtain displacement values at boundaries sufficiently away from the crack whose length is 2l. At these boundaries, values of stresses and strains are practically identical in both uncracked and cracked body. Using this principle, displacement values were recorded at distance of c = 6l, which has proven to be enough for crack problems [15]. These displacement values were then set on the edges of the sub-model sized c × c. The use of sub-modeling is encouraged because of significant computer memory and process time saving. Further into the analysis, sub-models with different crack sizes were used. Several crack sizes, a, were modeled on the inner edge of pressure vessel wall having thickness t, with ratios a/t = 0.25, 0.375, 0.5, 0.625. The same materials were taken into consideration again (20MnMoNi55 and 50CrMo4). Sub-models of pressure vessel were meshed with 20-node solid elements, Fig. 9. Because of the symmetry, only a quarter of the submodel is modeled with particular care in meshing the crack front. For every example, stress analysis results at constant pressure were recorded in the integration points. The crack growth was again simulated using node releasing technique. With this in mind, it is necessary to ensure that the size of the finite elements correspond to desired measure of crack extension, Δa.
5.1. Tensometric verification of FE model In order to verify suitability of the FE model, tensometric measurements on real life-size pressure vessel made of 50CrMo4 were conducted simultaneously. Measurements were made using HBM-DMCplus apparatus with rectangular strain gage rosettes placed on the pressure vessel
Fig. 11. 20MnMoNi55 steel: J values for crack extension at p = 1.6 MPa.
outer wall. The vessel was pressurized with the manual water pump, Fig. 10. First, a vessel without a crack was experimentally investigated in order to determine strain distribution. Resulting strains at working pressure of p = 1.6 MPa are compared to values obtained numerically and calculated analytically in Table 3. Results show good agreement which verifies suitability of the FE model and encourages the use of its displacement values for the sub-model. Tensometric measurements were also made on a pressure vessel with axial crack. A crack with a length of 2l = 100 mm, opening of d = 2 mm and a depth of a = 1 mm, Fig. 3, was machined in the pressure vessel inner wall with a precise lathe grinder. Strains were recorded on the pressure vessel outer wall with a strain gage positioned along the path of the crack. Resulting strains are presented in Table 4 along with numerical data and they also show good agreement giving confidence in further use of cracked FE sub-model. Experimental values are obtained for pressure p = 1.6 MPa. 5.2. Real structure—results and discussion Stress analysis results, in the integration points of selected elements along the path of the J-integral, are used to calculate the contribution of each element to the total value of J-integral. The path passes through two of the integration points in each element. Three paths were defined and their average value was taken as final. This procedure is repeated for crack extensions and resulting values of J as a measure of crack driving force are plotted against crack extension Δa for pressure vessels made of two considered materials at working pressure p = 1.6 MPa, Figs. 11 and 12. To additionally verify the suitability of the procedure and numerical models, resulting values are compared with those obtained by commercial FE software Ansys that uses Shih formulation [16] for
Table 3 The pressure vessel without crack—comparison of pressure vessel circumferential, εc, and axial strains, εa, obtained analytically, numerically and experimentally. Pressure [MPa] 1.6
Circumferential strain, εc [·10− 4]
Axial strain, εa [·10− 4]
Analytically
Numerically
Experimentally
Analytically
Numerically
Experimentally
2.85
2.752
2.888
0.67
0.648
0.679
34
G. Vukelic, J. Brnic / Journal of Constructional Steel Research 72 (2012) 29–34
Fig. 13. J values for crack size a/t = 0.5 in correlation to pressure variation. Fig. 12. 50CrMo4 steel: J values for crack extension at p = 1.6 MPa.
J-integral calculation. This similar approach relies on energy domain integral methodology [17] and gives a bit more conservative results, but good compatibility with results obtained from proposed method is still evident. An advantage of the numerical algorithm based on expression (2) is that it is relatively easy to implement and can be used as a supplement to user-developed FE programs. When comparing J-integral values for the same material obtained by using specimen or pressure vessel model, Figs. 5 and 6 with Figs. 11 and 12, a similar trend of J values distribution can be noticed. In comparison with J values obtained on specimen models, 20MnMoNi55 steel has higher resulting values of J-integral than 50CrMo4 steel when comparing their J values on cracked pressure vessel model. Here also higher a/W ratios correspond with lower J-integral values of materials and vice versa. Although it is obvious that J values obtained on specimen models cannot be transferred to real structures, J values calculated on pressure vessels prove useful in the design or inspection process of such vessels. Further, numerical research was conducted for pressure vessel containing inner axial crack of a/t = 0.5 for different pressures. Resulting J-integral in correlation to pressure variation values are presented in Fig. 13. It can be noticed that higher J values correspond with higher pressure when dealing with constant crack size. J values behavior is similar to previous work of other authors when dealing with stationary cracks in pressurized cylinders [18]. 6. Conclusion A numerical algorithm for J-integral calculation based on FE stress analysis results was developed in this work. Obtained results give an insight into the values of crack driving force for 20MnMoNi55 and 50CrMo4 steels using modeled SENB and DCT specimens and pressure vessels, both containing a range of crack sizes. Comparing J values obtained by SENB specimen models with available experimental results, good compatibility is visible, Fig. 7. This gives confidence in using developed algorithm for evaluating J values of 50CrMo4 steel for which experimental results were not available to authors. With this suggested method, extensive experimental procedures can be reduced when having numerical results as a starting point in the investigation of J values for new materials. Such results can be of great help in the process of material selection during the design of structures. This procedure is also used for predicting J values of mentioned materials on models of cracked pressure vessels. The results obtained by developed algorithm correspond to those obtained by commercial FE software.
Although J values obtained on specimen models cannot be transferred into real structures, the results calculated on pressure vessels have proved themselves useful in the designing process of the pressure vessels with dimension similar to those used in this work. Also, they are valuable for inspecting bodies in their process of assessing the condition of pressure vessels in use. The developed algorithm gives an optional mean of J-integral determination, useful as a supplement to in-house built FE programs without that capability.
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