SAFE STRUCTURAL DESIGN FOR FATIGUE AND CREEP USING CYCLIC YIELD STRENGTH

Transcription

1 SAFE STRUCTURAL DESIGN FOR FATIGUE AND CREEP USING CYCLIC YIELD STRENGTH Yevgen Gorash * and Donald MaKenzie Department of Mehanial and Aerospae Engineering, University of Strathlyde, Glasgow, Sotland, UK ABSTRACT This study proposes yli yield strength (CYS, σy) as a potential harateristi of safe design for strutures operating under fatigue and reep onditions. CYS is defined on a yli stress-strain urve (SSC), while monotoni yield strength (MYS, σy m ) is defined on a monotoni SSC. Both values of σy and σy m are identified using a 2-step fitting proedure of the experimental SSCs using Ramberg-Osgood and Chabohe material models. A typial S-N urve in stress-life approah for fatigue analysis has a distintive minimum stress lower bound, the fatigue endurane limit (FEL, σlim f ). Comparison of σ y and σlim f reveals that they are approximately equal. Thus, safe fatigue design is guaranteed in the purely elasti domain defined by the σy. A typial long-term strength (LTS) urve in time-to-failure approah for reep analysis has 2 infletions orresponding to the σy and σy m. These infletions separate 3 setions on a LTS urve, whih are haraterised by different reep frature modes and reep deformation mehanisms. Thus, safe reep design is guaranteed in the linear reep domain with brittle failure mode defined by the σy. These assumptions are onfirmed using 3 strutural steels for normal and high-temperature appliations. The advantage of using σy for haraterisation of fatigue and reep strength is a relatively quik experimental identifiation. The total duration of yli tests for a yli SSC identifiation is muh less than the typial durations of fatigue and reep rupture tests at the stress levels around the σy. Keywords: Creep, Fatigue, Failure, Plastiity, Softening, Steel, Yield Strength. INTRODUCTION Charaterisation of long-term strength of strutural materials is an important engineering task for prevention of potential atastrophi failures of ritial equipment. However, studies of this type are usually very long-lasting, tehnially hallenging and involve expensive experimental work. Thus, the main sope of this study is the formulation of a simple way to predit harateristis of the longterm material behaviour (reep and fatigue, in the first instane) using basi material properties. Based upon the extensive availability of experimental material data, a signifiant progress toward this hallenge has been ahieved so far and may be observed in the literature. Comparative study by Kim et al. [1] evaluated seven basi methods for estimating uniaxial fatigue properties (inluding σlim f ) from tensile properties or hardness. This study was based upon the fatigue test data for eight dutile steels under axial and torsional loading. Three of the evaluated methods were able to predit over 93% of test ases within a fator of 3 ompared with observed lives. The formulas for σlim f predition inluded mehanial properties suh as elastiity modulus E, ultimate tensile strength σ u and true frature dutility ε f. Among the variety of empirial formulations for σlim f predition with different ombinations of aforementioned mehanial properties, the simplest are based on σ u : σlim f = σ u (Universal slopes method); σlim f = 1.5 σ u (Uniform material law); and σlim f = σ u MPa (Mithell s method), whih shown an auray of R 2 = Another simple method in this omparison, proposed by Roessle & Fatemi [2], used a Brinnell hardness * Corresponding author. Tel.: yevgen.gorash@strath.a.uk

2 HB for predition as σlim f = 4.25 HB MPa. This approah showed a reasonable auray of R 2 = 0.86 for experimental data fit. The study by Casagrande et al. [3] investigated a relationship between σlim f and Vikers hardness HV in steels and developed a method to predit σlim f. A good orrelation was observed between HV and σlim f for for four kinds of steels in different metallurgial states. However, the proposed empirial method is not straightforward and involves a number of parameters and equations to ahieve a reasonable of auray of σlim f preditions. Reently, Bandara et al. [4] proposed a formula for prediting σlim f of steels in the gigayle regime. It uses a ombination of σ u and HV as material parameters and was verified using the experimental results for 45 steels. A different approah was developed by Li et al. [5], who estimated theoretially σy and σlim f using test data for 27 alloy steels. One formula expresses σy by two onventional mehanial performane parameters σ u and the redution in area ψ. The other formula expresses the FEL by the CYS with a reasonable auray of R 2 = as σlim f = 1.13 (σ y) 0.9. Despite the relative simpliity, the proposed relation an t be onsidered as mathematially elegant, most probably beause of the onventional assumption of 0.2% plasti strain offset for σy and σy m. Nevertheless, this formula by Li et al. [5] demonstrated the tendeny that σlim f is not too muh different from σ y. Less progress has been ahieved in methods for reep rupture strength evaluation, but reently an important observation was disovered by Kimura [6]. The reep strength of ferriti and austeniti steels has been investigated in [6] through the orrelation between reep rupture urve, presenting stress vs. reep rupture life, and 50% of 0.2% offset yield stress (half yield) at a wide range of temperatures. The infletion of the reep rupture urve at half yield was reognised for ferriti reep resistant steels with martensiti or bainiti mirostruture, e.g. T91, T92 and T122. This was explained in terms of different mehanisms of mirostrutural evolution during reep at high- and low-stress regimes. The purpose of this study was to point out a signifiant risk of overestimation of long-term reep rupture strength by extrapolating the data for martensiti and bainiti steels (e.g. ASTM T91/P91) in high-stress regime to low-stress regime, whih are separated by half yield. A similar problem with partiular appliation to ASTM P91 steel was investigated and disussed by Gorash et al. [7, 8] for the purpose of a reep onstitutive model development. In these works, apart from infletion of reep rupture urve, the simultaneous infletion of the minimum reep rate urve, presenting minimum reep rate vs. stress, was reognised. Alternation of minimum reep rate slope was explained in terms of different reep deformation mehanism (linear reep for low stress and power-law for high stress), while alternation of reep rupture life slope was explained in terms of different damage aumulation modes (brittle frature for low stress and dutile for high stress). The infletion of both urves was haraterised by the same value σ 0 alled transition stress, whih had the meaning of material parameter in the developed double-power-law reep model. However, σ 0 was identified in [7, 8] using minimum reep rate data, and no relation of σ 0 to basi mehanial properties of ASTM P91 steel was reognised. The prinipal aim of the present study is to investigate a link in haraterisation of long-term strength of strutural steel by finding a similar quantative feature in available experimental data. This establishes a straight relation between harateristis of reep and fatigue behaviour on one hand and yield strength as a basi material property and harateristi of plastiity on other hand. CONCEPT OF THE SAFE STRUCTURAL DESIGN Definition of the yield strength Dowling [9] disusses several methods to haraterise the yield strength σ y. The first is the proportional limit σy, p whih is the stress where the first departure from linearity ours. The seond is the elasti limit σy el, whih is the highest stress that does not ause plasti deformation. The third is the offset yield strength σy 0.2%, whih is the stress in the point on stress-strain urve typially defined by the plasti strain offset of 0.2% from elasti line. This value is generally the most pratial means

3 stress, ultimate strength, u monotoni S-S urve yli S-S urve m monotoni yield strength, y yli yield strength, y fatigue endurane limit fatigue S-N urve plasti behaviour mixed behaviour aused by softening elasti behaviour safe design low moderate high normal temperature design total strain, number of yles to failure, log( ) stress, log( ) ultimate strength, monotoni yield strength, u m y yli yield strength, minimum reep strain rate urve y power-law breakdown power-law reep linear reep safe design dutile frature mixed mode frature brittle frature reep rupture urve transition stress low moderate high high temperature design min. reet rate, log( ) time to failure, log( t ) Figure 1: Conept of the safe strutural design for fatigue and reep using yli yield strength of defining the yielding event for engineering metals. Therefore, σy 0.2% is usually meant to define the yield strength σ y in the literature. However, here the elasti limit σy el, defined in the sope of unified Chabohe model [10, 11], is used as the yield strength σ y. This study proposes σy as a key harateristi for the definition of safe design for engineering strutures operating under fatigue and reep onditions, as illustrated in Fig. 1. It is onventionally defined in ontext of a yli stress-strain urve (SSC), whih is obtained from results of yli tests for a number of different strain ranges. Eah yli test produes a stabilised stress response, whih is effeted either by hardening or by softening depending on the type of steel. In the ase of steels with a yli softening effet, σy separates the low stress range of purely elasti behaviour from moderate stress range of mixed elasto-plasti behaviour. Monotoni yield strength σy m, whih is onventionally defined in ontext of a monotoni SSC, separates the moderate stress range of mixed elasto-plasti behaviour from the high stress range of purely plasti behaviour. Both values of σy m and σy are identified using a 2-steps fitting proedure of the experimental S-S urves. The first step applies the Ramberg-Osgood material model, whih produes basi smoothing and extrapolation, to the both monotoni and yli SSCs separately. The seond step of fitting involves a typial rate-independent form of the Chabohe material model with 3 kinemati bakstresses. Fitting the Chabohe model with two separate sets of material onstants sequentially to the both SSCs provides the values of σy m and σy with minimum offset from the elasti line as elasti limits. Stress-strain urves fitting proedure As experimental SSCs usually demonstrate some level of satter, the first step in data fitting for the material parameters identifiation is basi urve smoothing. The onventional Ramberg-Osgood (R-O) equation [15] is optimal for suh urve smoothing sine it was formulated to desribe the

4 elasti 600 elasti Stress (MPa) mono. data mono. fit yli data Stress (MPa) mono. data Ch. mono σy mono 150 R-O yli 300 yli data 100 Ch. yli σy yli 200 R-O yli Ch. mono σy yli Total strain Figure 2: Fitting of monotoni and yli SSCs of ASTM A36 steel from [12] at RT Total strain Figure 3: Fitting of monotoni and yli SSCs of AISI 4340 steel from [13] at RT Stress (MPa) elasti R-O mono Ch. mono σy mono yli data R-O yli Ch. yli σy yli Stress (MPa) elasti R-O mono Ch. mono σy mono yli data R-O yli Ch. yli σy yli Total strain Figure 4: Fitting of monotoni and yli SSCs of ASTM P91 steel from [14] at 550 C Total strain Figure 5: Fitting of monotoni and yli SSCs of ASTM P91 steel from [14] at 600 C non-linear relationship between stress and strain in materials near their yield point. It is partiularly useful for metals that harden or soften with plasti deformation showing a smooth elasti-plasti transition. The equations for the monotoni and yli SSCs are: ε tot = σ E + ( σ B ) 1/β and ε tot 2 = σ 2 E + ( ) 1/β σ, (1) 2 B where ε tot is the total strain range and σ is the total stress range (MPa) for eah yli test respetively; B and β are the R-O material parameters; and Young s modulus E in MPa. Using the value of E, the total strain ε tot in the experimental urves is deomposed into elasti and plasti strain. Then the plasti omponent ε p of strain is fitted using the the least squares method by the following power-law relations, whih are derived from the Eq. (1): σ = B (ε p ) β and σ 2 = B ( ) ε p β. (2) 2

5 Table 1: Fitting parameters of the Ramberg-Osgood model (1) for different steels and temperatures Type of plasti Elasto-plasti onstants material response E (MPa) B (MPa) β σ y (MPa) ASTM A36 RT yl AISI 4340 RT yl ASTM P91 RT mono ASTM P91 RT yl ASTM P C m ASTM P C ASTM P C m ASTM P C ASTM P C m ASTM P C ASTM P C m ASTM P C Extended version of the R-O model (6) is used for data fitting. The resultant R-O fits for monotoni and yli urves are then used to identify the parameters for the Chabohe material model. The range of appliability for the R-O fit is usually quite narrow not exeeding 1% of ε tot depending on the grade of urvature grade for a SSC. The basi variant of the rate-independent Chabohe model [10, 11] is presented as a ombination of nonlinear kinemati hardening and nonlinear isotropi hardening models. The model allows the superposition of several independent bakstress tensors and an be ombined with any of the available isotropi hardening models. Sine in this study monotoni and yli SSCs are fitted separately only for the identifiation of σ y, only the kinemati hardening omponent is onsidered: X = N X i, with Ẋ i = C i ε p γ i X i ṗ, (3) i=1 where ε p is the plasti strain rate, and ṗ is its magnitude. The total bakstress X in Eq. (3) is given by the superposition of a number N of kinemati bakstresses X i with a orresponding evolution equation initially proposed by Armstrong & Frederik [16] for Ẋ i, where C i and γ i are kinemati material onstants. Chabohe et al. [10] reommended N = 3 in order to provide a good fit of experimental SSCs, whih inlude large strain areas. Therefore, three bakstresses are onsidered in this study providing an exellent math of the R-O fit (1) for a whole range of strains. The kinemati hardening onstants (C i, γ i ) and σ y, whih define the size of the yield surfae, are identified as reommended in [11]. The yli SSC is fitted by the following relation: σ 2 = σ y + N i=1 C i γ i tanh (γ i ε p 2 ), (4) whih is obtained in [11] by integrating Eq. (3) and onsidering ε p onst at the peak stresses for strain-ontrolled yli loading. Relation (4) is valid for the yli urve after stabilisation of the hardening or softening effets. Constants (C i, γ i and yli σ y) are identified by automati fitting Eq. (4) to the R-O extrapolation (2) with yli values of onstants B and β. The identifiation proedure is implemented in Mirosoft Exel using an add-in Solver [17]. The Solver searhes for an optimal (minimum in this ase) value for a formula in one ell alled the objetive ell subjet to onstraints, or limits, on the values of other formula ells on a worksheet. The Solver works with a group of ells, alled deision variables or simply variable ells, that partiipate in omputing the formulas in the objetive and onstraint ells. In this ase, the Solver adjusts the values in

6 Table 2: Fitting parameters of the Chabohe model (3)-(5) for different steels and temperatures Type of plasti Three kinemati hardening bakstresses Yield σ material response C 1 (MPa) γ 1 C 2 (MPa) γ 2 C 3 (MPa) γ 3 σ y (MPa) ASTM A36 RT yl AISI 4340 RT mono AISI 4340 RT yl ASTM P91 RT mono ASTM P91 RT yl ASTM P C m ASTM P C ASTM P C m ASTM P C ASTM P C m ASTM P C ASTM P C m ASTM P C the deision variable ells ontaining material onstants (C i, γ i and σy) in order to minimise the value in the objetive ell. This ell ontains an average value of the absolute differene between olumns ontaining σ 2 alulated by Eq. (2) and Eq. (4) orrespondingly in a partiular range of ε p. Applying this approah, an exellent math of Eqs (2) and (4) is ahieved. The monotoni SSC is fitted by the different relation in the following form [11]: σ = σ m y + N i=1 C i γ i [1 exp( γ i ε p )], (5) whih ontains the monotoni σ m y and different values of kinemati hardening onstants (C i, γ i ). These onstants are identified by fitting Eq. (5) to the R-O extrapolation (2) with monotoni values of the R-O parameters B and β. The identifiation proedure is implemented in Mirosoft Exel using an add-in Solver [17] in the same way as for yli SSC. An advaned step-by-step guideline for the estimation of the Chabohe visoplastiity model parameters with their further optimisation was developed by Hyde et al. [18]. Appliation to three strutural steels The above desribed fitting proedure is applied to SSCs of three strutural steels for the purpose of σy m and σy identifiation. The first is ASTM A36 steel, with mehanial properties reported in [19, 12], whih is a standard low arbon steel, without advaned alloying and is a prinipal arbon steel employed for bridges, buildings, and many other strutural uses. The monotoni SSC for this steel shown in Fig. 2 exhibits perfetly plasti behaviour when reahing the stress of 36 ksi = MPa in average, whih is onsidered as σy m. The perfetly plasti yielding lasts for approximately of ε p = 1 (%) of strain plateau, whih is followed by the strain hardening area, then gradually approahing failure at ε tot = 30 (%). The yli SSC for this steel shown in Fig. 2 from [12] is fitted by the 2-step proedure, and the obtained material parameters for the R-O (1) and Chabohe (3)-(5) models are listed in Tables 1 and 2 orrespondingly. The seond material is AISI 4340 steel [13], a high-strength alloy steel, whih has good mahinability features and used for a wide range of appliations inluding airraft landing gears, shafts or axels for power transmission, gears, high pressure pump housings, et. Both monotoni and yli SSCs shown in Fig. 3 and mehanial properties are taken from [13]. Sine it is available expliitly, the monotoni SSC is fitted by the Chabohe model (5) diretly, and the material parameters are listed in Table 2. The yli SSC for this steel shown in Fig. 3 from [13] is available at ten times

7 wider strain range than for the ASTM A36 steel. Therefore, the R-O model (1) is not able to provide an aurate fit of the yli SSC. In this ase, the following modifiation of the R-O equation (1) is used for fitting analysis: ε tot = σ ( ) 1/β σ E + σy and B ε tot 2 = σ 2 E + ( ) 1/β σ σy, (6) 2 B Compared to Eq. (1), this notation ontains an additional parameter of the yield strength σ y in the meaning of σy el, and an be applied for an aurate fitting of muh wider strain range than Eq. (1). Thus, the yli SSC is fitted by the 2-step proedure. The obtained material parameters for the modified R-O (6) and Chabohe (3)-(5) models are listed in Tables 1 and 2 orrespondingly. The third material is ASTM P91 (modified 9Cr-1Mo) steel [20, 14], an advaned ferriti steel with martensiti mirostruture, whih has already been widely used over the last 2 deades as tubes/pipes for heat exhangers, plates for pressure vessels, and other forged, rolled and ast omponents for high temperature servies. Both monotoni and yli SSCs shown in Figs 4 and 5 and mehanial properties at room temperature (RT), 500 C, 550 C, 600 C and 650 C are taken from [14]. Firstly, the monotoni SSCs are presented in [14] by the material parameters for the R-O model (1) listed in Table 1. The yli SSCs are presented in [14] by raw data, whih is fitted by the R-O model (1) with material parameters listed in Table 1. Seondly, both monotoni and yli R-O extrapolations are fitted by the Chabohe model (3)-(5) with material parameters listed in Table 2. RELATION IN MECHANICAL CHARACTERISTICS The next step is a hek for possible orrelations between the obtained yield strength values (σ m y and σ y) for ASTM A36, AISI 4340 and ASTM P91 steels and their fatigue and reep behaviour. This identifies a lear similarity for harateristi transition stresses in S-N fatigue, minimum reep strain rate and reep rupture urves, as explained below. Fatigue behaviour at normal temperature Engineering strutures operating under yli loading onditions at normal temperature are usually designed against fatigue failure using the onventional stress-life approah. This approah involves experimental fatigue S-N urves with number of yles to failure N vs. stress. A typial S-N urve is a straight line in double logarithmi oordinates with a distintive minimum stress lower bound, whih is alled a fatigue endurane limit (FEL, σlim f ). Referring to [9, 18], σf lim is observed for a number of strutural steels in benign environmental onditions and represents a stress level below whih the material does not fail and an be yled infinitely without fatigue damage. Comparison of σy defined as material onstant and experimentally observed σlim f reveals that they are lose. This assumption is onfirmed by high-yle fatigue (HCF) experimental data for ASTM A36 [21] and AISI 4340 [22, 23, 24] steels shown in Fig. 6. Comparison of σlim f with σ y summarised in Table 3 for ASTM A36 steel gives 27.6% auray and 5.5% auray for AISI 4340 steel. These observations indiate that safe fatigue design is guaranteed in the purely elasti domain defined by σy. Creep behaviour at elevated temperature Engineering strutures operating under onstant loading onditions at high temperature are usually designed against reep failure using the onventional time-to-failure approah. This approah involves experimental reep rupture urves with stress vs. time to failure t. A typial reep rupture urve is a trilinear smoothed urve in double logarithmi oordinates, with two infletions orresponding to σ y and σ m y. These infletions separate three setions on the reep rupture urve, whih are haraterised by three different reep damage aumulation modes brittle, dutile and mixed. Three setions with different reep deformations mehanisms an be typially observed

8 Alternating stress (MPa) exp., Atlas of Fatigue Curves exp., Dowling (2004) exp., Ragab et al. (1989) exp., did not fail 4340 fatigue fit (σ_lim = 350MPa) exp., Wang et al. (2010) traditional exp., Wang et al. (2010) energy A36 fatigue fit (σ_lim = 160MPa) Number of yles to failure Figure 6: S-N urve fits of ASTM A36 steel based on HCF data by Wang et al. [21] and AISI 4340 steel based on HCF data from Atlas of Fatigue Curves [22], Dowling [23] and Ragab et al. [24] on the minimum reep rate urve, presenting minimum reep strain rate vs. stress, whih is also a trilinear smoothed urve in double logarithmi oordinates. The deformations mehanism (linear reep, power-law reep and power-law breakdown) are separated by the same two infletions. This assumption is onfirmed by experimental observations for ASTM P91 steel at elevated temperatures. Data for reep rupture shown in Fig. 7 is all taken from the reent study by Kimura [6]. The infletions of orresponding urves were well observed at 600 and 650 C and explained in terms of half monotoni yield (σy 0.2% /2). In ontrast to [6], in urrent study, σy m and σy from Table 2 are used in ombination with test data [6] to provide a basi polylinear fitting. Data for min. reep strain rate shown in Fig. 8 is taken from studies by Sklenička et al. [25], Klo & Fiala [26] and Kimura [20]. The infletions of orresponding urves were observed at 550, 600 and 650 C and explained in terms of transition between different reep deformation mehanisms. As in the ase of reep rupture, here the same σy m and σy from Table 2 are used in ombination with test data [20, 26, 25] to provide a basi polylinear fitting. Sine the infletions are aptured reasonably well on both types of data in Figs 7 and 8, the orrespondene of transition stresses on reep rupture and min. reep rate urves proposed by Gorash et al. [7, 8] is proved by relating them to σy m and σy. It should be noted that Dimmler et al. [27] assoiated these infletions with mirostruturally determined threshold stresses (bak-stress onept). The appliability of this onept was shown using the experimental minimum reep rate and reep rupture urves for several 9-12%Cr heat resistant steels (P91, GX12, NF616, X20 and B2). Dimmler et al. [27] emphasised that the knowledge of these threshold stresses limits the range of experimentally based preditions, thus preventing from overestimation of long-term reep rate and reep strength from extrapolated short-term reep data. Therefore, these observations arise a onsideration that the most safe reep design is guaranteed in linear reep domain with brittle failure mode, whih is also defined by the σy. Finally, the fatigue performane of ASTM P91 steel is analysed using the HCF experimental data by Matsumori et al. [28] at three different temperatures (RT, 400 and 550 C) illustrated in Fig. 9.

9 Stress (MPa) 100 exp., 500 C, Kimura (2013) stress-based data fit, 500 C exp., 550 C, Kimura (2013) stress-based data fit, 550 C exp., 600 C, Kimura (2013) stress-based data fit, 600 C exp., 650 C, Kimura (2013) stress-based data fit, 650 C yli yield σy, all temperatures monotoni yield σym, all temp Time to failure (h) Figure 7: Stress vs. reep rupture life of ASTM P91 steel based on the data by Kimura [6] Minimum reep strain rate (1/h) 1.0E E E E E E E E E-07 exp., 550 C, Kimura et al. (2009) exp., 550 C, Sklenika et al. (1994) stress-based data fit, 550 C exp., 600 C, Klo & Fiala (2005) stress-based data fit, 600 C exp., 650 C, Klo & Fiala (2005) stress-based data fit, 650 C yli yield σy, all temperatures monotoni yield σym, all temperatures 1.0E E Stress (MPa) Figure 8: Min. reep rate vs. stress of ASTM P91 steel based on several sets of data [20, 26, 25]

10 Alternating stress (MPa) exp., RT exp., 400 C exp., 550 C exp., did not fail RT fit (σ_lim = 418MPa) C fit (σ_lim = 350MPa) 550 C fit (no σ_lim exists) Number of yles to failure Figure 9: S-N urve fits of ASTM P91 steel based on HCF data by Matsumori et al. [28] From these data, it an be onluded that at elevated temperatures heat-resistant steels don t exhibit σlim f on S-N fatigue urves, whih is usually observed at normal temperature. The reason for this is the elimination of purely elasti behaviour at high temperature, sine there is always some amount of inelasti strain, whih is aused by reep. Therefore, there is always a permanent aumulation of reep damage, even at low stress levels and high-strain rate, whih leads to inevitable failure. This fat is onfirmed by experimental observations [28], whih demonstrated the extintion of σlim f at 550 C for over 10 8 loading yles. However, a good math of σlim f with σm y with auray of 2.8% is observed at RT for this steel as shown in Table 3, whih makes advaned martensiti steels different from simple ferriti steels is σlim f predition. This effet an be explained by the assumption of Terent ev [29], who reognised two types of the fatigue endurane limit σlim f standard in HCF range (N = yles) and ultrahigh in gigayle fatigue (GCF) range (N = yles). The existene of ultrahigh σlim f was proved by the experimental data for high-strength steels (50CrV4, 54SiCrV6 and 54SiCr6), whih demonstrated two infletions of the fatigue urves followed by horizontal plateaus first in HCF area (N ), seond in GCF area (N ). The orrespondene of σy with ultrahigh σlim f for ASTM P91 steel is expeted to be found at N > 108 yles, but no experimental data is available for this range. CONCLUSIONS Kimura s [6] assumption of half monotoni yield (σ 0.2% y /2) agrees very well with the outomes of the urrent study. Aording to Table 3, the relation σ y σ m y /2 is valid for all temperatures exept the highest 650 C. This assumption is not relevant to AISI 4340 steel, whih exhibits σ y σ m y. The prinipal advantage of the σ y appliation to the haraterisation of fatigue and reep longterm strength is the relatively fast experimental identifiation. The total duration of all yli tests, whih are required to reah the stabilised stress response for the onstrution of yli SSC is muh less than the typial durations of fatigue and reep rupture tests at stress levels around σ y.

11 Table 3: Comparison of σy m, σy and σlim f for ASTM A36, AISI 4340 and ASTM P91 steels Steel ASTM A36 AISI 4340 ASTM P91 Temp., C RT RT RT m, MPa σy, MPa σy m /σy σlim f, MPa σ, % The ritial point in the work presented here is an appliation of the advaned material model (i.e. Chabohe model [10, 11]) to the estimation of a single value of elasti limit σy el, whih may seem to be ompleted. However, this approah is effetive in typial ases when experimental SSCs are unavailable in expliit form, but available in the form of R-O [15] fittings (1). In other ases, when all neessary experimental SSCs are available in form of raw data, the modified form (6) of the R-O model may redue the fitting proedure just to one step. Sine Eq. (6) ontains σ y as a material parameter, the appliation of Chabohe model equations (3)-(5) is no longer needed. REFERENCES [1] Kim, K. S. et al. Estimation methods for fatigue properties of steels under axial and torsional loading. Int. J. Fatigue, vol. 24, no. 7, (2002), pp [2] Roessle, M. L. and Fatemi, A. Strain-ontrolled fatigue properties of steels and some simple approximations. Int. J. Fatigue, vol. 22, no. 6, (2000), pp [3] Casagrande, A. et al. Relationship between fatigue limit and Vikers hardness in steels. Mater. Si. & Eng. A, vol. 528, no. 9, (2011), pp [4] Bandara, C. S. et al. Fatigue Strength Predition Formulae for Steels and Alloys in the Gigayle Regime. Int. J. Mater., Meh. & Manufaturing, vol. 1, no. 3, (2013), pp [5] Li, J. et al. Theoretial estimation to the yli yield strength and fatigue limit for alloy steels. Mehanis Researh Communiations, vol. 36, no. 3, (2009), pp [6] Kimura, K. Creep rupture strength evaluation with region splitting by half yield. Pro. ASME 2013 PVP Conf. PVP , ASME, Paris, Frane (Jul , 2013), pp [7] Gorash, Y. Development of a reep-damage model for non-isothermal long-term strength analysis of high-temperature omponents operating in a wide stress range. PhD thesis, Martin- Luther-University Halle-Wittenberg, Halle (Saale), Germany (Jul. 21, 2008). [8] Altenbah, H. et al. Steady-state reep of a pressurized thik ylinder in both the linear and the power law ranges. Ata Mehania, vol. 195, no. 1-4, (2008), pp [9] Dowling, N. E. Mehanial Behavior of Materials: Engineering Methods for Deformation, Frature, and Fatigue. Pearson Eduation Limited, Harlow, UK, 4th ed. (2013). [10] Chabohe, J.-L. et al. Modelization of the strain memory effet on the yli hardening of 316 stainless steel. Trans. 5th Int. Conf. on Strutural Mehanis in Reator Tehnology, no. L11/3 in SMiRT5, IASMiRT, Berlin, Germany (Aug. 1979), pp [11] Chabohe, J.-L. A review of some plastiity and visoplastiity onstitutive theories. Int. J. Plastiity, vol. 24, no. 10, (2008), pp

12 [12] Higashida, Y. and Lawrene, F. V. Strain ontrolled fatigue behavior of weld metal and heataffeted base metal in A36 and A514 steel welds. FCP Report No. 22, University of Illinois, Urbana, Illinois, USA (Aug. 1976). [13] Smith, R. W. et al. Fatigue behavior of materials under strain yling in low and intermediate life range. Tehnial Note No. D-1574, NASA, Cleveland, Ohio, USA (Jan. 1963). [14] Data sheets on elevated-temperature, time-dependent low-yle fatigue properties of ASTM A387 Grade 91 (9Cr-1Mo) steel plate for pressure vessels. NRIM Fatigue Data Sheet No. 78, National Researh Institute for Metals, Tokyo, Japan (De. 25, 1993). [15] Ramberg, W. and Osgood, W. R. Desription of stress-strain urves by three parameters. Tehnial Note No. 902, NASA, Washington DC, USA (Jul. 1943). [16] Armstrong, P. J. and Frederik, C. O. A mathematial representation of the multiaxial Baushinger effet. Report No. RD/B/N731, CEGB, Berkeley, UK (De. 1966). [17] MirosoftR Offie Professional Plus. Exel Help System // Analyzing data // What-if analysis // Define and solve a problem by using Solver. Mirosoft Corp., Release 2010 ed. (2009). [18] Hyde, T. et al. Applied Creep Mehanis. MGraw-Hill Eduation, New York, USA (2004). [19] ASTM Standard. Standard Speifiation for Carbon Strutural Steel. A36/A36M 08, West Conshohoken, USA (2008). [20] Kimura, K. et al. Long-term reep deformation property of modified 9Cr-1Mo steel. Mater. Si. & Eng. A, vol , (2009), pp [21] Wang, X. G. et al. Quantitative Thermographi Methodology for Fatigue Assessment and Stress Measurement. Int. J. of Fatigue, vol. 32, no. 12, (2010), pp [22] Boyer, H. E. Atlas of Fatigue Curves. ASM International, Materials Park, Ohio, USA (1986). [23] Dowling, N. E. Mean Stress Effets in Stress-Life and Strain-Life Fatigue. SAE Tehnial Paper,, no , (2004), pp [24] Ragab, A. et al. Corrosion Fatigue of Steel in Various Aqueous Environments. Fatigue Frat. Engng Mater. & Strut., vol. 12, no. 6, (1989), pp [25] Sklenička, V. et al. Creep Behaviour and Mirostruture of a 9Cr Steel. D. Coutsouradis et al., ed., Pro. Conf. Materials far Advaned Power Engineering Part I, Kluwer Aademi Publishers, Liège, Belgium (Nov. 3-6, 1994), pp [26] Klo, L. and Fiala, J. Visous reep in metals at intermediate temperatures. Kovové Materiály, vol. 43, no. 2, (2005), pp [27] Dimmler, G. et al. Extrapolation of short-term reep rupture data The potential risk of over-estimation. Int. J. of Pressure Vessels & Piping, vol. 85, (2008), pp [28] Matsumori, Y. et al. High Cyle Fatigue Properties of Modified 9Cr-1Mo Steel at Elevated Temperatures. Pro. ASME 2012 Int. Mehanial Engineering Congress & Exposition. IMECE , ASME, Houston, Texas, USA (Nov. 9-15, 2012), pp [29] Terent ev, V. F. Endurane limit of metals and alloys. Metal Si. & Heat Treatment, vol. 50, no. 1-2, (2008), pp