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Fracture strength of flawed cylindrical pressure vessels under cryogenic temperatures
Cryogenics 42 (2002) 661–673 www.elsevier.com/locate/cryogenics
Fracture strength of flawed cylindrical pressure vessels under cryogenic temperatures T. Christopher a, K. Sankarnarayanasamy b, B. Nageswara Rao
c,*
a
c
Faculty of Mechanical Engineering, Alagappa Chettiar College of Engineering & Technology, Karaikudi 630 004, India b Faculty of Mechanical Engineering, Regional Engineering College, Tiruchirapalli 620 015, India Structural Engineering Group, Propellants and Special Chemicals Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India Received 30 May 2002; accepted 12 August 2002
Abstract Damage tolerant and fail-safe approaches have been employed increasingly in the design of critical engineering components. In these approaches, one has to assess the residual strength of a component with an assumed pre-existing crack. In other cases, cracks may be detected during service. Then, there is a need to evaluate the residual strength of the cracked components in order to decide whether they can be continued safely or repair and replacement are imperative. A three-parameter fracture criterion is applied to correlate the fracture data on aluminium, titanium and steel materials from test results on cylindrical tanks/pressure vessels at cryogenic temperatures. Fracture parameters to generate the failure assessment diagram are determined for the materials considered in the present study. Failure pressure estimates were found to be in good agreement with test results. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Failure pressure; Aluminium; Titanium; Steels; Cylindrical vessels; Through-wall cracks; Surface cracks; Elastic stress intensity factor; Failure assessment diagram
1. Introduction The aerospace engineering profession has historically sought lighter structures through new materials, design techniques, and manufacturing operations to improve a vehicleÕs performance and propellant consumption. One technology for a fully reusable and permanently space based cryogenic upper stage rocket is the use of liquid hydrogen and liquid oxygen propellants in less than atmospheric condition––a ‘‘low-vapor pressure’’ system. Torre et al. [1] have discussed important issues relevant to structural design of very-thin-gauge tanks; e.g., fracture mechanics, thin-gauge formability, inspection, and weight. In cryogenic tanks, a crack through the thickness, while not of critical size structurally, can be catastrophic because leaking propellants pose a fire/ explosion hazard. Proof tests can establish the leak-free integrity of tanks and therefore verify the absence of through cracks in the tank wall. The choice of material is critical because the low-temperature environment may
*
Corresponding author. Fax: +91-471-415236. E-mail address: bnrao52@rediffmail.com (B. Nageswara Rao).
also reduce ductility and fracture toughness of the tank materials to unacceptably low levels. Fracture mechanics analyses are thus required on all cryogenic tanks to demonstrate that the maximum size of flaw or crack-like defect that could exist after proof testing and nondestructive evaluation will not grow to critical size and cause premature failure during the required service life. Fracture mechanics seeks to relate the three parameters (viz., size of the defect, material toughness, and applied stress), which combine to control the process of fracture. If any two of the three parameters are known then it is possible using the principles of fracture mechanics to estimate the value of the third parameter, which will give rise to failure of the structure. Gordon [2] has reviewed several of the major pipeline codes, which have incorporated fracture mechanics based fitness-forservice concepts. When applying fracture mechanics approaches it is important to adhere to their respective range of applicability. In each particular case a decision has to be made as to whether the assessment is to be founded on the methods of linear elastic fracture mechanics (LEFM), elastic plastic fracture mechanics (EPFM) or plastic collapse. In cases where brittle fracture occurs at low applied stresses, the concept of LEFM can
0011-2275/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 2 ) 0 0 1 1 4 - 5
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Nomenclature a c Do , Di KF , m, Kmax pb pbf pi
crack depth half the crack length outer and inner diameter of cylindrical vessel p three fracture parameters in Eq. (1) stress intensity factor at failure failure pressure of unflawed cylindrical vessel failure pressure of flawed cylindrical vessel internal pressure
be applied, i.e., a stress intensity factor approach [3]. At the other extreme, when the failure mechanism is plastic overload, assessments should be performed using limit load or plastic collapse analyses. Between these two extremes EPFM methods can be applied to assess the integrity of structures. Full-length EPFM analyses are often formidable in terms of time and economy and require a high degree of expertise to implement. A number of engineering methods have been proposed for the assessment of elastic plastic fracture behavior of cracked configurations [4–8]. These include the two-parameter R6 failure assessment method [9], J-integral estimation method [10], and the net section plastic collapse method [11]. In the failure assessment diagram (FAD) or the R6 method [12], the integrity of the structure is assessed and represented in a two-dimensional way: a function of the failure strength pursuant to LEFM is plotted as ordinate and that pursuant to plastic collapse as abscissa. The Dugdale model [13] established the stress limits for any transitional stages between linear elastic failure and plastic collapse. Experiments largely proved these limits to be conservative. The two-parameter fracture criterion (TPFC) of Newman [14,15] too, applies relations derived within the scope of LEFM. In this criterion, the two fracture parameters take account of the deviation of the stress to failure from the stress calculated pursuant to LEFM principles. These parameters have to be determined earlier in pretests, so-called base line tests, to be conducted under identical conditions of material. Keller et al. [16] have carried out fracture analysis of surface cracks in cylindrical vessels applying the TPFC. It was possible neither to determine satisfactorily the failure stresses of vessels by means of fracture parameters obtained from fracture mechanics specimens, nor to predict the loads to failure of the specimens by means of the vesselsÕ fracture parameters. Negeswara Rao and Acharya [17] have developed an empirical relationship between the failure stress and the elastic stress intensity factor at failure by means of three fracture parameters successfully and correlated the fracture data of Ref. [16]. However, systematic validation of this criterion to understand its conservatism based on experimental results is still very limited.
R t W rf ru rult rys
inner radius of cylinder thickness of SCT specimen/cylinder specimen width failure stress nominal stress required to produce a fully plastic region on the net section ultimate tensile strength yield strength or 0.2% proof stress
Zerbst et al. [18] have applied the recently developed European flaw assessment procedure structural integrity assessment procedures for European industry (SINTAP) to the published fracture data on steel pipes having through-wall and surface cracks subjected to internal pressure. The SINTAP procedure offers a crack driving force (CDF) and a FAD route. Both are complementary and give identical results. In the CDF route the determination of the crack tip loading in the component and its comparison with the fracture resistance of the material are two separate steps. In contrast to this philosophy, in the FAD route a failure line is constructed by normalizing the crack tip loading by the materialÕs fracture resistance. The assessment of the component is then based on the relative location of an assessment point with respect to this failure line. The predicted failure pressures were checked against a set of published experimental data. The differences from the experimental failure pressures were found to vary between 20% and 70%. They reported that the reason for this spread is due to their lower bound estimations of fracture toughness of the material from the Charpy energy values through empirical relations. They have applied different analysis levels of SINTAP by considering another set of fracture data and claimed that the real load carrying capacity was under estimated by about 20% or less for the higher analysis levels which demand high quality of the stress–strain curves of the material in the yield range. The extraordinary success of fracture mechanics lies in its ability to combine a theoretical framework with experimentally measured critical quantities. The objective of the present study is to consolidate the fracture data on various materials generated at cryogenic temperatures from cylindrical pressure vessels containing cracks, and examine how well the developed threeparameter fracture criterion correlates the test data.
2. Three-parameter fracture criterion The significant parameters affecting the size of a critical crack in a structure are the applied stress levels, the fracture toughness of the material, the location of
T. Christopher et al. / Cryogenics 42 (2002) 661–673
the crack and its orientation. Since the intensity of the stress at the crack-tip, K is a function of load, geometry and crack size, it is more appropriate to have a relationship between the stress intensity factor at failure (Kmax ) and the failure stress (rf ) from the fracture data of cracked specimens for the estimation/prediction of the fracture strength to any cracked configurations. The relationship between Kmax and rf can be of the form [17,19] p rf rf Kmax ¼ KF 1 m ð1 mÞ : ð1Þ ru ru Here, rf is the failure stress normal to the direction of crack in a body and ru is the nominal stress required to produce a fully plastic region (or hinge) on the net section. For center crack tension specimens, ru is equal to the ultimate strength (rult ) of the material. For cylindrical pressure vessels, ru is the hoop stress at the burst pressure level of the unflawed thin cylindrical shell. KF , m and p are the three fracture parameters to be determined from the fracture data. Kmax in Eq. (1) for center surface crack tension (SCT) specimens and cylindrical vessels with an axial surface crack (see Fig. 1) from the stress intensity factor expressions obtained from the finite element solutions [20,21] are given below: Kmax ¼ rf ðpaÞ rf ¼ rmax ¼ ¼
pbf R t
1=2
M=/;
Pmax tW
for SCT specimens;
for cylindrical pressure vessels;
ð2Þ
M¼
663
rffiffiffi pffiffi rffiffiffi a 2pffiffip c a p c fw M1 þ / þ / ðM2 1Þ M1 a t a t
for SCT specimens; ¼ Me fs for cylindrical pressure vessels;
ru ¼ rult for SCT specimens; pb R for cylindrical pressure vessels; ¼ t a 1:65 /2 ¼ 1 þ 1:464 for a 6 c; c c 1:65 ¼ 1 þ 1:464 for a > c; a a M1 ¼ 1:13 0:1 for a 6 c; c n c o c 1=2 for a > c; ¼ 1 þ 0:03 a a p 1=2 M2 ¼ for a 6 c; 4 c p 1=2 ¼1þ 1 for a > c; a 4 rffiffiffi c a q Me ¼ M1 þ / ; M1 a t sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc a 1=2 fw ¼ sec ; W t
1=2 fs ¼ 1 þ 0:52ks þ 1:29k2s 0:074k3s
Fig. 1. Cracked configurations.
for 0 6 ks 6 10;
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c a ks ¼ pffiffiffiffiffi ; t Rt
T. Christopher et al. / Cryogenics 42 (2002) 661–673
q¼2þ8
a 3 c
;
a is the depth and c is half the crack length of a surface crack, W is the width of the plate, R is the radius of the cylindrical vessel, t is the thickness, Pmax is the maximum tensile load, pb is the general collapse or bursting pressure of an unflawed cylindrical vessel, pbf is the failure pressure of a cylindrical vessel with an axial surface crack. When the depth (a) of the crack equals to the thickness (t), Kmax corresponds to the maximum stress intensity factor for through cracked specimens. A comparative study in Ref. [22] on failure pressure estimation of unflawed cylindrical vessels indicates the validity of the following FaupelÕs formula for steel vessels: h 2 rys ti pb ¼ pffiffiffi rys 2 ð3Þ ln 1 þ ; R rult 3 where, rys is the 0.2% proof stress or the yield strength of the material. Using Eqs. (1) and (2), one can set up the following equation to determine the nominal failure stress (rf ) for a specified crack size: ) p ( 1=2 rf ru ðpaÞ M rf þ mþ ð1 mÞ 1 ¼ 0: /KF ru ru ð4Þ The non-linear Eq. (4) is solved for rf using the Newton–Raphson iterative method. Failures of materials properly understood are, among the most important links in the design process. When dealing with a specific material for a particular application, it is not clearly established whether KIC (plane strain fracture toughness) or KC (fracture toughness when plane strain conditions are not met) values should be used. The values of KIC seem to be relevant in heavy sections like forgings or thick plates. Past experience suggests that design based on KIC will be unreasonably conservative in conventionally thick-sectioned structural members in aerospace application. In such circumstances it is necessary to carry out what is called KC tests corresponding to the thickness of intended structural application. The value of KC can be determined from the point of tangency between the crack growth resistance curve (R-curve) and the CDF curve appropriate for the loading geometry. ASTM-E561 standards suggest generation of a R-curve from through-crack test coupons like compact tension specimens, etc. It should be noted that KC is geometry dependant whereas the R-curve is considered to be a material property, independent of geometry (except thickness). Thus, the R-curve of the material will be useful for accurate determination of the critical load of the through cracked configurations. For part-through cracked configurations, fracture strength
estimations are not possible directly from the R-curve of the material because the part-through crack has two dimensions, namely, crack length and its depth. In such situations, the relationship between the failure stress (rf ) and the stress intensity factor (Kmax ) at failure by means of three fracture parameters (KF , m and p) in Eq. (1), will be useful for fracture strength evaluation of cracked configurations. In fact, Eq. (1) developed is following the basic concepts of the NewmanÕs TPFC [14]. The TPFC equation: rf Kmax ¼ KF 1 m ð5Þ ru was derived using two approaches. In the first approach the stress concentration factor for an ellipsoidal cavity was used with NeuberÕs equation to derive a relation between the local elastic–plastic stresses and strains, and remote loading. Assuming that fracture occurred when the notch-root stress and strain was equal to the fracture stress and strain, and that a crack has critical notch-root radius, the TPFC equation was derived. The second parameter, m, came from the unity term in the stress concentration equation. The second approach used the stress field equation for a crack and NeuberÕs equation to relate the elastic stresses to the elastic–plastic stresses and strains at a crack-tip. In this approach, it was again assumed that the fracture occurred at a critical distance ahead of the crack tip when the local stress and strain was equal to the fracture stress and strain. The second parameter, m, came from the next higher-order term in the stress-field expansion. The details on the derivation of TPFC Eq. (5) are given in Ref. [15]. It is well known fact that the tensile strength, rf , of a specimen decreases with increasing crack size. If rf < rys , then there exists a linear relationship between rf and Kmax . For small sizes of cracks where rys < rf < ru , the relationship between rf and Kmax is expected to be nonlinear. The idea of expressing Kmax as a function of rf in a single expression (1), is mainly for estimation of failure strength of a cracked body whether it contains throughthickness or part-through cracks which are small or large in size. In the absence of cracks, rf ! ru and Kmax ! 0. To account this limiting condition, the expression (1) is more appropriate. The exponential form of the third term in Eq. (1) is preferred to represent the non-linear variation of Kmax with rf , when rf > rys . An empirical relation is derived in Ref. [19] for the third fracture parameter, p, in terms of the second fracture parameter, m, as p¼
1 ð 1 þ fÞ 2 1 1 1 pffiffiffi ln 1 pffiffiffi ð1 þ fÞ þ 21 m ; ð1 mÞ f 2 2 ln
1
ð6Þ
T. Christopher et al. / Cryogenics 42 (2002) 661–673
where f¼
4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 3 þ 9 8m
ð7Þ
With this empirical relation (6), one needs to evaluate only KF and m in Eq. (1) utilizing the fracture strength values from two cracked configurations. To account the scatter in the experimental data, test results from more number of cracked configurations should be fitted in Eq. (1) for obtaining the material parameters. The material parameter, m, in general, is greater than zero and less than unity. If m is found to be less than zero due to large scatter in the fracture data, the parameter, m, has to be set to zero and the average of Kmax from the fracture data, yields the parameter, KF and the third parameter, p, from Eq. (6) gives the value close to 12. When, m is close to unity, the third term in Eq. (1) becomes insignificant. Whenever, m is found to be greater than unity, the parameter, m, has to be set to 1, by suitably modifying the parameter, KF , with the fracture data. If the fracture strength data is less than the yield strength of the material, KF and m in Eq. (1) can be obtained by fitting the fracture data in Eq. (1) and neglecting the third term on the right hand side of Eq. (1). The third parameter, p, is obtained using Eq. (6). If the fracture strength values are higher than the yield strength, one has to obtain KF value in an iterative process by specifying m (between 0 and 1), evaluating p from Eq. (6), and fitting the fracture data in Eq. (1). This iterative process should continue till Eq. (1) satisfactorily correlates the fracture data with the obtained fracture parameters, KF , m and p. Since there is no unique fracture criterion applicable for all materials, one has to select/establish a criterion suitable for the intended material. Generation of fracture data on different materials for validation of the selected fracture criterion is a difficult task for a single
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agency. In such circumstances, one has to rely on the published experimental data. An attempt is made here to correlate the fracture data on pressure vessels tested at cryogenic temperatures with the developed Eq. (1). For ready reference, expressions are provided for the stress intensity equation to cylindrical pressure vessels, which will be useful for analysts to verify the accuracy in their finite element models/solutions.
3. Results and discussion To examine the accuracy of the empirical relation (3) while estimating the collapse or bursting pressure of unflawed vessels, test data in Refs. [23,24] are considered. The relative error (%) is computed as Analysis result Relative error ðREÞ % ¼ 100 1 ; Test result ð8Þ A standard error (SE) is computed as " SE ¼
N X i¼1
Analysis result 1 Test result
2 , #1=2 ; N
ð9Þ
i
where N is the number of test specimens. Here a positive RE indicates an under-estimate while a negative RE implies an un-conservative over-estimate. Tables 1 and 2 give comparison of analytical and experimental failure pressure values of maraging steel and 15CDV6 steel rocket motor cases. The analytical results indicated that FaupelÕs formula (3) is accurate to 5%. Hrisafovic and Ledy [25] have performed tensile fracture tests on AZ5G aluminium alloy center surface crack specimens having 50 mm width and 4.7 mm thickness. The sizes of cracks in the weldment of
Table 1 Comparison of analytical and experimental failure pressure of maraging steel rocket motor cases rys (MPa) 2128 2128 2128
rult (MPa) 2155 2155 2155
Outer diameter, Do (mm) 93 94 93
Thickness, t (mm)
Failure pressure, pb (MPa)
1.6 1.8 1.7
Test [23]
Eq. (3)
Relative error (%)
87 94 92
88.5 97.2 94.1
)1.7 )3.4 )2.3
Table 2 Comparison of analytical and experimental failure pressure of 15CDV6 steel rocket motor cases rys (MPa)
rult (MPa)
Outer diameter, Do (mm)
Thickness, t (mm)
Failure pressure, pb (MPa) Test [24]
Eq. (3)
Relative error (%)
915 915 915
1060 1060 1060
212 212 162
2.6 2.6 3.8
29 30 58
29.9 29.9 56.9
)3.3 )0.9 2.5
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T. Christopher et al. / Cryogenics 42 (2002) 661–673
specimens are: crack length is five times to its depth and depth of the crack varies up to the thickness of the specimens. The fracture parameters (KF , m and p) in Eq. (1) are obtained from the rupture curve of Ref. [25]. The ultimate tensile strength of this material is close to the achieved weld properties of Afnor 7020 aluminium alloy [26]. Hence, the fracture parameters of AZ5G aluminium alloy are used in Eq. (4) for evaluating the fracture strength of Afnor 7020 aluminium alloy center SCT specimens. The analysis results in Table 3 are found to be in good agreement with the test results of Ref. [26]. The ultra high strength aluminium alloy Afnor 7020 sheets and forgings are used to fabricate liquid propellant and water tanks in upper stages of large size launch vehicles. Fracture data on steel cylinders having an axial surface crack under different pressure medium reported in Ref. [16] is examined. Table 4 gives failure pressure estimates within 15% of the test results. Peters and Kuhn [27] have generated fracture data from cylindrical pressure vessels made of 7075-T6 and
2024-T3 aluminium alloys. In the pressure test, cylinders were pressurized with air and with oil. The results indicate that the pressurizing medium has a negligible effect on the bursting pressure. The fracture parameters (KF , m and p) in Eq. (1) are determined. Tables 5 and 6 give comparison of failure pressure estimates with test results. It can be seen from Fig. 2 that most of the analytical results are within 10% of the test results. The large discrepancy in few cases is mainly due to scatter in test results. It is very interesting to note from Table 6 that values of the fracture toughness parameter, KF are found to decrease with increase in thickness. Anderson and Sullivan [28] have generated fracture data at cryogenic temperatures from cylindrical pressure vessels made of aluminium alloy (2014-T6 Al) and extralow-interstitial (ELI) titanium alloy (5Al–2.5 Sn–Ti). Tables 7 and 8 give fracture parameters for these materials. It can be seen from Figs. 3 and 4 that most of the failure pressure estimates are within 10% of the test results. The major discrepancy in the results of Table 8 is mainly due to scatter in test results.
Table 3 Comparison of analytical and experimental fracture strength results for AZ5G and Afnor 7020 aluminium alloy tensile specimens containing surface pffiffiffiffi cracks: (KF ¼ 83:7 MPa m; m ¼ 0:76; p ¼ 29:2) Specimen dimensions (mm)
Crack dimensions (mm)
W
a
2c
Test [25,26]
Analysis
MPa (Weldment) 0.47 0.94 1.41 1.88 2.35 2.82 3.29 4.23 4.70
2.35 4.70 7.05 9.40 11.75 14.10 16.45 21.15 23.50
344.8 340.6 333.7 324.0 311.5 296.2 278.1 233.7 207.3
337.2 330.7 322.4 311.0 296.6 281.0 265.4 234.6 219.3
2.2 2.9 3.4 4.0 4.8 5.1 4.6 )0.4 )5.8
Afnor 7020 aluminium alloy [26]: rys ¼ 332:4 MPa; rult ¼ 376:1 MPa (Parent Metal) 75 5.10 2.55 11.75 345.6 75 4.28 2.78 13.23 332.1 75 5.09 2.80 13.91 338.7 75 5.80 2.90 13.14 339.0 75 5.80 2.96 12.98 330.3 75 5.16 2.99 15.03 329.8 75 5.10 3.01 17.97 319.4 75 5.81 3.02 14.30 332.0 75 5.81 3.08 15.09 326.0 75 5.85 3.22 13.71 335.8
313.0 297.9 301.4 306.2 305.9 295.0 285.7 300.7 297.3 299.9
9.4 10.3 11.0 9.7 7.4 10.6 10.5 9.4 8.8 10.7
Afnor 7020 aluminium alloy [26]: rys ¼ 302:1 MPa; rult ¼ 343:4 MPa (Weldment) 75 4.34 2.56 14.59 321.5 75 4.73 2.65 14.59 284.0 75 5.67 2.95 10.99 303.6 75 4.84 2.95 11.29 321.1 75 5.19 3.06 13.94 306.8 75 5.87 3.17 14.64 295.6 75 5.33 3.89 20.32 256.7
281.9 283.1 294.7 289.8 281.4 281.4 252.9
12.3 0.3 3.0 9.8 8.3 4.8 1.4
t
AZ5G aluminium alloy [25]: rult ¼ 346:2 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70
Standard error ¼ 0:075
Fracture strength, rf (MPa)
Relative error (%)
T. Christopher et al. / Cryogenics 42 (2002) 661–673
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Table 4 Failure pressure estimates of steel cylinders (Do ¼ 560 mm) having an axial surface crack Strength properties rys (MPa)
rult (MPa)
Thickness, t (mm)
Pressure medium: water þ air cushion at 293 K (KF ¼ 462:9 798 922 18.4 778 925 18.0 703 847 18.0 751 886 17.8 878 990 20.4 866 979 21.7 813 944 17.6
Crack dimensions
Failure pressure, pbf (MPa)
a (mm) 2c (mm) pffiffiffiffi MPa m; m ¼ 0:823; p ¼ 36:8) 16.8 218 9.3 144 11.6 215 15.8 150 16.1 96 14.5 65 14.6 65
Test [16]
Analysis
21.57 42.17 31.09 32.75 49.03 54.42 47.76
21.80 42.46 30.19 27.75 46.14 59.29 42.87
)1.1 )0.7 2.9 15.3 5.9 )9.0 10.2
45.30 40.01 43.83 36.58 48.05 55.31
44.45 34.04 46.95 41.79 43.90 48.73
1.9 14.9 )7.1 )14.2 8.6 11.9
27.65 28.44
27.41 26.56
0.9 6.6
Relative error (%)
Standard error ¼ 0:081 pffiffiffiffi Pressure medium: water þ glycol at 253 K (KF ¼ 426:1 MPa m; m ¼ 0:6; p ¼ 20:4) 859 982 17.5 13.0 94 853 973 18.4 14.7 155 842 985 18.5 10.7 143 830 984 17.7 9.0 215 726 879 17.8 10.0 142 843 976 18.7 13.5 93 Standard error ¼ 0:108 pffiffiffiffi Pressure medium: air at 293 K (KF ¼ 407:1 MPa m; m ¼ 1:0) 831 947 17.7 13.1 832 951 17.6 11.6
160 205
Standard error ¼ 0:047
Table 5 Failure pressure estimates of aluminium alloy (7075-T6 Al) cylinders having an axial through crack (Do ¼ 182:9 mm; rys ¼ 450 MPa; pffiffiffiffi rult ¼ 550 MPa; KF ¼ 64:2 MPa m; m ¼ 1:0) Thickness, t (mm)
Crack length, 2c (mm)
Failure pressure, pbf (MPa) Test [27]
Analysis
0.41 0.41 0.41 0.64 0.64 0.64
16.76 32.77 65.02 25.40 51.31 101.60
0.59 0.36 0.17 0.79 0.40 0.18
0.659 0.343 0.157 0.797 0.377 0.172
Relative error (%) )11.6 4.7 7.6 )0.9 5.6 4.7
Standard error ¼ 0:067
Pressure vessels are often fabricated by starting with flat sheet material and producing the required curvature in the sheet by one of a number of forming methods. Most of the methods result in the retention of residual stresses in the formed sheet. Pressure vessels with small radii of curvature have higher residual stresses than those with larger radii if they are made from flat sheet of the same thickness and are not subjected to an annealing or stress-relieving process. Thus, if material properties determined for smaller tanks are to be applied to larger vessels, the effect of the residual stresses on the results should be determined. Also, the effect of stress-relief treatment on the failure pressures or stresses in tanks
containing flaws is of interest. To obtain some insight into the effect of residual stress on the fracture strength of vessels containing through cracks, tanks were fabricated from AISI 301 stainless steel and 5Al–2.5Sn–Ti ELI titanium. These tanks, containing through-cracks of various lengths, were pressurized to the burst point [29]. The tanks which were fabricated from AISI 301 stainless-steel 60% cold-reduced material were formed by spiral wrapping nominally 0.56 mm thick sheet to form an 11° helix which was then butt-welded (see Fig. 5). The cylinders were nominally 152 mm in diameter and 457 mm in length. The AISI 301 stainless steel tanks were tested in the non-stress-relieved and 50% stressrelieved conditions. Those tanks, which were stressrelieved, were heated for 8 h at 673 K and furnace cooled. The 5Al–2.5Sn–Ti ELI titanium tanks were also 152 mm in diameter and 457 mm in length and were fabricated from nominally 0.51 mm thick sheet using a single longitudinal butt weld (see Fig. 5). The material was mill annealed at 991 K for 4 h and furnace cooled. To prevent contamination of, and possible explosive damage to the cryogenic test facility, each cylinder was completely disassembled and thoroughly cleaned following the fatigue-cracking process. Before testing the cylinders in liquid hydrogen or liquid nitrogen, it was planned to close the cylinder ends and seal the through crack. The seal used was a
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T. Christopher et al. / Cryogenics 42 (2002) 661–673
Table 6 Failure pressure estimates of aluminium alloy (2024-T3Al) cylinders having an axial through crack (Do ¼ 182:9 mm, rys ¼ 250 MPa, rult ¼ 450 MPa) Crack length, 2c (mm) t ¼ 0:15 mm (KF 11.94 24.38 49.78
Failure pressure, pbf (MPa) Test [27] Analysis pffiffiffiffi ¼ 148:3 MPa m; m ¼ 1:0) 0.31 0.386 0.24 0.229 0.12 0.116
Relative error (%) )24.4 4.5 3.5
Standard error ¼ 0:145 t ¼ 0:30 mm (KF 6.10 6.10 12.70 12.70 24.38 24.38 24.38 47.50 97.03 24.89 50.29 102.10 24.38 48.51 97.28
pffiffiffiffi ¼ 124 MPa m; 0.92 0.95 0.70 0.61 0.46 0.47 0.48 0.25 0.11 0.53 0.29 0.13 0.52 0.23 0.12
m ¼ 1:0) 1.010 1.010 0.763 0.763 0.488 0.488 0.488 0.258 0.126 0.479 0.243 0.121 0.488 0.252 0.126
)10.0 )6.5 )9.0 )25.1 )6.1 )3.8 )1.6 )3.2 )14.8 9.5 16.2 6.9 6.2 )9.8 )5.0
Standard error ¼ 0:106 t ¼ 0:38 mm (KF 15.75 30.48 60.96
pffiffiffiffi ¼ 112 MPa m; 0.84 0.49 0.25
m ¼ 0:94; p ¼ 91:8) 0.863 )2.7 0.510 )4.1 0.250 0.2
Standard error ¼ 0:028 pffiffiffiffi t ¼ 0:64 mm (KF ¼ 83:7 MPa m; m ¼ 0:4827; p ¼ 17:3) 26.16 1.05 1.060 )0.9 50.80 0.50 0.509 )1.9 143.00 0.16 0.159 0.5 Standard error ¼ 0:012
composite patch designed to ensure proper sealing without reinforcing the area surrounding the crack. A layer of 0.05 mm Mylar pressure-sensitive tape was placed over the crack from inside the tank. A piece of 0.25 mm Mylar matter was placed over the layer of tape. To prevent leaking during pressurization or local effects on the crack, the Mylar patch was made large enough to cover the original crack plus an additional amount to allow for stable crack growth. Three additional overlapping layers of Mylar tape completed the patch. A cylindrical insert, which reduced the volume of liquid hydrogen, was used in the test tank. The insert minimized damage to the test specimen and test facility at failure. After assembly the specimen was placed in a cryostat, submerged in and filled with the desired
Fig. 2. Comparison of analytical and experimental failure pressure of aluminium alloy (7075-T6Al, 2024-T3Al) cylindrical vessels having an axial through crack. Table 7 Failure pressure estimates of aluminium alloy (2014-T6Al) cylinders having an axial through crack (Do ¼ 142:2 mm, t ¼ 1:52 mm) Crack length, 2c (mm)
Failure pressure, pbf (MPa) Test [28]
At 298 K: rys ¼ 470 MPa; rult m ¼ 0:691; p ¼ 24:3 2.90 9.52 12.70 5.02 25.40 3.04 50.80 1.43
Relative error (%) Analysis pffiffiffiffi ¼ 545 MPa; KF ¼ 60:3 MPa m; 8.94 4.95 2.93 1.42
6.1 1.3 3.5 0.6
Standard error ¼ 0:036 pffiffiffiffi At 78 K: rys ¼ 520 MPa; rult ¼ 610 MPa; KF ¼ 71:5 MPa m; m ¼ 0:677; p ¼ 23:6 2.87 10.58 10.43 1.5 3.81 10.43 9.64 7.5 5.08 9.41 8.80 6.5 6.35 8.64 8.12 6.0 7.62 7.71 7.54 2.2 10.16 7.00 6.58 6.0 12.70 5.92 5.81 1.8 19.05 4.46 4.41 1.2 25.40 3.41 3.46 )1.5 31.75 2.75 2.80 )1.7 44.45 2.13 1.96 8.1 50.80 1.67 1.68 )0.7 Standard error ¼ 0:046 pffiffiffiffi At 20 K: rys ¼ 560 MPa; rult ¼ 680 MPa; KF ¼ 68:9 MPa m; m ¼ 0:601; p ¼ 20:4 2.64 12.15 11.72 3.6 6.35 9.37 8.65 7.7 12.70 5.85 6.02 )2.9 19.05 4.73 4.48 5.3 25.40 3.10 3.48 )12.1 31.75 2.93 2.79 4.9 44.45 1.95 1.93 1.0 50.80 1.76 1.65 6.1 Standard error ¼ 0:063
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Table 8 Failure pressure estimates of ELI titanium alloy (5Al–2.5Sn–Ti) cylinders having an axial through crack (Do ¼ 152:4 mm, t ¼ 0:51 mm) Crack length, 2c (mm)
Failure pressure pbf (MPa) Test [28]
At 78 K: rys ¼ 1200 MPa; rult m ¼ 0:763, p ¼ 29:5 3.18 8.75 6.12 7.58 5.82 7.20 11.18 5.31 11.51 4.83 18.62 3.90 19.43 3.43 24.03 3.30 25.17 3.04 37.59 2.03 37.08 1.65
Relative error (%) pffiffiffiffi ¼ 1400 MPa; KF ¼ 274:8MPa m; Analysis
8.11 6.80 6.92 5.18 5.09 3.63 3.51 2.91 2.78 1.85 1.87
7.3 10.3 3.9 2.5 )5.4 6.8 )2.3 12.0 8.5 9.1 )13.5
Standard error ¼ 0:082 pffiffiffiffi At 20 K: rys ¼ 1525 MPa; rult ¼ 1675 MPa; KF ¼ 228:2 MPa m; m ¼ 1:0 2.36 7.88 7.64 3.0 3.94 7.40 6.82 7.9 4.83 6.13 6.44 )5.0 7.04 5.58 5.63 )0.9 7.11 5.25 5.61 )6.8 13.28 3.89 4.01 )3.0 12.24 3.49 4.23 )21.1 20.40 2.92 2.87 1.9 23.90 2.90 2.48 14.5 19.35 2.83 3.00 )6.0 25.02 2.37 2.37 )0.1 40.64 1.87 1.43 23.8 39.62 1.73 1.47 15.2
Fig. 4. Comparison of analytical and experimental failure pressure of ELI titanium alloy (5Al–2.5Sn–Ti) cylindrical vessels having an axial through crack.
Standard error ¼ 0:113
Fig. 5. AISI 301 stainless steel spiral-welded and titanium pressure vessels.
Fig. 3. Comparison of analytical and experimental failure pressure of aluminium alloy (2014-T6Al) cylindrical vessels having an axial through crack.
cryogen, and pressurized to burst. Gaseous helium was used as the pressurizing medium. Tables 9 and 10 give fracture parameters for these materials. Failure pressure estimates are found to be within 10% of the test results. Figs. 6 and 7 show the FADs for these materials. The applicability of the three-parameter fracture criterion is examined by considering the published fracture
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Table 9 Failure pressure estimates of AISI 301 stainless steel, 60% cold-reduced, 152 mm diameter tanks having axial through cracks Cylinder dimensions (mm) Do (Non-stress relieved) at 20 K: rys 153.0 152.4 153.0 153.0 153.0 152.4 151.8 152.4
Crack length, 2c (mm)
Failure pressure, pbf (MPa)
t Test [29] Analysis pffiffiffiffi ¼ 1830 MPa; rult ¼ 2220 MPa; KF ¼ 201:2 MPa m; m ¼ 0:614; p ¼ 20:9 0.564 4.34 10.32 9.95 0.561 4.78 9.79 9.52 0.566 10.57 6.33 6.06 0.597 10.80 6.52 6.35 0.584 17.93 3.90 4.02 0.577 24.43 3.03 2.89 0.584 37.52 1.97 1.80 0.587 37.64 1.65 1.80
Relative error (%)
3.6 2.8 4.3 2.6 )3.0 4.7 8.5 )9.4
Standard error ¼ 0:054 pffiffiffiffi MPa; KF ¼ 242:2 MPa m; m ¼ 0:942; p ¼ 95:2 4.83 9.55 9.27 10.77 6.59 6.33 17.68 3.89 3.90 24.82 2.93 3.07 37.26 1.91 1.80
2.9 4.0 )0.4 )4.8 5.7
pffiffiffiffi ¼ 1650 MPa; rult ¼ 2270 MPa; KF ¼ 208:1 MPa m; m ¼ 0:64; p ¼ 21:9 0.572 4.09 10.43 10.53 0.587 4.30 10.33 10.54 0.559 9.91 6.65 6.37 0.592 11.99 6.71 5.94 0.594 18.29 4.25 4.13 0.589 22.38 3.15 3.33 0.587 36.88 1.86 1.90
)1.0 )2.1 4.2 11.5 2.8 )5.8 )2.0
(Stress-relieved) at 20 K; rys ¼ 1630 MPa; rult ¼ 2250 152.4 0.587 153.4 0.587 153.0 0.533 152.4 0.566 153.4 0.528 Standard error ¼ 0:040 (Non-stress relieved) at 77 K: rys 152.4 153.4 153.4 153.0 153.0 153.4 153.4 Standard error ¼ 0:053
Table 10 Failure pressure estimates of ELI titanium alloy (5Al–2.5Sn–Ti), 152 mm diameter non-stress relieved tanks having axial through cracks Cylinder dimensions (mm) Do
t
Crack length, 2c (mm)
Failure pressure, pf (MPa) Test [29]
Relative error (%)
Analysis
pffiffiffiffi At 20 K: rys ¼ 1400 MPa; rult ¼ 1560 MPa; KF ¼ 143:3 MPa m; m ¼ 0:802; p ¼ 33:8 152.4 0.503 5.46 5.25 5.11 2.6 153.0 0.493 17.53 2.48 2.25 9.2 152.4 0.483 18.49 2.16 2.09 3.2 153.4 0.447 24.31 1.34 1.42 )6.1 152.4 0.503 38.13 1.02 1.01 1.5 Standard error ¼ 0:053 pffiffiffiffi At 77 K: rys ¼ 1190 MPa; rult ¼ 1240 MPa; KF ¼ 214:2 MPa m; m ¼ 0:623; p ¼ 21:2 152.4 0.483 5.54 6.66 6.14 7.7 153.0 0.503 12.12 4.25 4.25 0.0 152.4 0.485 18.59 2.90 2.90 0.0 152.4 0.488 25.37 2.41 2.17 10.1 153.0 0.516 37.36 1.51 1.53 )1.5 Standard error ¼ 0:057
Fig. 6. FAD for AISI 301 stainless steel.
data [30] on 2014-T6 aluminium and 2219-T87 aluminium, generated from the thin plates containing through and part-through cracks. The specimens were tested
T. Christopher et al. / Cryogenics 42 (2002) 661–673
671
tensile specimens having through-thickness cracks were found to be in good agreement with the published test data [30]. Figs. 8 and 9 show the FADs for 2014-T6 aluminium alloy and 2219-T87 aluminium alloy.
4. Concluding remarks
Fig. 7. FAD for ELI titanium alloy (5Al–2.5Sn–Ti).
at cryogenic temperatures, which were established by immersing the specimen in liquid nitrogen or liquid hydrogen. Fracture parameters for 2014-T6 aluminium were evaluated at 20 K. Table 11 gives a good comparison of analytical and experimental results of Ref. [30]. Fracture parameters for 2219-T87 aluminium were evaluated at cryogenic temperatures and were presented in Table 12. Failure load estimations on
Failure assessment has been made on tested flawed aluminium, titanium and steel cylinders under internal pressure at cryogenic temperatures. Fracture parameters to generate FAD, are determined from the published fracture data. Failure pressure estimates were found to be reasonably in good agreement with the tested flawed cylinders at cryogenic temperatures. The empirical relationship between Kmax and rf =ru through fracture parameters KF , m and p in Eq. (1), has been successfully applied to correlate the fracture data on various materials generated from the test tanks at cryogenic temperatures. The three-fracture parameters are found to be dependent on thickness and temperature. For fracture strength evaluation of any structural configuration, the stress intensity factor corresponding to that geometry is to be used in Eq. (1) to develop the necessary strength equation. If the values of applied
Table 11 Comparison of analytical and experimental results for 2014-T6 aluminium tensile specimens having through thickness and surface cracks at 20 K pffiffiffiffi (rys ¼ 554 MPa, rult ¼ 687 MPa, KF ¼ 61:6 MPa m, m ¼ 0:7, p ¼ 24:8) Specimen dimensions (mm)
Crack dimensions (mm)
Fracture strength, rmax (MPa)
Width, W
Thickness, t
Depth, a
Length, 2c
Test [30]
Analysis
76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2
1.56 1.58 1.57 1.52 1.56 1.55 1.53 1.53 1.54 1.55 1.58 1.57 1.56 1.56 1.52 1.56 1.54 1.55 1.53 1.52 1.53 1.57
7.04 7.06 14.15 19.68 21.49 26.21 30.15 30.58 35.53 36.37 5.23 5.41 7.90 10.24 5.89 6.73 11.61 15.47 32.38 28.32 34.62 5.03
408.9 405.4 320.6 288.9 272.4 235.8 222.0 217.2 195.8 189.6 497.1 488.2 437.1 379.9 456.5 435.1 356.5 329.6 216.5 231.7 204.1 459.9
393.8 393.5 311.5 273.5 263.2 239.8 222.7 220.9 201.6 198.5 487.4 480.1 436.3 399.0 443.2 432.2 371.7 327.0 233.2 244.4 220.7 438.9
Standard error ¼ 0:043
1.56 1.58 1.57 1.52 1.56 1.55 1.53 1.53 1.54 1.55 0.86 0.89 0.97 1.07 1.12 1.12 1.17 1.30 1.37 1.40 1.40 1.47
Relative error (%) 3.7 3.0 2.8 5.4 3.4 )1.7 )0.3 )1.7 )3.0 )4.7 1.9 1.7 0.2 )5.0 2.9 0.7 )4.3 0.8 )7.7 )5.5 )8.2 4.6
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Table 12 Comparison of analytical and experimental results for 2219-T87 aluminium tensile specimens having through-thickness cracks at cryogenic temperatures Specimen dimensions (mm) Width, W
Fracture strength, rmax (MPa) Thickness, t
Relative error (%)
Test [30]
Analysis
459.9 445.4 431.6 424.0 401.3 396.5 384.1 369.6
474.3 453.3 437.8 421.1 397.1 390.6 378.5 366.1
)3.1 )1.8 )1.4 0.7 1.1 1.5 1.5 1.0
pffiffiffiffi At 78 K: rys ¼ 445 MPa; rult ¼ 579 MPa; KF ¼ 149:6 MPa m; m ¼ 0:9; p ¼ 58:7 139.7 1.74 8.43 435.8 139.7 1.74 10.67 424.0 139.7 1.73 12.19 415.1 139.7 1.72 15.77 393.7 170.2 1.71 22.45 385.4 170.2 1.73 25.35 376.5 170.2 1.74 30.56 352.3
446.9 430.2 420.5 400.9 373.2 363.2 347.3
)2.6 )1.5 )1.3 )1.8 3.2 3.5 1.4
371.8 361.7 348.2 336.2 312.8 306.3 295.1
)2.7 )2.3 )1.4 )1.8 4.1 3.7 1.4
At 20 K: rys ¼ 487 MPa; rult 139.7 139.7 139.7 139.7 170.2 170.2 170.2 170.2
Crack length, 2c pffiffiffiffi ¼ 664 MPa; KF ¼ 160:1 MPa m; m ¼ 1 1.73 7.54 1.73 10.16 1.71 12.50 1.71 15.49 1.73 20.95 1.72 22.63 1.74 26.09 1.74 30.07
Standard error ¼ 0:017
Standard error ¼ 0:023 At 295 K: rys ¼ 379 MPa; rult ¼ 467 MPa; KF ¼ 134:5 139.7 1.72 139.7 1.71 139.7 1.71 139.7 1.72 170.2 1.70 170.2 1.73 170.2 1.73
pffiffiffiffi MPa m; m ¼ 0:9; p ¼ 58:7 8.48 362.0 10.21 353.7 13.08 343.4 15.82 330.3 23.09 326.1 25.50 317.9 30.12 299.2
Standard error ¼ 0:027
Fig. 8. FAD for 2014-T6 aluminium.
Fig. 9. FAD for 2219-T87 aluminium.
stress and corresponding stress intensity factor for the specified crack in a structure lie below the Kmax –rf curve
of the FAD, the structure for that loading condition is safe. Based on these studies, it is concluded that the
T. Christopher et al. / Cryogenics 42 (2002) 661–673
empirical relationship (1) is useful for fracture strength evaluation of cracked configurations.
References [1] Torre CN, Witham JA, Dennison EA, McCool RC, Rinker MW. An analysis of a low-vapor-pressure cryogenic propellant tankage system. AIAA paper 87-2068, AIAA/SAE/ASME/ASEE 23rd Joint Propulsion Conference, San Diego, CA. 29 June–2 July l987. [2] Gordon JR. A review of fracture assessment procedures and their applicability to welded structures. Can Metall Q 1993;32:195–203. [3] Nguyen-duy P, Blanchette Y, Lanouette C. Use of linear elastic fracture mechanics in the in-service fracture analysis of structural components. Can Metall Q 1993;32:213–22. [4] Agatonovic P, Windisch M. Role of combined numerical and experimental investigations in the justification of the structural integrity and damage tolerance of space structures. ESA-SP-321 1991:679–85. [5] Shin CS, Wu PL, Jao WT. Re-appraisal of the net section plastic collapse method in the assessment of elastic––plastic fracture. J Chin Soc Mech Eng 1992;13:198–204. [6] Shin CS, Wu PL, Jao WT. Engineering assessment methods in the prediction of elastic–plastic fracture behavior. J Chin Soc Mech Eng 1992;13:505–18. [7] Ainsworth RA, OÕdowd NP. Constraint in the failure assessment diagram approach for fracture assessment. Trans ASME, J Pressure Vessel Technol 1995;117:260–7. [8] Bloom JM. Deformation plasticity failure assessment diagram (DPFAD) for materials with non-Ramberg-Osgood stress–strain curves. Trans ASME, J Pressure Vessel Technol 1995;117:346–56. [9] Milne I, Ainsworth RA, Chell GG. Predicting the integrity of large diameter stainless steel pipes using the R6 procedures. Nucl Eng Des 1987;98:401–10. [10] Tada H, Paris PC, Gamble RM. A stability analysis of circumferential cracks for reactor piping systems. ASTM-STP-700 1980:296–313. [11] ASME Boiler and Pressure Vessel code, Section XI. Rules for inservice inspection of nuclear power plant components. Article IWB-3640, Evaluation produces and acceptance criteria for austenitic piping. 1983. [12] Milne I, Chell GG. Evaluation of flaw tolerance during stable crack growth. Inst Mech Eng C 1980;18:85–91. [13] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100–8. [14] Newman Jr JC. An evaluation of fracture analysis methods. ASTM-STP-896 1985:5–96.
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[15] Newman Jr JC. The merging of fatigue and fracture mechanics concepts: a historical perspective. Prog Aerospace Sci 1998;34: 347–90. [16] Keller HP, Junker G, Merker W. Fracture analysis of surface cracks in cylindrical pressure vessels applying the two parameter fracture criterion(TPFC). Int J Pressure Vessels Piping 1987;29: 113–53. [17] Nageswara Rao B, Acharya AR. Failure assessment on 34Cr Mo4 grade steel cylindrical pressure vessels with an axial surface crack. Int J Pressure Vessels Piping 1997;72:157–63. [18] Zerbst U, Hamann R, Wohlschlegel A. Application of the European flaw assessment procedure SINTAP to pipes. Int J Pressure Vessels Piping 2000;77:697–702. [19] Govindan Potti PK, Nageswara Rao B, Srivatsava VK. Residual strength of aluminium–lithium alloy center surface crack tension specimens at cryogenic temperatures. Cryogenics 2000;40:789– 95. [20] Newman Jr JC, Raju IS. Analysis of surface cracks under tension or bending loads. NASA-TP-1578, 1979. [21] Newman Jr JC. Fracture analysis of surface and through cracks in cylindrical pressure vessels. NASA-TND-8325, 1976. [22] Christopher T, Ramasarma BSV, Govindan Potti PK, Nageswara Rao B, Sankaranarayanasamy K. A comparative study on failure pressure estimations of unflawed cylindrical vessels. Int J Pressure Vessels Piping 2002;79:53–66. [23] Margetson J. Burst pressure predictions of rocket motors. AIAA paper No. 78 1569, AIAA/SAE 14th Joint Propulsion Conference, Las Vegas, NV, USA. 22–27 July 1978. [24] Beena AP, Sundaresan MK, Nageswara Rao B. Destructive tests of 15CDV6 steel rocket motor cases and their application to light weight design. Int J Pressure Vessels Piping 1995;62:313– 20. [25] Hrisafovic N, Ledey JP. Study of the fracture mechanics phenomena in the ARIANE program. Acta Astronautica 1980;7:839– 45. [26] Sharma VMJ, Diwakar V, Arumugham S, Lakshmanan TS. Residual strength of aluminium alloy Afnor 7020. Sci Technol Weld Join 1999;4:89–93. [27] Peter RW, Kuhn P. Bursting strength of unstiffened pressure cylinders with slits. NACA-TN-3993, 1957. [28] Anderson RB, Sullivan TL. Fracture mechanics of throughcracked cylindrical pressure vessels. NASA-TND-3252, 1966. [29] Calfo FD. Effect of residual stress on fracture strength of AISI 301 stainless steel and Ti–5Al–2.5Sn ELI titanium cracked thin-wall cylinders. NASA-TND-4777, 1968. [30] Orange TW, Sullivan TL, Calfo FD. Fracture of thin sections containing through and part through cracks. ASTM-STP-496 1971:61–81.
Fracture strength of flawed cylindrical pressure vessels under cryogenic temperatures T. Christopher a, K. Sankarnarayanasamy b, B. Nageswara Rao
c,*
a
c
Faculty of Mechanical Engineering, Alagappa Chettiar College of Engineering & Technology, Karaikudi 630 004, India b Faculty of Mechanical Engineering, Regional Engineering College, Tiruchirapalli 620 015, India Structural Engineering Group, Propellants and Special Chemicals Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India Received 30 May 2002; accepted 12 August 2002
Abstract Damage tolerant and fail-safe approaches have been employed increasingly in the design of critical engineering components. In these approaches, one has to assess the residual strength of a component with an assumed pre-existing crack. In other cases, cracks may be detected during service. Then, there is a need to evaluate the residual strength of the cracked components in order to decide whether they can be continued safely or repair and replacement are imperative. A three-parameter fracture criterion is applied to correlate the fracture data on aluminium, titanium and steel materials from test results on cylindrical tanks/pressure vessels at cryogenic temperatures. Fracture parameters to generate the failure assessment diagram are determined for the materials considered in the present study. Failure pressure estimates were found to be in good agreement with test results. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Failure pressure; Aluminium; Titanium; Steels; Cylindrical vessels; Through-wall cracks; Surface cracks; Elastic stress intensity factor; Failure assessment diagram
1. Introduction The aerospace engineering profession has historically sought lighter structures through new materials, design techniques, and manufacturing operations to improve a vehicleÕs performance and propellant consumption. One technology for a fully reusable and permanently space based cryogenic upper stage rocket is the use of liquid hydrogen and liquid oxygen propellants in less than atmospheric condition––a ‘‘low-vapor pressure’’ system. Torre et al. [1] have discussed important issues relevant to structural design of very-thin-gauge tanks; e.g., fracture mechanics, thin-gauge formability, inspection, and weight. In cryogenic tanks, a crack through the thickness, while not of critical size structurally, can be catastrophic because leaking propellants pose a fire/ explosion hazard. Proof tests can establish the leak-free integrity of tanks and therefore verify the absence of through cracks in the tank wall. The choice of material is critical because the low-temperature environment may
*
Corresponding author. Fax: +91-471-415236. E-mail address: bnrao52@rediffmail.com (B. Nageswara Rao).
also reduce ductility and fracture toughness of the tank materials to unacceptably low levels. Fracture mechanics analyses are thus required on all cryogenic tanks to demonstrate that the maximum size of flaw or crack-like defect that could exist after proof testing and nondestructive evaluation will not grow to critical size and cause premature failure during the required service life. Fracture mechanics seeks to relate the three parameters (viz., size of the defect, material toughness, and applied stress), which combine to control the process of fracture. If any two of the three parameters are known then it is possible using the principles of fracture mechanics to estimate the value of the third parameter, which will give rise to failure of the structure. Gordon [2] has reviewed several of the major pipeline codes, which have incorporated fracture mechanics based fitness-forservice concepts. When applying fracture mechanics approaches it is important to adhere to their respective range of applicability. In each particular case a decision has to be made as to whether the assessment is to be founded on the methods of linear elastic fracture mechanics (LEFM), elastic plastic fracture mechanics (EPFM) or plastic collapse. In cases where brittle fracture occurs at low applied stresses, the concept of LEFM can
0011-2275/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 2 ) 0 0 1 1 4 - 5
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Nomenclature a c Do , Di KF , m, Kmax pb pbf pi
crack depth half the crack length outer and inner diameter of cylindrical vessel p three fracture parameters in Eq. (1) stress intensity factor at failure failure pressure of unflawed cylindrical vessel failure pressure of flawed cylindrical vessel internal pressure
be applied, i.e., a stress intensity factor approach [3]. At the other extreme, when the failure mechanism is plastic overload, assessments should be performed using limit load or plastic collapse analyses. Between these two extremes EPFM methods can be applied to assess the integrity of structures. Full-length EPFM analyses are often formidable in terms of time and economy and require a high degree of expertise to implement. A number of engineering methods have been proposed for the assessment of elastic plastic fracture behavior of cracked configurations [4–8]. These include the two-parameter R6 failure assessment method [9], J-integral estimation method [10], and the net section plastic collapse method [11]. In the failure assessment diagram (FAD) or the R6 method [12], the integrity of the structure is assessed and represented in a two-dimensional way: a function of the failure strength pursuant to LEFM is plotted as ordinate and that pursuant to plastic collapse as abscissa. The Dugdale model [13] established the stress limits for any transitional stages between linear elastic failure and plastic collapse. Experiments largely proved these limits to be conservative. The two-parameter fracture criterion (TPFC) of Newman [14,15] too, applies relations derived within the scope of LEFM. In this criterion, the two fracture parameters take account of the deviation of the stress to failure from the stress calculated pursuant to LEFM principles. These parameters have to be determined earlier in pretests, so-called base line tests, to be conducted under identical conditions of material. Keller et al. [16] have carried out fracture analysis of surface cracks in cylindrical vessels applying the TPFC. It was possible neither to determine satisfactorily the failure stresses of vessels by means of fracture parameters obtained from fracture mechanics specimens, nor to predict the loads to failure of the specimens by means of the vesselsÕ fracture parameters. Negeswara Rao and Acharya [17] have developed an empirical relationship between the failure stress and the elastic stress intensity factor at failure by means of three fracture parameters successfully and correlated the fracture data of Ref. [16]. However, systematic validation of this criterion to understand its conservatism based on experimental results is still very limited.
R t W rf ru rult rys
inner radius of cylinder thickness of SCT specimen/cylinder specimen width failure stress nominal stress required to produce a fully plastic region on the net section ultimate tensile strength yield strength or 0.2% proof stress
Zerbst et al. [18] have applied the recently developed European flaw assessment procedure structural integrity assessment procedures for European industry (SINTAP) to the published fracture data on steel pipes having through-wall and surface cracks subjected to internal pressure. The SINTAP procedure offers a crack driving force (CDF) and a FAD route. Both are complementary and give identical results. In the CDF route the determination of the crack tip loading in the component and its comparison with the fracture resistance of the material are two separate steps. In contrast to this philosophy, in the FAD route a failure line is constructed by normalizing the crack tip loading by the materialÕs fracture resistance. The assessment of the component is then based on the relative location of an assessment point with respect to this failure line. The predicted failure pressures were checked against a set of published experimental data. The differences from the experimental failure pressures were found to vary between 20% and 70%. They reported that the reason for this spread is due to their lower bound estimations of fracture toughness of the material from the Charpy energy values through empirical relations. They have applied different analysis levels of SINTAP by considering another set of fracture data and claimed that the real load carrying capacity was under estimated by about 20% or less for the higher analysis levels which demand high quality of the stress–strain curves of the material in the yield range. The extraordinary success of fracture mechanics lies in its ability to combine a theoretical framework with experimentally measured critical quantities. The objective of the present study is to consolidate the fracture data on various materials generated at cryogenic temperatures from cylindrical pressure vessels containing cracks, and examine how well the developed threeparameter fracture criterion correlates the test data.
2. Three-parameter fracture criterion The significant parameters affecting the size of a critical crack in a structure are the applied stress levels, the fracture toughness of the material, the location of
T. Christopher et al. / Cryogenics 42 (2002) 661–673
the crack and its orientation. Since the intensity of the stress at the crack-tip, K is a function of load, geometry and crack size, it is more appropriate to have a relationship between the stress intensity factor at failure (Kmax ) and the failure stress (rf ) from the fracture data of cracked specimens for the estimation/prediction of the fracture strength to any cracked configurations. The relationship between Kmax and rf can be of the form [17,19] p rf rf Kmax ¼ KF 1 m ð1 mÞ : ð1Þ ru ru Here, rf is the failure stress normal to the direction of crack in a body and ru is the nominal stress required to produce a fully plastic region (or hinge) on the net section. For center crack tension specimens, ru is equal to the ultimate strength (rult ) of the material. For cylindrical pressure vessels, ru is the hoop stress at the burst pressure level of the unflawed thin cylindrical shell. KF , m and p are the three fracture parameters to be determined from the fracture data. Kmax in Eq. (1) for center surface crack tension (SCT) specimens and cylindrical vessels with an axial surface crack (see Fig. 1) from the stress intensity factor expressions obtained from the finite element solutions [20,21] are given below: Kmax ¼ rf ðpaÞ rf ¼ rmax ¼ ¼
pbf R t
1=2
M=/;
Pmax tW
for SCT specimens;
for cylindrical pressure vessels;
ð2Þ
M¼
663
rffiffiffi pffiffi rffiffiffi a 2pffiffip c a p c fw M1 þ / þ / ðM2 1Þ M1 a t a t
for SCT specimens; ¼ Me fs for cylindrical pressure vessels;
ru ¼ rult for SCT specimens; pb R for cylindrical pressure vessels; ¼ t a 1:65 /2 ¼ 1 þ 1:464 for a 6 c; c c 1:65 ¼ 1 þ 1:464 for a > c; a a M1 ¼ 1:13 0:1 for a 6 c; c n c o c 1=2 for a > c; ¼ 1 þ 0:03 a a p 1=2 M2 ¼ for a 6 c; 4 c p 1=2 ¼1þ 1 for a > c; a 4 rffiffiffi c a q Me ¼ M1 þ / ; M1 a t sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc a 1=2 fw ¼ sec ; W t
1=2 fs ¼ 1 þ 0:52ks þ 1:29k2s 0:074k3s
Fig. 1. Cracked configurations.
for 0 6 ks 6 10;
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c a ks ¼ pffiffiffiffiffi ; t Rt
T. Christopher et al. / Cryogenics 42 (2002) 661–673
q¼2þ8
a 3 c
;
a is the depth and c is half the crack length of a surface crack, W is the width of the plate, R is the radius of the cylindrical vessel, t is the thickness, Pmax is the maximum tensile load, pb is the general collapse or bursting pressure of an unflawed cylindrical vessel, pbf is the failure pressure of a cylindrical vessel with an axial surface crack. When the depth (a) of the crack equals to the thickness (t), Kmax corresponds to the maximum stress intensity factor for through cracked specimens. A comparative study in Ref. [22] on failure pressure estimation of unflawed cylindrical vessels indicates the validity of the following FaupelÕs formula for steel vessels: h 2 rys ti pb ¼ pffiffiffi rys 2 ð3Þ ln 1 þ ; R rult 3 where, rys is the 0.2% proof stress or the yield strength of the material. Using Eqs. (1) and (2), one can set up the following equation to determine the nominal failure stress (rf ) for a specified crack size: ) p ( 1=2 rf ru ðpaÞ M rf þ mþ ð1 mÞ 1 ¼ 0: /KF ru ru ð4Þ The non-linear Eq. (4) is solved for rf using the Newton–Raphson iterative method. Failures of materials properly understood are, among the most important links in the design process. When dealing with a specific material for a particular application, it is not clearly established whether KIC (plane strain fracture toughness) or KC (fracture toughness when plane strain conditions are not met) values should be used. The values of KIC seem to be relevant in heavy sections like forgings or thick plates. Past experience suggests that design based on KIC will be unreasonably conservative in conventionally thick-sectioned structural members in aerospace application. In such circumstances it is necessary to carry out what is called KC tests corresponding to the thickness of intended structural application. The value of KC can be determined from the point of tangency between the crack growth resistance curve (R-curve) and the CDF curve appropriate for the loading geometry. ASTM-E561 standards suggest generation of a R-curve from through-crack test coupons like compact tension specimens, etc. It should be noted that KC is geometry dependant whereas the R-curve is considered to be a material property, independent of geometry (except thickness). Thus, the R-curve of the material will be useful for accurate determination of the critical load of the through cracked configurations. For part-through cracked configurations, fracture strength
estimations are not possible directly from the R-curve of the material because the part-through crack has two dimensions, namely, crack length and its depth. In such situations, the relationship between the failure stress (rf ) and the stress intensity factor (Kmax ) at failure by means of three fracture parameters (KF , m and p) in Eq. (1), will be useful for fracture strength evaluation of cracked configurations. In fact, Eq. (1) developed is following the basic concepts of the NewmanÕs TPFC [14]. The TPFC equation: rf Kmax ¼ KF 1 m ð5Þ ru was derived using two approaches. In the first approach the stress concentration factor for an ellipsoidal cavity was used with NeuberÕs equation to derive a relation between the local elastic–plastic stresses and strains, and remote loading. Assuming that fracture occurred when the notch-root stress and strain was equal to the fracture stress and strain, and that a crack has critical notch-root radius, the TPFC equation was derived. The second parameter, m, came from the unity term in the stress concentration equation. The second approach used the stress field equation for a crack and NeuberÕs equation to relate the elastic stresses to the elastic–plastic stresses and strains at a crack-tip. In this approach, it was again assumed that the fracture occurred at a critical distance ahead of the crack tip when the local stress and strain was equal to the fracture stress and strain. The second parameter, m, came from the next higher-order term in the stress-field expansion. The details on the derivation of TPFC Eq. (5) are given in Ref. [15]. It is well known fact that the tensile strength, rf , of a specimen decreases with increasing crack size. If rf < rys , then there exists a linear relationship between rf and Kmax . For small sizes of cracks where rys < rf < ru , the relationship between rf and Kmax is expected to be nonlinear. The idea of expressing Kmax as a function of rf in a single expression (1), is mainly for estimation of failure strength of a cracked body whether it contains throughthickness or part-through cracks which are small or large in size. In the absence of cracks, rf ! ru and Kmax ! 0. To account this limiting condition, the expression (1) is more appropriate. The exponential form of the third term in Eq. (1) is preferred to represent the non-linear variation of Kmax with rf , when rf > rys . An empirical relation is derived in Ref. [19] for the third fracture parameter, p, in terms of the second fracture parameter, m, as p¼
1 ð 1 þ fÞ 2 1 1 1 pffiffiffi ln 1 pffiffiffi ð1 þ fÞ þ 21 m ; ð1 mÞ f 2 2 ln
1
ð6Þ
T. Christopher et al. / Cryogenics 42 (2002) 661–673
where f¼
4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 3 þ 9 8m
ð7Þ
With this empirical relation (6), one needs to evaluate only KF and m in Eq. (1) utilizing the fracture strength values from two cracked configurations. To account the scatter in the experimental data, test results from more number of cracked configurations should be fitted in Eq. (1) for obtaining the material parameters. The material parameter, m, in general, is greater than zero and less than unity. If m is found to be less than zero due to large scatter in the fracture data, the parameter, m, has to be set to zero and the average of Kmax from the fracture data, yields the parameter, KF and the third parameter, p, from Eq. (6) gives the value close to 12. When, m is close to unity, the third term in Eq. (1) becomes insignificant. Whenever, m is found to be greater than unity, the parameter, m, has to be set to 1, by suitably modifying the parameter, KF , with the fracture data. If the fracture strength data is less than the yield strength of the material, KF and m in Eq. (1) can be obtained by fitting the fracture data in Eq. (1) and neglecting the third term on the right hand side of Eq. (1). The third parameter, p, is obtained using Eq. (6). If the fracture strength values are higher than the yield strength, one has to obtain KF value in an iterative process by specifying m (between 0 and 1), evaluating p from Eq. (6), and fitting the fracture data in Eq. (1). This iterative process should continue till Eq. (1) satisfactorily correlates the fracture data with the obtained fracture parameters, KF , m and p. Since there is no unique fracture criterion applicable for all materials, one has to select/establish a criterion suitable for the intended material. Generation of fracture data on different materials for validation of the selected fracture criterion is a difficult task for a single
665
agency. In such circumstances, one has to rely on the published experimental data. An attempt is made here to correlate the fracture data on pressure vessels tested at cryogenic temperatures with the developed Eq. (1). For ready reference, expressions are provided for the stress intensity equation to cylindrical pressure vessels, which will be useful for analysts to verify the accuracy in their finite element models/solutions.
3. Results and discussion To examine the accuracy of the empirical relation (3) while estimating the collapse or bursting pressure of unflawed vessels, test data in Refs. [23,24] are considered. The relative error (%) is computed as Analysis result Relative error ðREÞ % ¼ 100 1 ; Test result ð8Þ A standard error (SE) is computed as " SE ¼
N X i¼1
Analysis result 1 Test result
2 , #1=2 ; N
ð9Þ
i
where N is the number of test specimens. Here a positive RE indicates an under-estimate while a negative RE implies an un-conservative over-estimate. Tables 1 and 2 give comparison of analytical and experimental failure pressure values of maraging steel and 15CDV6 steel rocket motor cases. The analytical results indicated that FaupelÕs formula (3) is accurate to 5%. Hrisafovic and Ledy [25] have performed tensile fracture tests on AZ5G aluminium alloy center surface crack specimens having 50 mm width and 4.7 mm thickness. The sizes of cracks in the weldment of
Table 1 Comparison of analytical and experimental failure pressure of maraging steel rocket motor cases rys (MPa) 2128 2128 2128
rult (MPa) 2155 2155 2155
Outer diameter, Do (mm) 93 94 93
Thickness, t (mm)
Failure pressure, pb (MPa)
1.6 1.8 1.7
Test [23]
Eq. (3)
Relative error (%)
87 94 92
88.5 97.2 94.1
)1.7 )3.4 )2.3
Table 2 Comparison of analytical and experimental failure pressure of 15CDV6 steel rocket motor cases rys (MPa)
rult (MPa)
Outer diameter, Do (mm)
Thickness, t (mm)
Failure pressure, pb (MPa) Test [24]
Eq. (3)
Relative error (%)
915 915 915
1060 1060 1060
212 212 162
2.6 2.6 3.8
29 30 58
29.9 29.9 56.9
)3.3 )0.9 2.5
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specimens are: crack length is five times to its depth and depth of the crack varies up to the thickness of the specimens. The fracture parameters (KF , m and p) in Eq. (1) are obtained from the rupture curve of Ref. [25]. The ultimate tensile strength of this material is close to the achieved weld properties of Afnor 7020 aluminium alloy [26]. Hence, the fracture parameters of AZ5G aluminium alloy are used in Eq. (4) for evaluating the fracture strength of Afnor 7020 aluminium alloy center SCT specimens. The analysis results in Table 3 are found to be in good agreement with the test results of Ref. [26]. The ultra high strength aluminium alloy Afnor 7020 sheets and forgings are used to fabricate liquid propellant and water tanks in upper stages of large size launch vehicles. Fracture data on steel cylinders having an axial surface crack under different pressure medium reported in Ref. [16] is examined. Table 4 gives failure pressure estimates within 15% of the test results. Peters and Kuhn [27] have generated fracture data from cylindrical pressure vessels made of 7075-T6 and
2024-T3 aluminium alloys. In the pressure test, cylinders were pressurized with air and with oil. The results indicate that the pressurizing medium has a negligible effect on the bursting pressure. The fracture parameters (KF , m and p) in Eq. (1) are determined. Tables 5 and 6 give comparison of failure pressure estimates with test results. It can be seen from Fig. 2 that most of the analytical results are within 10% of the test results. The large discrepancy in few cases is mainly due to scatter in test results. It is very interesting to note from Table 6 that values of the fracture toughness parameter, KF are found to decrease with increase in thickness. Anderson and Sullivan [28] have generated fracture data at cryogenic temperatures from cylindrical pressure vessels made of aluminium alloy (2014-T6 Al) and extralow-interstitial (ELI) titanium alloy (5Al–2.5 Sn–Ti). Tables 7 and 8 give fracture parameters for these materials. It can be seen from Figs. 3 and 4 that most of the failure pressure estimates are within 10% of the test results. The major discrepancy in the results of Table 8 is mainly due to scatter in test results.
Table 3 Comparison of analytical and experimental fracture strength results for AZ5G and Afnor 7020 aluminium alloy tensile specimens containing surface pffiffiffiffi cracks: (KF ¼ 83:7 MPa m; m ¼ 0:76; p ¼ 29:2) Specimen dimensions (mm)
Crack dimensions (mm)
W
a
2c
Test [25,26]
Analysis
MPa (Weldment) 0.47 0.94 1.41 1.88 2.35 2.82 3.29 4.23 4.70
2.35 4.70 7.05 9.40 11.75 14.10 16.45 21.15 23.50
344.8 340.6 333.7 324.0 311.5 296.2 278.1 233.7 207.3
337.2 330.7 322.4 311.0 296.6 281.0 265.4 234.6 219.3
2.2 2.9 3.4 4.0 4.8 5.1 4.6 )0.4 )5.8
Afnor 7020 aluminium alloy [26]: rys ¼ 332:4 MPa; rult ¼ 376:1 MPa (Parent Metal) 75 5.10 2.55 11.75 345.6 75 4.28 2.78 13.23 332.1 75 5.09 2.80 13.91 338.7 75 5.80 2.90 13.14 339.0 75 5.80 2.96 12.98 330.3 75 5.16 2.99 15.03 329.8 75 5.10 3.01 17.97 319.4 75 5.81 3.02 14.30 332.0 75 5.81 3.08 15.09 326.0 75 5.85 3.22 13.71 335.8
313.0 297.9 301.4 306.2 305.9 295.0 285.7 300.7 297.3 299.9
9.4 10.3 11.0 9.7 7.4 10.6 10.5 9.4 8.8 10.7
Afnor 7020 aluminium alloy [26]: rys ¼ 302:1 MPa; rult ¼ 343:4 MPa (Weldment) 75 4.34 2.56 14.59 321.5 75 4.73 2.65 14.59 284.0 75 5.67 2.95 10.99 303.6 75 4.84 2.95 11.29 321.1 75 5.19 3.06 13.94 306.8 75 5.87 3.17 14.64 295.6 75 5.33 3.89 20.32 256.7
281.9 283.1 294.7 289.8 281.4 281.4 252.9
12.3 0.3 3.0 9.8 8.3 4.8 1.4
t
AZ5G aluminium alloy [25]: rult ¼ 346:2 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70 50 4.70
Standard error ¼ 0:075
Fracture strength, rf (MPa)
Relative error (%)
T. Christopher et al. / Cryogenics 42 (2002) 661–673
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Table 4 Failure pressure estimates of steel cylinders (Do ¼ 560 mm) having an axial surface crack Strength properties rys (MPa)
rult (MPa)
Thickness, t (mm)
Pressure medium: water þ air cushion at 293 K (KF ¼ 462:9 798 922 18.4 778 925 18.0 703 847 18.0 751 886 17.8 878 990 20.4 866 979 21.7 813 944 17.6
Crack dimensions
Failure pressure, pbf (MPa)
a (mm) 2c (mm) pffiffiffiffi MPa m; m ¼ 0:823; p ¼ 36:8) 16.8 218 9.3 144 11.6 215 15.8 150 16.1 96 14.5 65 14.6 65
Test [16]
Analysis
21.57 42.17 31.09 32.75 49.03 54.42 47.76
21.80 42.46 30.19 27.75 46.14 59.29 42.87
)1.1 )0.7 2.9 15.3 5.9 )9.0 10.2
45.30 40.01 43.83 36.58 48.05 55.31
44.45 34.04 46.95 41.79 43.90 48.73
1.9 14.9 )7.1 )14.2 8.6 11.9
27.65 28.44
27.41 26.56
0.9 6.6
Relative error (%)
Standard error ¼ 0:081 pffiffiffiffi Pressure medium: water þ glycol at 253 K (KF ¼ 426:1 MPa m; m ¼ 0:6; p ¼ 20:4) 859 982 17.5 13.0 94 853 973 18.4 14.7 155 842 985 18.5 10.7 143 830 984 17.7 9.0 215 726 879 17.8 10.0 142 843 976 18.7 13.5 93 Standard error ¼ 0:108 pffiffiffiffi Pressure medium: air at 293 K (KF ¼ 407:1 MPa m; m ¼ 1:0) 831 947 17.7 13.1 832 951 17.6 11.6
160 205
Standard error ¼ 0:047
Table 5 Failure pressure estimates of aluminium alloy (7075-T6 Al) cylinders having an axial through crack (Do ¼ 182:9 mm; rys ¼ 450 MPa; pffiffiffiffi rult ¼ 550 MPa; KF ¼ 64:2 MPa m; m ¼ 1:0) Thickness, t (mm)
Crack length, 2c (mm)
Failure pressure, pbf (MPa) Test [27]
Analysis
0.41 0.41 0.41 0.64 0.64 0.64
16.76 32.77 65.02 25.40 51.31 101.60
0.59 0.36 0.17 0.79 0.40 0.18
0.659 0.343 0.157 0.797 0.377 0.172
Relative error (%) )11.6 4.7 7.6 )0.9 5.6 4.7
Standard error ¼ 0:067
Pressure vessels are often fabricated by starting with flat sheet material and producing the required curvature in the sheet by one of a number of forming methods. Most of the methods result in the retention of residual stresses in the formed sheet. Pressure vessels with small radii of curvature have higher residual stresses than those with larger radii if they are made from flat sheet of the same thickness and are not subjected to an annealing or stress-relieving process. Thus, if material properties determined for smaller tanks are to be applied to larger vessels, the effect of the residual stresses on the results should be determined. Also, the effect of stress-relief treatment on the failure pressures or stresses in tanks
containing flaws is of interest. To obtain some insight into the effect of residual stress on the fracture strength of vessels containing through cracks, tanks were fabricated from AISI 301 stainless steel and 5Al–2.5Sn–Ti ELI titanium. These tanks, containing through-cracks of various lengths, were pressurized to the burst point [29]. The tanks which were fabricated from AISI 301 stainless-steel 60% cold-reduced material were formed by spiral wrapping nominally 0.56 mm thick sheet to form an 11° helix which was then butt-welded (see Fig. 5). The cylinders were nominally 152 mm in diameter and 457 mm in length. The AISI 301 stainless steel tanks were tested in the non-stress-relieved and 50% stressrelieved conditions. Those tanks, which were stressrelieved, were heated for 8 h at 673 K and furnace cooled. The 5Al–2.5Sn–Ti ELI titanium tanks were also 152 mm in diameter and 457 mm in length and were fabricated from nominally 0.51 mm thick sheet using a single longitudinal butt weld (see Fig. 5). The material was mill annealed at 991 K for 4 h and furnace cooled. To prevent contamination of, and possible explosive damage to the cryogenic test facility, each cylinder was completely disassembled and thoroughly cleaned following the fatigue-cracking process. Before testing the cylinders in liquid hydrogen or liquid nitrogen, it was planned to close the cylinder ends and seal the through crack. The seal used was a
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Table 6 Failure pressure estimates of aluminium alloy (2024-T3Al) cylinders having an axial through crack (Do ¼ 182:9 mm, rys ¼ 250 MPa, rult ¼ 450 MPa) Crack length, 2c (mm) t ¼ 0:15 mm (KF 11.94 24.38 49.78
Failure pressure, pbf (MPa) Test [27] Analysis pffiffiffiffi ¼ 148:3 MPa m; m ¼ 1:0) 0.31 0.386 0.24 0.229 0.12 0.116
Relative error (%) )24.4 4.5 3.5
Standard error ¼ 0:145 t ¼ 0:30 mm (KF 6.10 6.10 12.70 12.70 24.38 24.38 24.38 47.50 97.03 24.89 50.29 102.10 24.38 48.51 97.28
pffiffiffiffi ¼ 124 MPa m; 0.92 0.95 0.70 0.61 0.46 0.47 0.48 0.25 0.11 0.53 0.29 0.13 0.52 0.23 0.12
m ¼ 1:0) 1.010 1.010 0.763 0.763 0.488 0.488 0.488 0.258 0.126 0.479 0.243 0.121 0.488 0.252 0.126
)10.0 )6.5 )9.0 )25.1 )6.1 )3.8 )1.6 )3.2 )14.8 9.5 16.2 6.9 6.2 )9.8 )5.0
Standard error ¼ 0:106 t ¼ 0:38 mm (KF 15.75 30.48 60.96
pffiffiffiffi ¼ 112 MPa m; 0.84 0.49 0.25
m ¼ 0:94; p ¼ 91:8) 0.863 )2.7 0.510 )4.1 0.250 0.2
Standard error ¼ 0:028 pffiffiffiffi t ¼ 0:64 mm (KF ¼ 83:7 MPa m; m ¼ 0:4827; p ¼ 17:3) 26.16 1.05 1.060 )0.9 50.80 0.50 0.509 )1.9 143.00 0.16 0.159 0.5 Standard error ¼ 0:012
composite patch designed to ensure proper sealing without reinforcing the area surrounding the crack. A layer of 0.05 mm Mylar pressure-sensitive tape was placed over the crack from inside the tank. A piece of 0.25 mm Mylar matter was placed over the layer of tape. To prevent leaking during pressurization or local effects on the crack, the Mylar patch was made large enough to cover the original crack plus an additional amount to allow for stable crack growth. Three additional overlapping layers of Mylar tape completed the patch. A cylindrical insert, which reduced the volume of liquid hydrogen, was used in the test tank. The insert minimized damage to the test specimen and test facility at failure. After assembly the specimen was placed in a cryostat, submerged in and filled with the desired
Fig. 2. Comparison of analytical and experimental failure pressure of aluminium alloy (7075-T6Al, 2024-T3Al) cylindrical vessels having an axial through crack. Table 7 Failure pressure estimates of aluminium alloy (2014-T6Al) cylinders having an axial through crack (Do ¼ 142:2 mm, t ¼ 1:52 mm) Crack length, 2c (mm)
Failure pressure, pbf (MPa) Test [28]
At 298 K: rys ¼ 470 MPa; rult m ¼ 0:691; p ¼ 24:3 2.90 9.52 12.70 5.02 25.40 3.04 50.80 1.43
Relative error (%) Analysis pffiffiffiffi ¼ 545 MPa; KF ¼ 60:3 MPa m; 8.94 4.95 2.93 1.42
6.1 1.3 3.5 0.6
Standard error ¼ 0:036 pffiffiffiffi At 78 K: rys ¼ 520 MPa; rult ¼ 610 MPa; KF ¼ 71:5 MPa m; m ¼ 0:677; p ¼ 23:6 2.87 10.58 10.43 1.5 3.81 10.43 9.64 7.5 5.08 9.41 8.80 6.5 6.35 8.64 8.12 6.0 7.62 7.71 7.54 2.2 10.16 7.00 6.58 6.0 12.70 5.92 5.81 1.8 19.05 4.46 4.41 1.2 25.40 3.41 3.46 )1.5 31.75 2.75 2.80 )1.7 44.45 2.13 1.96 8.1 50.80 1.67 1.68 )0.7 Standard error ¼ 0:046 pffiffiffiffi At 20 K: rys ¼ 560 MPa; rult ¼ 680 MPa; KF ¼ 68:9 MPa m; m ¼ 0:601; p ¼ 20:4 2.64 12.15 11.72 3.6 6.35 9.37 8.65 7.7 12.70 5.85 6.02 )2.9 19.05 4.73 4.48 5.3 25.40 3.10 3.48 )12.1 31.75 2.93 2.79 4.9 44.45 1.95 1.93 1.0 50.80 1.76 1.65 6.1 Standard error ¼ 0:063
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Table 8 Failure pressure estimates of ELI titanium alloy (5Al–2.5Sn–Ti) cylinders having an axial through crack (Do ¼ 152:4 mm, t ¼ 0:51 mm) Crack length, 2c (mm)
Failure pressure pbf (MPa) Test [28]
At 78 K: rys ¼ 1200 MPa; rult m ¼ 0:763, p ¼ 29:5 3.18 8.75 6.12 7.58 5.82 7.20 11.18 5.31 11.51 4.83 18.62 3.90 19.43 3.43 24.03 3.30 25.17 3.04 37.59 2.03 37.08 1.65
Relative error (%) pffiffiffiffi ¼ 1400 MPa; KF ¼ 274:8MPa m; Analysis
8.11 6.80 6.92 5.18 5.09 3.63 3.51 2.91 2.78 1.85 1.87
7.3 10.3 3.9 2.5 )5.4 6.8 )2.3 12.0 8.5 9.1 )13.5
Standard error ¼ 0:082 pffiffiffiffi At 20 K: rys ¼ 1525 MPa; rult ¼ 1675 MPa; KF ¼ 228:2 MPa m; m ¼ 1:0 2.36 7.88 7.64 3.0 3.94 7.40 6.82 7.9 4.83 6.13 6.44 )5.0 7.04 5.58 5.63 )0.9 7.11 5.25 5.61 )6.8 13.28 3.89 4.01 )3.0 12.24 3.49 4.23 )21.1 20.40 2.92 2.87 1.9 23.90 2.90 2.48 14.5 19.35 2.83 3.00 )6.0 25.02 2.37 2.37 )0.1 40.64 1.87 1.43 23.8 39.62 1.73 1.47 15.2
Fig. 4. Comparison of analytical and experimental failure pressure of ELI titanium alloy (5Al–2.5Sn–Ti) cylindrical vessels having an axial through crack.
Standard error ¼ 0:113
Fig. 5. AISI 301 stainless steel spiral-welded and titanium pressure vessels.
Fig. 3. Comparison of analytical and experimental failure pressure of aluminium alloy (2014-T6Al) cylindrical vessels having an axial through crack.
cryogen, and pressurized to burst. Gaseous helium was used as the pressurizing medium. Tables 9 and 10 give fracture parameters for these materials. Failure pressure estimates are found to be within 10% of the test results. Figs. 6 and 7 show the FADs for these materials. The applicability of the three-parameter fracture criterion is examined by considering the published fracture
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Table 9 Failure pressure estimates of AISI 301 stainless steel, 60% cold-reduced, 152 mm diameter tanks having axial through cracks Cylinder dimensions (mm) Do (Non-stress relieved) at 20 K: rys 153.0 152.4 153.0 153.0 153.0 152.4 151.8 152.4
Crack length, 2c (mm)
Failure pressure, pbf (MPa)
t Test [29] Analysis pffiffiffiffi ¼ 1830 MPa; rult ¼ 2220 MPa; KF ¼ 201:2 MPa m; m ¼ 0:614; p ¼ 20:9 0.564 4.34 10.32 9.95 0.561 4.78 9.79 9.52 0.566 10.57 6.33 6.06 0.597 10.80 6.52 6.35 0.584 17.93 3.90 4.02 0.577 24.43 3.03 2.89 0.584 37.52 1.97 1.80 0.587 37.64 1.65 1.80
Relative error (%)
3.6 2.8 4.3 2.6 )3.0 4.7 8.5 )9.4
Standard error ¼ 0:054 pffiffiffiffi MPa; KF ¼ 242:2 MPa m; m ¼ 0:942; p ¼ 95:2 4.83 9.55 9.27 10.77 6.59 6.33 17.68 3.89 3.90 24.82 2.93 3.07 37.26 1.91 1.80
2.9 4.0 )0.4 )4.8 5.7
pffiffiffiffi ¼ 1650 MPa; rult ¼ 2270 MPa; KF ¼ 208:1 MPa m; m ¼ 0:64; p ¼ 21:9 0.572 4.09 10.43 10.53 0.587 4.30 10.33 10.54 0.559 9.91 6.65 6.37 0.592 11.99 6.71 5.94 0.594 18.29 4.25 4.13 0.589 22.38 3.15 3.33 0.587 36.88 1.86 1.90
)1.0 )2.1 4.2 11.5 2.8 )5.8 )2.0
(Stress-relieved) at 20 K; rys ¼ 1630 MPa; rult ¼ 2250 152.4 0.587 153.4 0.587 153.0 0.533 152.4 0.566 153.4 0.528 Standard error ¼ 0:040 (Non-stress relieved) at 77 K: rys 152.4 153.4 153.4 153.0 153.0 153.4 153.4 Standard error ¼ 0:053
Table 10 Failure pressure estimates of ELI titanium alloy (5Al–2.5Sn–Ti), 152 mm diameter non-stress relieved tanks having axial through cracks Cylinder dimensions (mm) Do
t
Crack length, 2c (mm)
Failure pressure, pf (MPa) Test [29]
Relative error (%)
Analysis
pffiffiffiffi At 20 K: rys ¼ 1400 MPa; rult ¼ 1560 MPa; KF ¼ 143:3 MPa m; m ¼ 0:802; p ¼ 33:8 152.4 0.503 5.46 5.25 5.11 2.6 153.0 0.493 17.53 2.48 2.25 9.2 152.4 0.483 18.49 2.16 2.09 3.2 153.4 0.447 24.31 1.34 1.42 )6.1 152.4 0.503 38.13 1.02 1.01 1.5 Standard error ¼ 0:053 pffiffiffiffi At 77 K: rys ¼ 1190 MPa; rult ¼ 1240 MPa; KF ¼ 214:2 MPa m; m ¼ 0:623; p ¼ 21:2 152.4 0.483 5.54 6.66 6.14 7.7 153.0 0.503 12.12 4.25 4.25 0.0 152.4 0.485 18.59 2.90 2.90 0.0 152.4 0.488 25.37 2.41 2.17 10.1 153.0 0.516 37.36 1.51 1.53 )1.5 Standard error ¼ 0:057
Fig. 6. FAD for AISI 301 stainless steel.
data [30] on 2014-T6 aluminium and 2219-T87 aluminium, generated from the thin plates containing through and part-through cracks. The specimens were tested
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tensile specimens having through-thickness cracks were found to be in good agreement with the published test data [30]. Figs. 8 and 9 show the FADs for 2014-T6 aluminium alloy and 2219-T87 aluminium alloy.
4. Concluding remarks
Fig. 7. FAD for ELI titanium alloy (5Al–2.5Sn–Ti).
at cryogenic temperatures, which were established by immersing the specimen in liquid nitrogen or liquid hydrogen. Fracture parameters for 2014-T6 aluminium were evaluated at 20 K. Table 11 gives a good comparison of analytical and experimental results of Ref. [30]. Fracture parameters for 2219-T87 aluminium were evaluated at cryogenic temperatures and were presented in Table 12. Failure load estimations on
Failure assessment has been made on tested flawed aluminium, titanium and steel cylinders under internal pressure at cryogenic temperatures. Fracture parameters to generate FAD, are determined from the published fracture data. Failure pressure estimates were found to be reasonably in good agreement with the tested flawed cylinders at cryogenic temperatures. The empirical relationship between Kmax and rf =ru through fracture parameters KF , m and p in Eq. (1), has been successfully applied to correlate the fracture data on various materials generated from the test tanks at cryogenic temperatures. The three-fracture parameters are found to be dependent on thickness and temperature. For fracture strength evaluation of any structural configuration, the stress intensity factor corresponding to that geometry is to be used in Eq. (1) to develop the necessary strength equation. If the values of applied
Table 11 Comparison of analytical and experimental results for 2014-T6 aluminium tensile specimens having through thickness and surface cracks at 20 K pffiffiffiffi (rys ¼ 554 MPa, rult ¼ 687 MPa, KF ¼ 61:6 MPa m, m ¼ 0:7, p ¼ 24:8) Specimen dimensions (mm)
Crack dimensions (mm)
Fracture strength, rmax (MPa)
Width, W
Thickness, t
Depth, a
Length, 2c
Test [30]
Analysis
76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2
1.56 1.58 1.57 1.52 1.56 1.55 1.53 1.53 1.54 1.55 1.58 1.57 1.56 1.56 1.52 1.56 1.54 1.55 1.53 1.52 1.53 1.57
7.04 7.06 14.15 19.68 21.49 26.21 30.15 30.58 35.53 36.37 5.23 5.41 7.90 10.24 5.89 6.73 11.61 15.47 32.38 28.32 34.62 5.03
408.9 405.4 320.6 288.9 272.4 235.8 222.0 217.2 195.8 189.6 497.1 488.2 437.1 379.9 456.5 435.1 356.5 329.6 216.5 231.7 204.1 459.9
393.8 393.5 311.5 273.5 263.2 239.8 222.7 220.9 201.6 198.5 487.4 480.1 436.3 399.0 443.2 432.2 371.7 327.0 233.2 244.4 220.7 438.9
Standard error ¼ 0:043
1.56 1.58 1.57 1.52 1.56 1.55 1.53 1.53 1.54 1.55 0.86 0.89 0.97 1.07 1.12 1.12 1.17 1.30 1.37 1.40 1.40 1.47
Relative error (%) 3.7 3.0 2.8 5.4 3.4 )1.7 )0.3 )1.7 )3.0 )4.7 1.9 1.7 0.2 )5.0 2.9 0.7 )4.3 0.8 )7.7 )5.5 )8.2 4.6
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Table 12 Comparison of analytical and experimental results for 2219-T87 aluminium tensile specimens having through-thickness cracks at cryogenic temperatures Specimen dimensions (mm) Width, W
Fracture strength, rmax (MPa) Thickness, t
Relative error (%)
Test [30]
Analysis
459.9 445.4 431.6 424.0 401.3 396.5 384.1 369.6
474.3 453.3 437.8 421.1 397.1 390.6 378.5 366.1
)3.1 )1.8 )1.4 0.7 1.1 1.5 1.5 1.0
pffiffiffiffi At 78 K: rys ¼ 445 MPa; rult ¼ 579 MPa; KF ¼ 149:6 MPa m; m ¼ 0:9; p ¼ 58:7 139.7 1.74 8.43 435.8 139.7 1.74 10.67 424.0 139.7 1.73 12.19 415.1 139.7 1.72 15.77 393.7 170.2 1.71 22.45 385.4 170.2 1.73 25.35 376.5 170.2 1.74 30.56 352.3
446.9 430.2 420.5 400.9 373.2 363.2 347.3
)2.6 )1.5 )1.3 )1.8 3.2 3.5 1.4
371.8 361.7 348.2 336.2 312.8 306.3 295.1
)2.7 )2.3 )1.4 )1.8 4.1 3.7 1.4
At 20 K: rys ¼ 487 MPa; rult 139.7 139.7 139.7 139.7 170.2 170.2 170.2 170.2
Crack length, 2c pffiffiffiffi ¼ 664 MPa; KF ¼ 160:1 MPa m; m ¼ 1 1.73 7.54 1.73 10.16 1.71 12.50 1.71 15.49 1.73 20.95 1.72 22.63 1.74 26.09 1.74 30.07
Standard error ¼ 0:017
Standard error ¼ 0:023 At 295 K: rys ¼ 379 MPa; rult ¼ 467 MPa; KF ¼ 134:5 139.7 1.72 139.7 1.71 139.7 1.71 139.7 1.72 170.2 1.70 170.2 1.73 170.2 1.73
pffiffiffiffi MPa m; m ¼ 0:9; p ¼ 58:7 8.48 362.0 10.21 353.7 13.08 343.4 15.82 330.3 23.09 326.1 25.50 317.9 30.12 299.2
Standard error ¼ 0:027
Fig. 8. FAD for 2014-T6 aluminium.
Fig. 9. FAD for 2219-T87 aluminium.
stress and corresponding stress intensity factor for the specified crack in a structure lie below the Kmax –rf curve
of the FAD, the structure for that loading condition is safe. Based on these studies, it is concluded that the
T. Christopher et al. / Cryogenics 42 (2002) 661–673
empirical relationship (1) is useful for fracture strength evaluation of cracked configurations.
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