## 1 Introduction

Characterisation of long-term strength of structural materials is an important engineering task for prevention of potential catastrophic failures of critical equipment. However, studies of this type are usually very long-lasting, technically challenging and involve expensive experimental work. Thus, the main scope of this study is the formulation of a simple way to predict characteristics of the longterm material behaviour under creep and fatigue conditions using basic material properties. Based upon the extensive availability of experimental material data, a significant progress toward this challenge has been achieved so far and may be observed in the literature as indicated below.

One should start with the fundamental work of Bäumel and Seeger [1], who discussed a few methods for estimating fatigue behaviour of metals using strain-life approach on the basis of monotonic test results. They included the Method of Universal Slopes (MUS) proposed by Manson [2] and modification of MUS developed by Muralidharan & Manson [3], which are applicable to all metals. However, this approach contained a critical limitation regardless of a good accuracy, which is the requirement of reduction in area *ψ* availability.

The flaw of MUS formulations induced Bäumel and Seeger [1] to develop the Uniform Material Law (UML) using as its basis the extensive collection of fatigue data [4] with over 1500 experimental results. Unalloyed and low alloy steels were analysed separately from aluminium and titanium alloys resulting into two sets of equations both being based on only elasticity modulus *E* and ultimate tensile strength (UTS) *σ*_{u}, which can be easily correlated with Vickers hardness *HV*. Later, the applicability of UML concept was extended to high-strength steels by Korkmaz [5].

Comparative study by Kim *et al*. [6] evaluated seven basic methods for estimating uniaxial fatigue properties (including fatigue limit *E*, *σ*_{u} and true fracture ductility ɛ_{f}. Among the variety of empirical formulations for *σ*_{u} : *σ*_{u} (MUS); *σ*_{u} (UML); and *σ*_{u} + 345 MPa (Mitchell’s method), which showed an accuracy of *R*^{2} = 0.88. Another simple method in this comparison, proposed by Roessle & Fatemi [7], used a Brinnell hardness *HB* for prediction as *HB* + 225 MPa. This approach showed a reasonable accuracy of *R*^{2} = 0.86 for the experimental data fit.

Casagrande *et al*. [8] investigated a relationship between *HV* in steels and developed a method to predict *HV* and *et al*. [9] proposed a formula for predicting *σ*_{u} and *HV* as material parameters and was verified using the experimental results for 45 steels.

An alternative approach was suggested by Li *et al*. [10], who estimated theoretically the cyclic yield strength *σ*_{u} and the reduction in area *ψ*. The other formula expresses the fatigue endurance limit through the cyclic yield strength with a reasonable accuracy of *R*^{2} = 0.883 as of *et al*. [10] demonstrated the tendency that

A computational approach was developed by Tomasella *et al*. [11], who applied the Artificial Neural Networks to estimation of the cyclic material properties used in strain-life fatigue approach from a set of monotonic material properties. This approach was implemented into the software called Artificial Neural Strain Life Curves (ANSLC), and has been tested on a large database of steels [1, 4]. In comparison with the largely used UML [1], the results of the estimation with ANSLC program, even without including the support of real experimental tests in the regression, showed a considerably higher accuracy in the life-time estimation.

Recently, Pang *et al*. [12] did a comprehensive review of the relations between *HV* or *HB*), strength (*σ*_{u} and *σ*_{y}) and toughness. Analysis of the numerous fatigue data resulted in the General Fatigue Formula (GFF): *σ*_{u} (*C* − *P* · *σ*_{u}), where *C* and *P* are fitting parameters. GFF was found applicable to *σ*_{u} in a wide range for many materials such as conventional metals and newly developed alloys. Pang *et al*. [12] suggested that GFF can provide a new clue to predict *P* and *C* adequately.

The concept of the fatigue limit

Less progress has been achieved in the methods for creep rupture strength evaluation, but recently an important observation was discovered by Kimura [14]. The creep strength of ferritic and austenitic steels has been investigated in [14] through the correlation between creep rupture curve, presenting stress vs. creep rupture life, and 50% of 0.2% offset yield stress (half yield) at a wide range of temperatures. The inflection of the creep rupture curve at half yield was recognised for ferritic creep resistant steels with martensitic or bainitic microstructure, e.g. T91, T92 and T122. This was explained in terms of different mechanisms of microstructural evolution during creep at high- and low-stress regimes. The purpose of this study was to point out a significant risk of overestimation of long-term creep rupture strength by extrapolating the data for martensitic and bainitic steels (e.g. ASTM T91/P91) in high-stress regime to low-stress regime, which are separated by half yield.

A similar problem with particular application to ASTM P91 steel was investigated and discussed earlier by Gorash *et al*. [15-17] for the purpose of a creep constitutive model development. In these works, apart from inflection of the creep rupture curve, the simultaneous inflection of the minimum creep rate curve, presenting minimum creep rate vs. stress, was recognised. Alternation of the minimum creep rate slope was explained in terms of different creep deformation mechanism (linear creep for low stress and power-law for high stress), while alternation of the creep rupture life slope was explained in terms of different damage accumulation modes (brittle fracture for low stress and ductile for high stress). The inflection of both curves was characterised by the same value

The principal aim of the present study is to investigate a link in characterisation of long-term strength of structural steel by finding a similar quantative feature in available experimental data. This work establishes relationships between characteristics of creep and fatigue behaviour on one hand and yield strength as a basic material property and characteristic of plasticity on other hand.

## 2 Concept of the safe structural design

### 2.1 Definition of the yield strength

Dowling [18] discusses several methods to characterise the yield strength *σ*_{y}. The first is the *proportional limit**elastic limit**offset yield strength**σ*_{y} in the literature. However, here the *elastic limit**σ*_{y}.

This study proposes the cyclic yield strength

### 2.2 Stress-strain curves fitting procedure

As experimental SSCs usually demonstrate some level of scatter, the first step in data fitting for the material parameters identification is the basic curve smoothing. The conventional Ramberg-Osgood (R-O) equation [24] is optimal for such curve smoothing since it was formulated to describe the non-linear relationship between stress and strain in materials near their yield point. It is particularly useful for metals that harden or soften with plastic deformation showing a smooth elastic-plastic transition. The equations for monotonic and cyclic SSCs are:

*Δɛ*

_{tot}is the total strain range and

*Δσ*is the total stress range (MPa) for each cyclic test respectively;

*B*and

*β*are the R-O material parameters; and Young’s modulus

*E*in MPa. Using the value of

*E*, the total strain

*ɛ*

^{tot}in the experimental curves is decomposed into elastic and plastic strain. Then the plastic component

*ɛ*

^{p}of strain is fitted using the the least squares method by the following power-law relations, which are derived from the Eq. (1):

*ɛ*

^{tot}depending on the grade of curvature grade for a SSC.

The basic variant of the rate-independent Chaboche model [19, 20] is presented as a combination of nonlinear kinematic hardening and nonlinear isotropic hardening models. The model allows the superposition of several independent backstress tensors and can be combined with any of the available isotropic hardening models. Since in this study monotonic and cyclic SSCs are fitted separately only for the identification of *σ*_{y}, only the kinematic hardening component is considered:

*ṗ*is its magnitude. The total backstress

*X*in Eq. (3) is given by the superposition of a number

*N*of kinematic backstresses

*X*

_{i}with a corresponding evolution equation initially proposed by by Armstrong & Frederick [25] for

*Ẋ*, where

_{i}*C*

_{i}and

*γ*are kinematic material constants. Chaboche

_{i}*et al*. [19] recommended

*N*= 3 in order to provide a good fit of experimental SSCs, which include large strain areas. Therefore, three backstresses are considered in this study providing an excellent match of the R-O fit (1) for a whole range of strains.

The kinematic hardening constants (*C _{i}*,

*γ*) and

_{i}*σ*

_{y}, which define the size of the yield surface, are identified as recommended in [20]. The cyclic SSC is fitted by the following relation:

*ɛ*

^{p}≈ const at the peak stresses for strain-controlled cyclic loading. Relation (4) is valid for the cyclic curve after stabilisation of the hardening or softening effects. Constants (

*C*,

_{i}*γ*and cyclic

_{i}*B*and

*β*. The identification procedure is implemented in Microsoft Excel using an add-in Solver [26]. The Solver searches for an optimal (minimum in this case) value for a formula in one cell - called the objective cell - subject to constraints, or limits, on the values of other formula cells on a worksheet. The Solver works with a group of cells, called decision variables or simply variable cells, that participate in computing the formulas in the objective and constraint cells. In this case, the Solver adjusts the values in the decision variable cells containing material constants (

*C*,

_{i}*γ*

_{i}and

*Δɛ*

^{p}. Applying this approach, an excellent match of Eqs (2) and (4) is achieved.

The monotonic SSC is fitted by the different relation in the following form [20]:

*C*

_{i},

*γ*). These constants are identified by fitting Eq. (5) to the R-O extrapolation (2) with “monotonic” values of the R-O parameters

_{i}*B*and

*β*. The identification procedure is implemented in Microsoft Excel using an add-in Solver in the same way as for cyclic SSC. An advanced step-by-step guideline for the estimation of the Chaboche viscoplasticity model parameters with their further optimisation was developed by Hyde

*et al*. [27].

### 2.3 Application to three structural steels

The above described fitting procedure is applied to SSCs of three structural steels for the purpose of *ɛ*^{p} = 1(%) of strain plateau, which is followed by the strain hardening area, then gradually approaching failure at *ɛ*^{tot} = 30(%). The cyclic SSC for this steel shown in Fig. 2a from [21] is fitted by the 2-step procedure, and the obtained material parameters for the R-O (1) and Chaboche (3)-(5) models are listed in Tables 1 and 2 correspondingly.

Fitting parameters of the Ramberg-Osgood model (1) and (6) for different steels and temperatures

Type of plastic material response | Elasto-plastic constants | |||
---|---|---|---|---|

E (MPa) | B (MPa) | β | σ_{y} (MPa) | |

ASTM A36 RT cycl. | 189606 | 1015.61 | 0.2362 | − |

AISI 4340 RT cycl.^{*} | 193053 | 1897.94 | 0.5175 | 320 |

ASTM P91 RT mono. | 215000 | 710 | 0.047 | − |

ASTM P91 RT cycl. | 1180 | 0.155 | − | |

ASTM P91 500°C m. | 180000 | 594 | 0.066 | − |

ASTM P91 500°C c. | 763 | 0.15 | − | |

ASTM P91 550^{°}C m. | 172000 | 482 | 0.054 | − |

ASTM P91 550^{°}C c. | 613 | 0.144 | − | |

ASTM P91 600^{°}C m. | 158000 | 330 | 0.042 | − |

ASTM P91 600^{°}C c. | 446 | 0.123 | − | |

ASTM P91 650^{°}C m. | 140000 | 269 | 0.071 | − |

ASTM P91 650^{°}C c. | 343 | 0.125 | − |

Fitting parameters of the Chaboche model (3)-(5) for different steels and temperatures

Type of plastic material response | Three kinematic hardening backstresses | Yield σ | |||||
---|---|---|---|---|---|---|---|

C (MPa)_{1} | γ_{1} | C_{2} | γ_{2} | C_{3} | γ_{3} | σ_{y} (MPa) | |

ASTM A36 RT cycl. | 87345.7 | 984.7 | 14013.4 | 111.78 | 3918.32 | 13.477 | 115.792 |

AISI 4340 RT mono. | 205524.6 | 535.8 | 8966.94 | 92.268 | 782.893 | 1.0739 | 341.153 |

AISI 4340 RT cycl. | 35912.1 | 650.7 | 6972.29 | 53.297 | 4221.72 | 5.7356 | 330.727 |

ASTM P91 RT mono. | 1120466 | 23911 | 125301.9 | 2539.9 | 17295.23 | 227.86 | 406.098 |

ASTM P91 RT cycl. | 1030320 | 11608 | 136282.4 | 1254.6 | 29535.03 | 148.08 | 197.493 |

ASTM P91 500°C m. | 1059420 | 23359 | 122317.7 | 2469.7 | 17631.89 | 219.49 | 270.687 |

ASTM P91 500°C c. | 659430 | 11229 | 87028.5 | 1248.7 | 19146.80 | 149.22 | 134.541 |

ASTM P91 550°C m. | 1059420 | 23359 | 122317.7 | 2469.7 | 17631.89 | 219.49 | 270.687 |

ASTM P91 550°C c. | 659430 | 11229 | 87028.5 | 1248.7 | 19146.80 | 149.22 | 134.541 |

ASTM P91 600°C m. | 511703 | 24975 | 56536.0 | 2630.3 | 7588.97 | 232.90 | 199.970 |

ASTM P91 600°C c. | 444752 | 12216 | 11344.6 | 160.13 | 56238.9 | 1347.6 | 107.731 |

ASTM P91 650°C m. | 498277 | 23543 | 56252.6 | 2433.8 | 8263.19 | 217.10 | 115.346 |

ASTM P91 650°C c. | 353928 | 12801 | 44816.6 | 1396.6 | 8916.41 | 162.14 | 80.6307 |

The second material is AISI 4340 steel [22], a high-strength alloy steel, which has good machinability features and used for a wide range of applications including aircraft landing gears, shafts or axels for power transmission, gears, high pressure pump housings, etc. Both monotonic and cyclic SSCs shown in Fig. 2b and mechanical properties are taken from [22]. Since it is available explicitly, the monotonic SSC is fitted by the Chaboche model (5) directly, and the material parameters are listed in Table 2. The cyclic SSC for this steel shown in Fig. 2b from [22] is available at ten times wider strain range than for the ASTM A36 steel. Therefore, the R-O model (1) is not able to provide an accurate fit of the cyclic SSC. In this case, the following modification of the R-O equation (1) proposed by Lemaitre & Chaboche [29] is used for fitting analysis:

*σ*

_{y}in the meaning of

The third material is ASTM P91 (modified 9Cr-1Mo) steel [23, 30], an advanced ferritic steel with martensitic microstructure, which has already been widely used over the last 2 decades as tubes/pipes for heat exchangers, plates for pressure vessels, and other forged, rolled and cast components for high temperature services. Both monotonic and cyclic SSCs shown in Figs 2c and 2d and mechanical properties at room temperature (RT), 500°C, 550°C, 600°C and 650°C are taken from [23]. Firstly, the monotonic SSCs are presented in [23] by the material parameters for the R-O model (1) listed in Table 1. The cyclic SSCs are presented in [23] by raw data, which is fitted by the R-O model (1) with material parameters listed in Table 1. Secondly, both monotonic and cyclic R-O extrapolations are fitted by the Chaboche model (3)-(5) with material parameters listed in Table 2.

## 3 Relation in mechanical characteristics

The next step is a search for possible correlations between the experimentally obtained yield strength values (

### 3.1 Fatigue behaviour at normal temperature

Engineering structures operating under cyclic loading conditions at normal temperature are usually designed against the fatigue failure using the conventional stress-life approach. This approach involves experimental fatigue S-N curves with a number of cycles to failure *N*_{*} vs. stress. A typical S-N curve is a straight line in double logarithmic coordinates, which conventionally described by the Basquin model [36]. Referring to [18, 27], a distinctive minimum stress lower bound, which is called a fatigue endurance limit ^{7}) are chosen to represent the fatigue life of the material.

Comparison of

In general, each S-N curve exhibits two limits: one, when stress tends towards the static fracture *σ*_{u} (fracture in a quarter of the cycle), and the other, when stress tends towards the fatigue limit *σ*_{lim}. The most known concepts able to describe a reverse sigmoidal shape of a generic S-N curve are presented by two models. The conventional one is the Bastenaire model [37,38]:

*σ*

_{a}is an alternating stress. Three material parameters

*A*,

*B*,

*C*and fatigue limit

*σ*

_{lim}are derived from fitting the experimental raw data.

A more advanced formulation was developed by Lemaitre & Chaboche [29] using the damage mechanics approach:

*σ̄*is the mean stress of the cycle;

*σ*

_{max}is the maximum stress in the cycle; and the other variables are material parameters defined in the material property set:

*a*- non-linear damage sensitivity,

*b*- mean stress correction factor,

*c*- the Chaboche equation coefficient, and

*α*- the Chaboche equation exponent. Because the Chaboche concept incorporates its own mean stress

*σ̄*correction [29] resembling the Goodman method, the Eq. (8) is fitted to a family of S-N curves with different stress ratios

*R*.

For the purpose of this study, the following equation for a S-N curve proposed in [39] was used as a basis:

*N*

_{*}is a number of cycles to fatigue failure,

*σ*

_{a}is an alternating stress, and

*f*

_{1},

*f*

_{2}and

*f*

_{3}are fitting parameters. This equation provides a smooth transition of the S-N curve into the horizontal plateau of

*σ*

_{u}, which is implemented as

The proposed modification (10) is more convenient than the original formulation (9), and it is a reasonable alternative to previously available equations (7) and (8) because:

- –it produces a fully adjustable reverse sigmoidal shape with a mathematical minimum of fitting parameters;
- –it contains an upper and lower bounds as
*σ*_{u}and ;${\sigma}_{lim}^{\text{f}}$ - –it is fully reversible.

Using a previously suggested assumption, a S-N curve lower bound *σ*_{u} and *σ*_{u} from [21] and AISI 4340 steel with *σ*_{u} from [22]. For both steels *f*_{1}, *f*_{2} and *f*_{3}) is implemented in Microsoft Excel using an add-in Solver in the same way as described previously for fitting of cyclic and monotonic SSCs by the Chaboche model. All parameters for the S-N curves are reported in Table 3. Finally, the whole range of mathematical formulations for S-N curves is discussed by Castillo & Fernández-Canteli [40].

Material parameters of the S-N curves for ASTM A36, AISI 4340 and ASTM P91 steels.

Steel | σ_{u} | f_{1} | f_{2} | f_{2} | |
---|---|---|---|---|---|

ASTM A36 RT | 413.7 | 115.8 | 0.23405 | 4778.8 | 1.0 |

AISI 4340 RT | 827.4 | 330.7 | 33.187 | 2955.7 | 0.3795 |

ASTM P91 RT | 658.0 | 406.1 | 29.037 | 10258 | 0.5154 |

ASTM P91 400°C | 534.0 | [350]^{*} | 28.827 | 10303 | 0.4738 |

ASTM P91 550°C | 380.0 | [0.0]^{**} | 0.0375 | 7806.7 | 1.1472 |

### 3.2 Creep behaviour at elevated temperature

Engineering structures operating under constant loading conditions at high temperature are usually designed against the creep failure using the conventional time-to-failure approach. This approach involves experimental creep rupture curves with stress vs. time to failure *t*_{*}. A typical creep rupture curve is a trilinear smoothed curve in double logarithmic coordinates, with two inflections corresponding to

Previously, the creep modelling using different creep exponent values in different stress ranges was studied by Gorash *et al*. [15-17]. These studies were devoted to the formulation of constitutive creep model, called double power law and applied to ASTM P91 steel at 600°C:

*n*= 12 and

*C*= 2.5 • 10

^{-7}MPa

^{-1}/h.

To implement two transitions into the constitutive model, Eq. (11) is modified by adding the third “power-law breakdown” component to become the triple power law:

*C*and

*n*are secondary creep parameters, which have the temperature dependence expressed by Arrhenius-type functions as follows [43]:

*σ*

_{*}= 0.0587 [MPa] and

*Q*

_{σ}= 54100 [J/mole],

*σ*= 1.9916 [1/h] and

_{C}*Q*= 8757.1 [J/mole],

_{C}*n*

_{0}= 0.2479 and

*Q*= 28648.4 [J/mole].

_{n}In notations (13)-(15): *T* is a temperature in K; *Q*_{σ}, *Q*_{C} and *Q*_{n} are creep activation energies and *R* = 8.314 [J • mol^{-1} • K^{-1}] is the universal gas constant. The transition stress *T*) and creep parameters *C*(*T*) and *n*(*T*) were obtained by fitting the data for minimum creep strain rate from studies by Sklenička *et al*. [42], Kloc & Fiala [41] and Kimura [30] and shown in Fig. 4. The inflections of corresponding curves are observed at 550, 600 and 650°C and explained in terms of transition between different creep deformation mechanisms. Comparison of ^{°}C.

Comparison of

Steel | ASTM A36 | AISI 4340 | ASTM P91 | |||||
---|---|---|---|---|---|---|---|---|

Temp., °C | RT | RT | RT | 400 | 500 | 550 | 600 | 650 |

248.2 | 341.2 | 406.1 | – | 270.7 | 253.0 | 200.0 | 115.3 | |

115.8 | 330.7 | 197.5 | – | 134.5 | 116.6 | 107.7 | 80.6 | |

2.1 | 1.0 | 2.1 | – | 2.0 | 2.2 | 1.9 | 1.4 | |

160.0 | 350.0 | 418.0 | 350.0 | – | – | – | – | |

– | – | – | – | – | 159.2 | 101.2 | 67.6 | |

Δσ, % | 27.6 | 5.5 | 2.8 | – | – | 26.7 | 6.4 | 19.2 |

Another important observation was done in [15-17] for creep rupture behaviour of this steel. The creep-rupture curve, which describes the dependence of time to rupture on stress, exhibits the distinctive inflection too. This inflection is explained as a transition from ductile character of rupture to brittle. This transition was observed at the approximately same stress as the inflection of min. creep rate − *t*_{*}:

*k*

_{1}= 0.23 [1/h]

^{k2-1}and

*k*

_{2}= 0.83 are the tertiary creep constants for the ASTM P91 steel, which are identified by fitting equation (16) to the creep-rupture experimental data [14] for the temperature range from 550° C to 650°C.

It should be noted that Dimmler *et al*. [44] associated these inflections with the microstructurally determined threshold stresses (back-stress concept). The applicability of this concept was shown using the experimental minimum creep rate and creep rupture curves for several 9-12%Cr heat resistant steels (P91, GX12, NF616, X20 and B2). Dimmler *et al*. [44] emphasised that the knowledge of these threshold stresses limits the range of experimentally based predictions, thus preventing from overestimation of the long-term creep rate and creep strength from extrapolated short-term creep data.

Since the inflections are captured reasonably well on both types of creep data in Figs 4 and 5, the transition stresses on min. creep rate curves and creep rupture curves proposed by Gorash *et al*. [15-17] can be explained by relating them to

### 3.3 Fatigue behaviour at elevated temperature

The fatigue performance of ASTM P91 steel is analysed using the HCF experimental data by Matsumori *et al*. [35] at three different temperatures (RT, 400 and 550° C) illustrated in Fig. 6. From these data, it can be concluded that at elevated temperatures the heat-resistant steels don’t exhibit ^{8} loading cycles. However, a good match of *N* = 10^{2}-10^{7} cycles) and ultrahigh in giga-cycle fatigue (GCF) range (*N* = 10^{7}-10^{11} cycles). The existence of ultrahigh *N* ≈ 10^{5}-10^{6}), second in GCF area (*N* ≈ 10^{8}-10^{9}). The correspondence of *N* > 10^{8} cycles, but no experimental data is available for this range.

Following these assumptions, experimental S-N curves for ASTM P91 steel by Matsumori *et al*. [35] are described by the Eq. (10), where *σ*_{u} for all temperatures are taken from [35]. The value of

Finally, the values of

*T*

_{eut}= 1000 K is a typical eutectic temperature for steel alloys;

*T*

_{eut}= 1000 K is proved experimentally to be an eutectic temperature in the iron-carbon system, which characterises the coexistence of solid and liquid phases on iron-carbon phase diagram.

## 4 Conclusions

This study explains the existence of the fatigue limit

Creep transition stress

An important finding is that the temperature dependence of yield strengths (

The principal advantage of the

The critical point in the work presented here is an application of the advanced material model [19,20] to the estimation of a single value of elastic limit *σ*_{y} as a material parameter, the application of Chaboche model equations (3)-(5) may no longer be needed.

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Abbreviations

GCF | Giga-Cycle Fatigue |

HCF | High-Cycle Fatigue |

MUS | Method of Universal Slopes |

R-O | Ramberg-Osgood |

RT | room temperature |

SSC | Stress-Strain Curve |

UML | Uniform Material Law |

UTS | Ultimate Tensile Strength |

Variables, Constants

σ | stress |

Δσ | stress range |

σ_{a} | alternating stress |

σ̄ | mean stress of the cycle |

σ_{max} | maximum stress in the cycle |

σ_{y} | yield strength |

σ_{y0} | yield strength at absolute zero |

cyclic yield strength | |

monotonic yield strength | |

proportional limit | |

elastic limit | |

offset yield strength | |

fatigue endurance limit | |

σ_{u} | ultimate tensile strength |

creep transition stress | |

ɛ | strain |

ɛ̇ | strain rate |

Δɛ | strain range |

ɛ_{f} | true fracture ductility |

ɛ^{tot} | total strain |

ɛ^{p} | plastic strain |

ψ | reduction in area |

HV | Vickers hardness |

HB | Brinnell hardness |

R^{2} | coefficient of determination |

E | Young’s (elasticity) modulus |

B, β | R-O model constants |

X_{i} | kinematic backstresses |

C_{i}, γ_{i} | kinematic material constants |

N_{*} | number of cycles to fatigue failure |

t_{*} | time to creep failure |

A, B, C | fatigue parameters for Bastenaire model |

a, b, c, α | fatigue parameters for Chaboche model |

f_{1}, f_{2}, f_{3} | fatigue parameters for Chaboche model |

C, n | secondary creep parameters |

Q_{σ}, Q_{C}, Q_{n} | creep activation energies |

k_{1}, k_{2} | secondary creep parameters |

T | temperature |

T_{eut} | eutectic temperature |

Subscripts, Superscripts

y | yield |

c | cyclic |

a | alternating |

m | monotonic |

cr | creep |

f | fatigue |

el | elastic |

p | plastic |

* | failure |

tot | total |

lim | limit |

u | UTS |

eut | eutectic |

## Footnotes

^{*}

Extended version of the R-O model (6) is used for data fitting.

^{*}

^{**}

no