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英文翻譯Ratcheting Behavior of T225NG Titanium Alloy under Uniaxial Cyclic Stressing: Experiments and ModelingAbstract: In order to investigate the ratcheting behavior of T225NG alloy, a series of ratcheting tests under uniaxial longcyclic stressing were performed. The results show that the ratchet ing strain of this alloy can get into shakedown after tens ( or hundreds) of thousand cycles. After the ratcheting strain is saturated under the condition that stress amplitude is half of peak stress, it will bring about subsequent fatigue failure, and relationship between fat igue life and one of peak stress and saturated ratcheting ( SR) strain meets pow er law . As the alloy is under stress jiggling with stress amplitude that is 1%25% of peak stress, the ratchet ing strain still become remarkable and goes into shakedown after several hundreds of thousand cycles but there exists little accessional strain caused by creep effect. It is notable that, w hen t he peak stress is 85%100% of yield stress, t he longcyclic stressing will lead SR strain to be from 14% to 2 5% even if the initial ratio of ratcheting strain is zero. Based on ratcheting threshold proper ty of peak stress and monotonicity of relationship between the peak stress and SR strain, a saturated ratcheting model ( SRM ) is developed to predict SR strain and to estimate saturated creep strain also. In addition, t he classes of ratchet ing evolutions of metals are discussed.Key words: T225NG; t itanium alloy; shakedown; ratcheting; cycle; plasticity; stress; strain; fa tigue; creepGradual increase plastic strain may be pro duced for metal materials under cyclic stressing and such kind of deformation w ith plastic strain accumulation is named as the ratcheting deform ation. The durative accumulation of ratcheting deforma tion can cause structures to exceed their size limit or to speed up their failure. The ratcheting strain can be defined by(1)w here the subscript T denotes the true state of strain and stress. For the true strain T and the true stress T , there are tw o expressions: T = ln( 1+ ) and T = ( 1 + ) w here and are corresponding to the uniaxial engineering strain and nominal stress ( axial force P / initial crosssection area A of specimens) , respectively. In Eq. ( 1) , T , m ax and T, min are the maximum and the m ini mum of true strain in each cycle, respectively; and T, R and T , L are the true stresses corresponding to T , m ax and T, min, respectively. T he ratcheting strain rate is given by（2) where N is the stress cyclic number. During the past tw o decades, scholars general ly believed that the m ain parameters controlling the uniaxial ratcheting behavior are stress am plitude a, mean stress m, stress rate , cyclic number N , temperature T , stress history and tem pera ture history !. Hence the ratcheting strain can be expressed as （3）Because of the complexity of the control pa rameters, researchers gave up systematic tests needed for given material and their traditional re search attention w as turned to the constitutive modeling on the basis of a few specific ratcheting experiments. In this w ay, the ratcheting strain is to be predicted theoretically[ 13] . In the existing constitutive models based on the yield surface theo ry, there are too m any undetermined parameters including some to be determined only by manm ade redressal. Therefore, the predictions of ratcheting deformation on the existing models are much more difficult. As the precision of these models can not satisfy various loading conditions, they still have long w ay to be developed for engineering applica tion. From a new angle, the authors performed series of uniaxial ratcheting tests under cyclic stress ing on 304, 1Cr18Ni9Ti stainless steel and T225NG titanium alloy [ 711] and studied the influ ences of a, m, v ( valley stress ) , p ( peak stress) , N , T , and !on ratcheting strain. A parabolic constitutive relation to describe the satu rated ratcheting ( SR ) strain based on unitary ratcheting stress r and tw o sets of m ethods to de scribe the saturated and nonsaturated ratcheting revolution w ere presented. Then a singlespecimen method to build the ratcheting constitutive rela tionships at room and elevated temperature w ere given. T hese developed methods provide new know ledge to control the ratcheting deformations of engineering materials and to study further on com plex ratcheting deformations. From the experimen tal and theoretical studies in the Refs. [ 211] , it is know n that the ratcheting behaviors of materials all are limited to cycle range w ithin tens or hundreds of cycles. Ratcheting deformation can cause structures to exceed their size limit, to rapture or to lose their stability, and this kind of problems has been taken into account in the ASME standard[ 12] . How ever, the ratcheting items in the standard are not com plete due to existing unknow n ratcheting and ratchetingfatigue behaviors. Experiments in the past disclosed that under cyclic stressing, som e ma terials w ill get into ratcheting shakedown only within hundreds of cycles[ 8, 9] but others can not do so [ 2, 4, 10] . U p to now , there is still lack of system atic experimental investigations on the ratcheting behaviors of some materials under longcyclic stressing in thousands of or hundreds of thousand cycles. A series of uniaxial ratcheting tests on T225NG alloy are performed in this paper. It is shown that under some stressing conditions this al loy has ratcheting shakedown after tens of thousand or hundreds of thousand cycles. Some behaviors of T225NG about ratcheting evolution, fatigue dam age, m icroamplitude ratcheting and ratcheting shakedow n in thousands of cycles are investigated and the saturatedratcheting constitutive relation ships based on unitary peak stress are presented. 1 Material, Facility and Test MethodExperiments are performed on uniaxial speci mens of T 225NG titanium alloy. The chem ical compositions ( mass percentage, %) of the alloy are C: 002, Si< 004, Al: 20, Zr: 245, Fe: 003, N: 0016,H : 0001 and O: 008 and the specific elon gation ?25 is 24% 。 In order to make microstructure of the alloy homogeneous and bring better stability of the me chanical properties, the specimens are treated by solid solution w ay w ith 120min heat preservation at 750 ? and subsequent cooling spontaneously in a tube vacuum furnace. The diam eter of the speci mens is 60 # 002m m and the w orking length is 18mm.T he test facility is M TS809 ( 25kN) , an elec trohydraulic servo material testing system, and its control system is TestStarII. The softw are used for data acquisition and test control is MT S790. 10. T he precision grades of both the M TS load cell and M TS axial extensometer are 02. All ratcheting tests are under uniaxial cyclic stress controlling with triangular waveshaped load ing signals: 0? p p- 2a , w here p and a are the peak stress and the stress amplitude, respec tively. In addition, the stressing rate is 200M Pa/ s. T able 1 gives the stressing conditions at room temperature and one specimen is tested for each stressing condition. T he stressing condition is nam ed in the form of p v , for example, for a specimen of T225NG alloy 500 200 means the cyclic stressing condition w ith 500 M Pa peak stress and 200MPa valley stress.Table 1 Cyclic stressing conditions for T225NG alloy2 Uniaxial Properties and the Const itutive Relat ionship of T225NG AlloyYoung% s modulus E , yield stress s and the strength limit b will be used to determine ratchet ing test parameters. From the monotonic tension tests performed on two specim ens of T 225NG al loy, it is know n that tw o curves are close to each other and have obvious yield plateau w ith 5000#strain range, therefore, the uniaxial consti tutive relationship can be expressed aswhere Young% s modulus E is equal to 107GPa; the hardening coefficient ?and the hardening exponent % are 770MPa and 01365, respectively; and the yield stress and the strength lim it are 398MPa and 540M Pa, respectively3 Ratcheting Evolution and Fatigue Damage of T225NG Alloy3.1 Ratcheting evolutionExperimental results under loading conditions in Tab. 1 show that plastic deformation w ill happen if only the peak stress is not less than 360 M Pa. Fig1 gives the Nr experimental curves of T225NG alloy as mean stresses are various from 185M Pa to 410MPa and the peak stress is eitherFig. 1 Experimental relationships between N and r?420M Pa or 390MPa. From Fig1, it is clear that the ratcheting strain rate decreases greatly at the beginning of the evolution and goes dow n gradually to small( 01#/ cycle) after tens of thousand cycles. T he ratcheting deformations of the alloy under loading conditions listed in Table 1 all can reach ratcheting shakedow n ( saturated ratcheting defor mation) if the shakedown criterion is defined as the ratcheting state when the ratcheting strain rate is less than 10- 2#/ cycle during cyclic stressing. T he additional tests performed show that the specimens w ill lose their ratcheting stability if p is greater than 430MPa. Fig1 can also shows that during ratcheting w ithin five tens of thousand cycles, the Nr experimental results of the T225NG speci mens under loadings w ith the sam e peak stress and different mean stresses form a bundle of curves in a sm all dispersive zone. In other words, peak stress except for mean and amplitude stresses controls the ratcheting evolutions of the specimens. As the alloy is subjected to elastic loading w ith 360 MPa peak stress, there still exists a large SR strain about 1. 62% ( see Fig. 2) . Abnormally, theFig. 2 Experimental relat ionship between N and r with low er stress levelinitial ratcheting strain rate is almost zero in first 20000 cycles and increases quickly in the cyclic pe riod from 20000 to 30000 cycles and then decreases quickly to zero in the next period from 30000 to 40000 cycles ( see Fig. 3) . In this case, variation of the ratcheting strain rate has such a process as: in creasing very slowly ?increasing quickly ? decreas ing to zero ( shakedown) and the cyclic number at the beginning stage of this kind of ratcheting revolution should be random.Fig. 3 Evolution law of ratchet ing strain ratio w ith lower stress level3. 2 Fatigue damageU nder the loading conditions w ith the peak stresses from 375 to 430 M Pa in T ab. 1, the ratch eting evolutions of all specimens show n in Fig. 1 will get into shakedown no matter how the ampli tude stress varies. How ever, as the stress ampli tude a is equal to p/ 2, all the tested specimens have the accumulation of fatigue damage follow ing the plastic deformation accumulation and get into failure after ratcheting shakedow n ( see Fig4) .Fig. 4 Experimental relationships betw een N and r w ith fatigue damageFig. 5 gives the experimental results of NE (unloading elastic modulus) of tw o specimens un der the cyclic stressing w ith the same peak stress: 420M Pa and different amplitude stresses: p/ 2 and p/ 6. It show s that the unloading elastic modulus decreases gradually w ith the cycles as a equals to p/ 2. This show s that greater stress amplitude w ill cause the fatigue damage and make the m aterial cyclic softening and the stiffness reducing. When the stiffness seldom changes, w hich indicates that the 6toequls pa??fatigue dam age is so little that the cyclic softening is not noticeable. Like the effect of the strain amplitude in the case of strain fatigue, the stress amplitude in the case of ratchet ing loading is a important parameter to control fa tigue damage, and the greater the amplitude is, the more serious the damage w ill be. It also should be noticed that the initial unloading modulus for the ratcheting case in Fig. 5 ( stress rate 6GPa/ s) goes dow n 8. 4% relatively to Young% s modulus.Fig. 5 Evolution laws of unloading elastic modulus of the alloy wit h two different stress amplitudesExperimental relationship between 2N f and p ( Fig. 6) w hen a = p/ 2 exhibits its monotonic property well. So, if failure happens to T 225NG specimens when a = p/ 2, the fatigue life of T 225NG is determined by the peak stress. In this case, the following power law equation can be used to describe the experimental relationship 2N f- p of T225NG.Fig. 6 Variation curves of saturated ratcheting strain and neak stress w ith twice lifeThe correlation coefficient R of this equation to ex perimental data is 0936. Fig6 also show s the ex perimental relationship between SR strain r, s and the tw ice life 2N f and it meets the power lawThe correlation coefficient R of Eq. ( 6) is 0985. Clearly, this better monotonic behavior is due to the plastic deformation accumulations which direct ly influence the fatigue damage evolution of the al loy. All the results above lead to the conclusion that under the ratcheting conditions, the fatigue life of T 225NG can be predicted by stress ampli tude and especially by the SR strain if only the re lationship betw een peak stress and amplitude stress is fixed.3.3 Uniaxial ratchetingevolution classification of metallic materialsU p to now , know ledge of ratcheting evolution laws of metallic materials is not systematical be cause most researches were mainly focused on the ratcheting behavior in a short cyclic period from tens to hundreds of cycles[ 2, 6, 10] . T o predict sub sequent ratcheting strain according to know ledge from ratcheting tests in a short cyclic period may result in w rong conclusion.Based on the test results[ 9, 10] of 304 and 1Cr18Ni9Ti stainless steel and the present results of T225NG alloy in this paper, the uniaxial ratch eting strain evolution law s can be divided into three types: ( , ) and ? ( see Fig. 7) . For type I of ratcheting, ratcheting strain rate at beginning w ith larger value w ill decrease to below a small value in a cyclic period, e. g, 10- 2 #/ per cycle, and in this final cycling, ratcheting gets into shakedow n ( see subtypes: I0 and I1 in Fig. 7) . As cyclic stressing is at elastic stage, the special ratcheting evolution shown in Fig. 2 may happen. This case belongs to subtype I2. For subtype I3 of ratcheting ( see Fig4) , ratcheting deformation will get into shake dow n at first cyclic stage and low cycle fatigue un der cyclic stressing w ill m ake a specimen failed at subsequent cyclic stage.Fig. 7 Uniaxial evolution behaviors of ratcheting strain of metallic materialsAccording to the equation: T, p= ( 1+ max ) ! ( Fp/ A ) w here both peak force Fp and crosssector area A are fixed, the true peak stress w ill become larger ow ing to that max w ill vary to be larger w ith plastic strain accumulation and then it returns to enlarge the plastic strain accumulation. In this cyclic stressing ( see Fig. 7) , ratcheting strain rate w ill decrease from larger value at beginning to a constant and increase gradually to lead specimen faild in subsequent longcyclic period. T his ratchet ing is defined as type II. Type III of ratcheting stands for those of w hich ratcheting strain rate de clines quickly to a certain value and then goes up rapidly to lead a specimen failed ( see Fig. 7) .T he discussions above are helpful to under stand the ratcheting strain evolution law s and the fatigue dam age behavior in different cyclicstressing periods and to find w ay to control plastic strain ac cumulation of materials.4 Ratcheting Shakedow n Properties of T 225NG4. 1 The effect of peak, amplitude and mean stressesAs is show n in Fig. 1, the ratcheting strains of the T225NG specimens can reach shakedow n or ratcheting saturation after several tens of thousand cycles. Fig. 8 gives all the experimental data ( p, r, s) and the corresponding initial data ( p, r, 0) . It is indicated from Fig. 8 that peak stress is the u nitary control parameter for the uniaxial SR strain of T225NG and the effects of amplitude and mean stress are not remarkable. Therefore, the peak stress p is the key parameter that determ ines the ratcheting shakedow n. It is w orth noticing that al though the initial ratcheting strain r, 0 is zero when the peak stress is below the yield stress 398M Pa, the SR strain is able to reach a large value about 15 000- 30 000#, and even w hen the peak stress is 360M Pa ( 95% below yield stress) , the alloy specimens still have the SR strain about 15 000#. This phenomena of metallic materials should be paid attention to in engineering field.Fig. 8 Experimental results of pr, s and pr, 04. 2 The effect of microamplitude cyclic stressingSome structures such as tubes and prestress steel bars are subjected to the pulse load w ith micro amplitude and certain mean stress, but such kind of ratcheting deformation is still not to be paid atten tion to by engineers yet. The experimental results ( see Fig9) under the microamplitude loadingFig. 9 Effects of stress jiggling on relationship betw een peak stress T, p and saturated ratcheting Strain r, sconditions listed in Tab. 1 indicate that there still exist larger ratcheting strains w hich can reachshakedown when specimens are subjected to the microamplitude stressing. Fig. 9 also shows that the T, pr, s curve under microamplitude stressing is somew hat higher than that under largeam plitude stressing. This is because the cyclic periods of the microamplitude stressings are 29 times than what the largeamplitude stressings have and longer cyclic periods may m ake the plastic strain accumu lations more completely. Ow ing to that the experi mental data ( p, r, s) have sm all disperive zone, the capability driving plastic strain accumulation by peak stress is invariable, so that the accessional ef fect of creep on the ratcheting shakedow n under microam plitude stressing is not remarkable, i. e. , there is little influence of stress amplitude on the plastic strain accumulation.Creep is a special stressing condition as the stress amplitude is closed to zero. Now that the ef fect of stress amplitude is not evident to SR strain, traditional creep tests can be replaced by ratcheting tests to predict plastic strain accumulation in shake dow n. This cognition m ay bring an improvement on the experimental study of creep in method.4. 3 Ratcheting threshold and saturatedratchet ing constitutive relationshipAccording to the saturatedratcheting proper ties of T225NG at room temperature for longcyclic stressing and the monotonicity of T , pr, s show n in Fig9, the concepts of ratcheting threshold stress r, th and ratcheting stress r given in papers, and the parabolic law betw een the unitary stress r and the SR strain all are still available. As can be seen from Fig. 10, the experimental relationship ofFig. 10 SRM and experimental relationship betw een T, p and r, sof T225NG alloy still has a parabolic trend, but the regressed parabolic curve does not intersect the stress coordinate, w hich is different from the law for stainless steels present in Ref. [ 9] . This indicates that once the plastic accumula tion starts, the SR strain can achieve larger value about 14500#. and under the cyclic stressing at room temperature, the saturatedratcheting consti tutive model ( SRM ) for T225NG alloy can be ex pressed as follow ingswhere &r, kr, r, th, k and c 0 are m aterial con stants. By regressing the experimental data, it can be got that r, th, &r, kr , r, th, k and c 0 are equal to 340M Pa, 05655mm/ mm , 0001433mm/ mm, 00145mm/ mm, 10415 and r, th/ E , respectively. T he strain r, th is the extremum of the parabolic SRM that is corresponding to the ratcheting threshold stress r, t h. For some materials such as stainless steels that would shakedow n after tens of or hundreds of cycles, r, th is zero.T he ratcheting threshold stress of T 225NG, 340M Pa, is dow n 146% from the yield stress, so that a large ratcheting strain for the alloy w ill still be accumulated even though the peak stress is low er in the range from 854% s to s. It is got ever that the ratcheting threshold stress is 384M Pa[ 10] according to the ratc