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) σu, 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
Casagrande et al. [8] investigated a relationship between
An alternative approach was suggested by Li et al. [10], who estimated theoretically the cyclic yield strength
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
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 σy. The first is the proportional limit
This study proposes the cyclic yield strength

Concept of the safe structural design for fatigue and creep using cyclic yield strength.
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
2.2 Stress-strain curves fitting procedure
2.3 Application to three structural steels
The above described fitting procedure is applied to SSCs of three structural steels for the purpose of

Fitting of monotonic and cyclic SSCs for: a) ASTM A36 steel from [21] at RT; b) AISI 4340 steel from [22] at RT c) ASTM P91 steel from [23] at 550°C and d) at 600°C.
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
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 | σy (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 | |||||
---|---|---|---|---|---|---|---|
C1 (MPa) | γ1 | C2 | γ2 | C3 | γ3 | σy (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 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
Comparison of

S-N curve fits of ASTM A36 steel based on HCF data by [31] and AISI 4340 steel based on HCF data by [32], [33] and [34]
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
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 σu and
; - –it is fully reversible.
Using a previously suggested assumption, a S-N curve lower bound
Material parameters of the S-N curves for ASTM A36, AISI 4340 and ASTM P91 steels.
Steel | σu | f1 | f2 | f2 | |
---|---|---|---|---|---|
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
In notations (13)-(15): T is a temperature in K; Qσ, QC and Qn are creep activation energies and R = 8.314 [J • mol-1 • K-1] is the universal gas constant. The transition stress

Min. creep rate vs. stress of ASTM P91 steel based on several sets of data [30, 41, 42]
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
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 |

Stress vs. creep rupture life of ASTM P91 steel based on the data by [14]
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
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

S-N curve fits of ASTM P91 steel based on HCF data by Matsumori et al. [35]
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
Following these assumptions, experimental S-N curves for ASTM P91 steel by Matsumori et al. [35] are described by the Eq. (10), where

Elliptic yield surfaces of ASTM P91 steel using temperature-dependent
Citation: Open Engineering 7, 1; 10.1515/eng-2017-0019
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
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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 |
σa | alternating stress |
σ̄ | mean stress of the cycle |
σmax | maximum stress in the cycle |
σy | yield strength |
σy0 | yield strength at absolute zero |
cyclic yield strength | |
monotonic yield strength | |
proportional limit | |
elastic limit | |
offset yield strength | |
fatigue endurance limit | |
σu | ultimate tensile strength |
creep transition stress | |
ɛ | strain |
ɛ̇ | strain rate |
Δɛ | strain range |
ɛf | true fracture ductility |
ɛtot | total strain |
ɛp | plastic strain |
ψ | reduction in area |
HV | Vickers hardness |
HB | Brinnell hardness |
R2 | coefficient of determination |
E | Young’s (elasticity) modulus |
B, β | R-O model constants |
Xi | kinematic backstresses |
Ci, γi | 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 |
f1, f2, f3 | fatigue parameters for Chaboche model |
C, n | secondary creep parameters |
Qσ, QC, Qn | creep activation energies |
k1, k2 | secondary creep parameters |
T | temperature |
Teut | 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 |