Fatigue life prediction for finite ratchetting of bellows at cryogenic temperatures

Transcription

1 LHC Project Note Fatigue life rediction for finite ratchetting of bellows at cryogenic temeratures B. Skoczen, T. urtyka, J.C. Brunet, A. Poncet, A. Jacquemod MT/ESH eywords: bellows, cyclic loads, low-cycle fatigue, ratchetting, cryogenic temeratures Summary In the resent work the rogressive deformation (ratchetting) of bellows subjected to a sustained load (internal ressure) and to a suerimosed cyclic deflection rogram at cryogenic temeratures is examined. In order to estimate the number of cycles to failure a generalized Manson-Coffin equation was develoed. The model is based on two arameters: the ratchetting induced mean lastic strain and the lastic strain amlitude. The finite element simulation of the initial 20 cycles leads to an estimation of the accumulated lastic strains and enables the calculation of the fatigue life of bellows. An exerimental stand for cryogenic fatigue tests is also resented and the first verification tests are reorted. 1. Introduction The exansion bellows lay an imortant role in the design of interconnections for the suerconducting magnets that constitute the new article accelerator, to be built at CERN. In general, corrugated, thin-walled axisymmetric shells are rovided to comensate for the axial, the lateral and the angular dislacements of ies and iing systems. The bellows for LHC are made of austenitic stainless steel 16 L and are designed for severe exloitation conditions comrising cryogenic temeratures, high internal ressures and large cyclic deflections. Since exansion joints are widely used as integral arts of heat exchangers and ressure vessels working at room or elevated temeratures, there is a well documented design rocedure for bellows (cf. standards of EJMA or ASME) based on large number of tests and theoretical studies. On the other hand, there is a significant lack of data for the bellows resonse at cryogenic temeratures. One of the most imortant roblems consists in estimating the fatigue life of bellows at low temerature. Usually it turns out to be much higher than the fatigue life at room temerature as a result of a significant change of the material roerties (yield oint, ultimate strength, hardening modulus). Thus, the alication of an existing standard design rocedure (justified for room or higher temeratures) may lead to an inaroriate solution and increased cost. This note is aimed at develoing a comutational model for simlified calculations of the lastic strain accumulation in the U-bellows (U-rofiled convolution, Fig. 1) subjected to constant internal ressure and cyclic axial deflections at low temeratures (77 and 4 ). Since the stainless steel retains its ductility at cryogenic temeratures, it seems justified to

2 aly the model successfully used at room temerature to describe the low cycle fatigue henomena. The model is based on the extended Manson-Coffin equation 1,2 accounting for the effect of the lastic strain range and the mean lastic strain over cycle on the fatigue life. In order to calculate the evolution of the lastic strains a finite element simulation of the initial 20 cycles is erformed. This number is usually sufficient to obtain a fairly stable lastic behavior of the structure. The material model is based on the elastic-lastic bilinear resonse with the kinematic hardening. The fatigue tests at cryogenic temeratures,4 indicate that the stainless steels exhibit a significant cyclic hardening in the initial cycles that leads to a raid change of the material roerties. This effect has been accounted for in the simulation, searately for each temerature level. The material data were extracted from the reorts on the fatigue roerties of the steels at cryogenic temeratures,4,5,6. The results of the theoretical analysis were checked exerimentally by using a suitable fatigue test stand, equied with a neumatic servo- mechanism to cycle the bellows in liquid nitrogen or liquid helium bath. 2. Extended Manson-Coffin equation for low-cycle fatigue/ ratchetting There are two main arameters of lastic deformation that influence the initiation and roagation of a fatigue crack at some oint of a structure: the lastic strain range (width of hysteresis loo) and the mean lastic strain over cycle (osition of the center of hysteresis). If the mean lastic strain is small comared to the fracture ductility (true strain to fracture) the fatigue life may be calculated by using the Manson-Coffin arametrical relationshi 1 : N β f 0 = C, (1) where N f 0 denotes the fatigue life, the lastic strain range, C and β are material constants. The constant C is related to the true fracture strain in simle tension f 0 : C = ( 0. 25) β 0. (2) However, if the mean lastic strain turns out to be of the order of the fracture ductility, a correction term accounting for its influence must be added 2 : f N f = N f 0 1 m f 0 α, () where m denotes the mean lastic strain (for a stabilized hysteresis) and α is a material constant. Such a situation is usually observed as a result of a rogressive deformation (ratchetting) accomanied by a large accumulation of lastic strains. Consequently, in order to calculate the fatigue life of a structure at low temerature the evolution of both the lastic strain range and the mean lastic strain must be calculated. Since at each oint of the bellows wall a multi-dimensional strain state occurs, a substitution of and m with the equivalent values ( ij ij) eq = 2, (4) 2

3 m m ( ij ij ) m eq = 2, (5) is roosed. Here, the lastic strain range for each strain comonent, calculated for a stabilized hysteresis, is defined as: ij = ij max ij min, (6) where ij max and ij min denote the strain comonent values at the reverse oints of the cycle. Similarly, the mean lastic strain comonents over cycle are given by the equation: m ij = ij min + 2 ij max. (7) These equations ermit to reduce a comlex multi-dimensional case to a simle onedimensional case. It is worth ointing out that the same sensitivity with resect to tension and comression has been assumed (i.e. lack of the strength differential effect).. Finite element model of the bellows convolution In order to simulate the evolution of strains in the bellows under the combined loadings a finite element model of a half convolution (Fig. 2) was built. Since the bellows is an axisymmetric shell structure a model based on axisymmetric elements (Fig. ) is suitable for the axisymmetric deformation attern. Moreover, it is assumed that the deformation of each convolution is identical (the boundary effect between the ie and the bellows is neglected), so that only a half convolution with the aroriate boundary conditions (Fig. 2) is considered. The finite element model was built u using the 8-node isoarametric axisymmetric lane elements. A smooth distribution of strains through thickness has been achieved by alying 4 elements in the direction normal to the midsurface of the shell. Furthermore, 20 divisions down each arc and 10 divisions down the straight art were introduced to ensure a sufficient accuracy of the solution. Since the concentration of strains occurs around the root and the crest of the convolution 7, the mesh in these zones is finer than in the straight art. The combined loadings are comosed of the constant comonent (internal ressure) and the cyclic comonent (eriodic axial movement), which act simultaneously and cause exclusively axisymmetric deformations. 4. The material model for stainless steel at low temerature The material model used, based on references,4,5,6, concerns the cryogenic fatigue roerties of austenitic stainless steels, with secial attention aid to the grade 16 L. It is worth ointing out that the grades 04 and 16 exhibit very similar fatigue roerties, therefore the roosed model may be regarded as alicable to both of them. The mechanical roerties of 0. mm thick sheet material of grade 16 L are given in Table 1 (data for 00 were measured at CERN, whereas data for 77 and 4 were deduced from the other tests, referenced above, as the average values). Stainless steels exhibit strong cyclic hardening at low temerature in the initial tens of cycles. The curves of maximum stress versus number of cycles indicate three hases of cyclic hardening: a slow increase (u

4 to 5-10 cycles), a raid increase (from 5-10 cycles to cycles) and a stabilization (above cycles u to ruture). In order to account for the effect of cyclic hardening a simlified stewise linear model has been alied. Three hases (I, II, III) are reresented by three lines of different inclination with resect to the horizontal axis (Figs. 4 and 5). Simlified curves are lotted for the total strain range of 1%, 1.5% and 2%. The evolution of the hysteresis loos for a stainless steel at low temerature 5 (Fig. 6) indicates clearly that before the hysteresis stabilizes, which usually takes lace after 20 cycles, the material roerties change quickly during the hase II of cyclic hardening. In view of this fact a bilinear elastic-lastic model (linear kinematic hardening) with changing cycle-to-cycle moduli (in hase II) was alied (Fig. 7). It has been assumed that the average modulus of elasticity over the first 20 cycles is about 10% lower than the initial modulus and that the yield strength σ 0 evolves in the same way as the maximum stress σ max. Furthermore, it has been assumed that the hardening modulus H also evolves linearly during the hase II, however its final value is calculated taking into account the decrease of the lastic strain range (aroximately by 40%) after 20 cycles. This simlified model is regarded as a good aroximation for the first 20 cycles and a rough aroximation for the subsequent cycles of the fatigue test (stabilization of the hysteresis is assumed). Table 1. Data for grade 16 L fine gauge sheet material. Temerature Young s Yield strength modulus Hardening modulus for 5% strain range Tensile Strength Finite element simulation of strains evolution at low temerature The U-tye bellows selected for tests and calculations were manufactured from initially 0. mm thick stainless steel sheets. The outer and the inner radii R out, R in (Fig. 2) were equal to 5.8 mm and 27.5 mm, resectively. The bellows itch was equal to 5 mm, with a convolution radius of 1.25 mm. The bellows was subjected to a constant internal ressure of 0.8 (lower than the instability ressure) suerimosed with symmetric axial deflection cycles of ±116. mm er convolution. The calculations were erformed by using the ANSYS 51 finite element code. The comutational strategy for this roblem was based on the incremental lastic analysis and consisted in two general stes: incremental increase of ressure u to the required level alication of suitable axial deflection increments under the constant ressure. Since the most intense straining was observed at the root and at the crest of the convolution 7, the strain data were stored for both sections. The evolutions of the lastic strain intensity, defined as: i = 2 ij ij, (8) 4

5 are lotted for the inner and the outer faces at the root (A, B) and the crest (C, D) in Figs. 8, 9 and 10 for the temeratures of 00, 77 and 4, resectively. The equivalent lastic t strain range eq (cf. Eq. (6)) and the equivalent total strain range eq (defined analogically to Eq. (6)), reached after 20 cycles, are resented for various temerature levels in Table 2. Table 2. The equivalent strain ranges Temerature t eq eq Since the mean lastic strain is of the order of the lastic strain range, its influence on the fatigue life is negligible (the correction factor imosed on the Manson-Coffin equation is close to 1). Finally, the numbers of cycles to failure, calculated using the Eq. (1) to (7) for various temerature levels, are resented in Table. The constants β and C in the Manson- Coffin equation were roosed and verified by Hamada and Tanaka 7 for stainless steel bellows at room temerature. For the cryogenic temeratures we roose to kee the same value of β and to modify the value of C by using Eq. (2) according to the reduction of f 0,4. The values of β and C are listed in Table 4. Table. Numbers of cycles to failure for 0.8 internal ressure. Temerature N f EJMA tem. EJMA N f N f from fatigue factor T Manson-Coffin f with factor T f curves for 16 L ~ ~ > Table 4. Constants in the Manson-Coffin equation. Temerature β C Another sequence of comutations has been erformed for the same bellows subjected to internal ressure of 2.0 (ossible for the bellows with a rotection sleeve) suerimosed with the same cyclic axial deflection as before. This numerical test has been dedicated to the effect of the mean lastic strain (resulting from the relatively high internal ressure) on the fatigue life of U-bellows at room and cryogenic temeratures. The lastic strain intensity evolutions (at root and crest of the convolution, Fig. 2) are resented in Figs. 11, 12 and 1 for the temeratures of 00, 77 and 4, resectively. Here, the resence of the so called local shakedown henomenon (local decay of lastic cycling) is observed at 77 and 4, as well. The numbers of cycles to failure, calculated without (the classical Manson-Coffin (1)) and with (the modified Manson-Coffin ()) the effect of the mean lastic strain on the fatigue life, are given in Table. 5. The arameter α in the ower law () has been assumed equal to.5. This assumtion, based on some revious studies 1,2, should be regarded as a hyothesis and must be exerimentally verified. 5

6 Table 5. Numbers of cycles to failure for 2.0 internal ressure. Temerature N f 0 N f f 0 classical Manson- modified Manson- Coffin (1) Coffin () Tests of fatigue of U-bellows at low temeratures For testing the stainless steel bellows at cryogenic temeratures a secial cryostat equied with a neumatic servo-mechanism was designed (Fig. 14). A secial chamber (volume B) containing the bellows was installed inside the cryostat (volume C). Two bellows are linked along the same axis and cycled simultaneously (one of them is extended, the other one is comressed) in order to kee the total volume inside the bellows (volume A) constant. This system ermits to kee the internal ressure at a stable monitored level during testing. The tests may be erformed at the temerature of liquid nitrogen or liquid helium. A lateral offset between the two bellows may also be introduced, if required. The bellows described in the revious ages, comosed of 22 convolutions, were cycled u to ruture under internal ressure of 0.8 and cyclic axial movement of ± 25 mm. A first test was erformed with liquid nitrogen (77). A raid decrease of the ressure in volume A after 7986 cycles indicated the develoment of a crack in one of the bellows. It turned out that the circumferential crack occurred at the crest of the second convolution (influence of the boundary effect), next to the welding seam. The other bellows was further tested u to 8580 cycles with no ressure inside, and without any visible crack develoment. Second test, erformed at room temerature under the same loading conditions, gave 58 cycles to failure. Further tests are laned for the nearest future. 7. Conclusions The method of evaluation of the lastic strains evolution in the bellows wall, combined with the extended Manson-Coffin equation, aears to be a ractical engineering tool when designing exansion joints for cryogenic temeratures. The fatigue life obtained by using this method should be regarded as a reference value and, if ossible, confirmed by suitable tests. The calculated fatigue life is comared in Table with the results of alication of the standard formulae (EJMA 8 ) with the temerature correction factor T f extended to cryogenic temeratures. This factor is defined as the ratio of the ultimate strength at low temerature to the ultimate strength at room temerature, calculated for the bellows material (16 L). Additionally, the fatigue life of bellows at low temeratures was evaluated using the fatigue curves for the total strain range, resented by Suzuki et al. for grade 16 L stainless steel. The latter estimates seem rather to confirm the values obtained by alying the Manson-Coffin equation. However, a comlete verification of the roosed calculation technique requires a large number of tests. Eventually, it can be concluded from the above resented analysis that design of bellows for cryogenic temeratures, based on the existing standards, without alication of an aroriate temerature correction factor may lead to too essimistic results, thus influencing the design and the cost of the whole junction. 6

7 References 1. L.F. Coffin, The deformation and fracture of a ductile metal under suerimosed cyclic and monotonic strain, in: Achievement of High Fatigue Resistance in Metals and Alloys, ASTM STP 467, American Society for Testing and Materials, 1970, B. Skoczen, Generalization of the Manson-Coffin equation with resect to the effect of large mean lastic strain, sent to Journ. of Eng. Mat. Techn. (1994)... Suzuki, J. Fakakura, and H. ashiwaya, Cryogenic fatigue roerties of 04 L and 16 L stainless steels comared to mechanical strength and increasing magnetic ermeability, Journ. of Test. and Evaluation 16: 190 (1988). 4. T. Ogata,. Ishikawa,. Nagai, O. Umezawa, and T. Yuri, Low cycle fatigue and other mechanical roerties of aged 16 LN stainless steel at liquid helium temerature, Adv. in Cryogenic Eng. (Mat.) 6: 1249 (1990). 5. A. Sadough-Vanini, P. Lehr, Low cycle fatigue behaviour of 18% Cr- 10% Ni stainless steel (04 L) at room and liquid nitrogen temeratures, La Revue de Met. -CIT/ Sci. et Genie des Mat., 1994, Engineering roerties of austenitic stainless steels, in: Materials for Cryogenic Service, INCO databooks, International Nickel Limited, M. Hamada and M. Tanaka, A consideration on the low-cycle fatigue life of bellows ( the fatigue life under comletely- reversed- deflection cycles), Bull. JSME 17:41 (1974). 8. Exansion Joint Manufacturers Association, Standards of the EJMA, Inc., 25 North Broadway, Tarrytown, New York