Statistical evaluation of low cycle durability for corrosion and heat-resistant steels welded joints materials at room and elevated temperature.

Daunys, M. ; Stulpinaite, A.

1. Introduction

Many structural components contain the zones of geometrical parameters change, shoulders, keyways, oil holes, welded joints and termed notches. When such elements are loaded, local stress and strain concentrations are generated in such zones of geometrical parameters change [1, 2]. Plastic strain in these areas appears in small material volumes. During cyclic loading the cyclic plastic deformation in the area of stress and strain concentrations can severely reduce the durability of the construction. Plastic strains in these areas are limited by adjacent elastic strained zones, therefore the conditions of loading with limited strains in these areas are very similar.

The low cycle loading experiments are very complicated and expensive, particular at elevated temperature because of temperature control and stress strain curves recording. That is why the attempts to obtain characteristics of low cycle loading from monotonous tension curves, hardness or other parameters, but without cyclic loading are made. Heat treatment is widely used in nuclear power equipment and other engineering components. Under low cycle loading tempered or normalized steels cyclically stabilized or hardened, therefore high strength steels are cyclically softened [3].

It is very important to analyze the reliability of structures under low cycling loading. The probabilistic methods of reliability determination of corrosion and heat-resistant steels are analyzed in this work. Durability calculation is based on the statistical method by the use mechanical characteristics and low cyclic loading parameters.

The parameters of low cycle loading fatigue curves according plastic strain [m.sub.p] and [C.sub.p] and elastic strain [m.sub.e] and [C.sub.e] for corrosion and heat-resistant steels at room and elevated temperatures were determined under tension compression and at symmetric (R = -1) strain limited conditions. In works [4, 5] detailed statistical analysis showed that parameters of Coffin curves the best correlate with modified plasticity ([[sigma].sub.u]/[[sigma].sub.y])Z , i.e. the parameter depending on ultimate tensile and yield strengths and reduction of the area at fracture, at room and elevated temperatures. The relationship between the parameters of fatigue curves and mechanical characteristics was conformed according to normal distribution. Only the results of experiments of Kaunas University of Technology laboratory were analyzed. In this work the results of the investigation of 55 corrosion and heat-resistant steels, 36 their weld metals at room (20[degrees]C) and 46 corrosion steels at elevated (250[degrees]C550[degrees]C) temperatures were selected from materials investigated in the laboratories of Kaunas University of Technology and other countries (Slovakia, Russia, Hungary).

2. Mechanical characteristics and low cycle fatigue curves parameters

The low cycle fatigue characteristics of materials are significant for estimating the reliability and durability of construction elements during exploitation [6]. The parameters of low cycle loading with limited strain are understood as the durability or low cycle fatigue curves, which are composed in coordinates lg [epsilon] - lg N and lgS- lg N according the number of cycles till crack Nc or fracture N f appears. The durability of the material under loading with limited strain is expressed by Coffin's equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [delta] is the range of plastic strain or the width of plastic hysteresis loop; N is the number of cycles up to crack formation or fracture; [m.sub.p] and [C.sub.p] are characteristics of the material, which are proposed by Coffin: [m.sub.p] = 0.5 and [C.sub.p] = 0.5ln (1/(1-Z)), where Z is reduction of the area at fracture, while S.S. Manson [8] proposed the expression [C.sub.p] = [(ln/1-Z).sup.0.6].

The hysteresis loop describes cyclic behavior of the material and its resistance to fatigue. Under loading with limited strain, the cyclic hardened, softened or stable materials are damaged of fatigue, because at this loading there is no quasistatic damage. The shape of hysteresis loop vary during the low cycle loading with limited strain for hardened and softened materials, therefore it is proposed to calculate equivalent plastic strain by expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], 8k where kc is the number of semicycles up to crack, or to the applied width of plastic hysteresis loop for durability kJ2 [3].

In work [3] it was proposed to change plastic deformation [delta] in Eq. (1) by [epsilon], because the range of total strain s remains constant at cyclic loading with limited strain therefore the durability is proposed to be evaluated by the equation

[epsilon][N.sup.m] = C (2)

This equation, when [epsilon] >(3.0-3.5)[e.sub.pr], is correct for the majority of materials, then, [m.sub.p] > m, [C.sub.p] > C [3], and when s < (3.0 -3.5)[e.sub.pr] the durability greatly increases, therefore low cycle fatigue curves are defined in this work by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [epsilon] is total elastic plastic strain; [m.sub.e], [C.sub.e], [m.sub.p], [C.sub.p] are parameters of low cycle fatigue curves according toelastic and plastic strains accordingly.

3. Statistic evaluation of low cycle fatigue parameters

The corrosion and heat-resistant steels and their weld metals [4, 5] investigated in this work were divided according temperature into 2 groups: 1) at room temperature; 2) at elevated temperature. Dependences of the parameters of low cycle fatigue curve [m.sub.p] , [C.sub.p] according plastic strain and [m.sub.e] , [C.sub.e] according elastic strain on modified plasticity ([[sigma].sub.u]/[[sigma].sub.y])Z for steels at room temperature and 95% confidence interval ranges (dotted line) to theoretic line are given in Fig. 1. Fig. 2 represent the results of parameters [m.sub.p] , [C.sub.p] dependences on modified plasticity for steels at elevated temperature.

[FIGURE 1 OMITTED]

Figs. 1 and 2 show that the 95% confidence interval ranges (dotted line) to theoretic line are narrower at room temperature comparing with the results at elevated temperature.

Rectangular diagrams of parameters [m.sub.p] for steels and their weld metals (Fig. 3) show that the scatter interval of the results is not wide (within limits [x.sub.min] / [x.sub.maxx]). In these diagrams the median values [x.sub.me] for the investigated n number of materials are also represented, which divides the scatter of the results into two equal parts. Defined area (within quartiles limits [x.sub.0.25] / [x.sub.0.75]) describes the 50% scatter of the middle values. Statistical characteristics of low cycle fatigue curve parameters [m.sub.p], [C.sub.p], [m.sub.e], [C.sub.e] according to elastic and plastic strain at room (20[degrees]C) and elevated (250[degrees]C - 550[degrees]C) temperatures are given in Table 1. Mean values of the parameters are similar to median values; the implication is that here are no strongly outstanding materials. The mean values of scatter results of parameters [m.sub.p] and [C.sub.p] according elastic and plastic strain for corrosion and heat-resistant steels at room temperature are greater comparing with the results at elevated temperature, however parameters [m.sub.e] and [C.sub.e] are smaller at room temperature than at elevated temperature.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Parameter [m.sub.p] of low cycle fatigue curves for analyzed steels and their weld metals at room and elevated temperatures has left skewness compared with normal distribution, while parameter [m.sub.e] for those steels at elevated temperature has right skewness. Kurtosis coefficient shows that the results of parameters [m.sub.p] and [m.sub.e] for corrosion and heat-resistant steels and weld metals at room and elevated temperatures are spread wider interval comparing with normal distribution.

Correlation analysis is statistical relation strength between analyzed variables, which is expressed by correlation coefficient. Pearson correlation coefficient measures the linear relation strength. Correlation analysis is not used to determine nonlinear correlations. When linear model is not adequate, it is necessary to use nonlinear model.

In previous works the accomplished statistical analysis conformed that the parameters of Coffin curves the best correlate with modified plasticity ([[sigma].sub.u]/[[sigma].sub.y]) z at room and elevated temperatures. The results in Table 2 confirm that the parameters of low cycle fatigue curves me, [C.sub.e], [m.sub.p] , [C.sub.p] for steels, their weld metals and modified plasticity ([[sigma].sub.u]/[[sigma].sub.y]) z at room and elevated temperatures are correlated. Pearson correlation coefficient has the minimum value |- 0.402 for corrosion and heat-resistant steels coefficient [C.sub.p] at elevated temperature and the maximum value |0.747| for those steels coefficient [C.sub.e] at room temperature.

Analytical dependences of Coffin parameters on modified plasticity for analyzed steels and weld metals at room and elevated temperatures are given in Table 3. The dependences of [m.sub.p] , [C.sub.p] , [m.sub.e], [C.sub.e] are used for forecasting the preliminary durability of a material by Eq. (3).

For the comparison of the results of experimental and calculated durability by Eq. (3), which are distributed according to normal low, there were determined such scatter limits: fourfold, ninefold, sixteenfold. The scatter between experimental and calculated durability results for steels and their weld metals (according analytical dependences given in Table 3) at room and elevated temperatures is presented in Table 4. Scatter of the results between experimental [N.sup.exp.sub.f] and calculated [N.sup.cal.sub.f] durability for steels at elevated (250[degrees]C - 550[degrees]C) temperature is 19% greater than the scatter of the results at room (20[degrees]C) temperature. The scatter of comparison results of experimental and calculated durability for steels and their weld metals are similar at room temperature. The comparison between their durability at room temperature is shown in Fig. 4. When [N.sup.cal.sub.f]f >10000 the relation [N.sup.cal.sub.f] /[N.sup.exp.sub.f] is greater than 10, it means that in Table 3 proposed analytical dependences are correct to use when [N.sup.cal.sub.f]l <10000.

[FIGURE 4 OMITTED]

4. Conclusions

1. The mean value of parameter [m.sub.p] for corrosion

and heat-resistant steels and their weld metals at room at elevated temperatures is greater than Coffin's suggested constant m = 0.5 . The obtained mean value at room temperature for steels [m.sub.p]=0.791, for weld metals [m.sub.p] = 0.608, at elevated temperature for steels [m.sub.p] = 0.735.

2. The parameters of low cycle fatigue curves me, [C.sub.e], [m.sub.p] , [C.sub.p] for steels and their weld metals are correlated with modified plasticity ([[sigma].sub.u]/[[sigma].sub.y])z by linear regression at room and elevated temperatures.

3. The scatter of the results between experimental [N.sup.exp.sub.f] and calculated [N.sup.cal.sub.f] durability for steels at elevated temperature is 19% greater than the scatter of the results at room temperature. The scatter of the results between experimental and calculated durability for steels and their weld metals are similar at room temperature.

4. Analytical dependences of low cycle fatigue curve parameters on modified plasticity for corrosion and heat-resistant steels and their weld metals are enough correct to figure out the durability at room and elevated temperatures. The scatter of the results is 2-3 times greater than for one material experimental durability at low cycle loading.

5. Dependencies proposed in this work may be used for preliminary durability evaluation of corrosion and heat-resistant steels and their weld metals at low cycle loading.

Received October 11, 2008

Accepted February 11, 2009

References

[1.] Krenevicius, A., Leonavicius, M., Petraitis, G., Gutauskas, M., Stonkus, R. Strength of differently cooled cast iron subjected to cyclic loading. -Mechanika. -Kaunas: Technologija, 2007, Nr.4(66), p.18-22.

[2.] Daunys, M., Bazaras, Z., Timofeev, B.T. Low cycle fatigue of materials in nuclear industry. -Mechanika. -Kaunas: Technologija, 2008, Nr.5(73), p.12-17.

[3.] Daunys, M. Cycle Strength and Durability of Structures. -Kaunas: Technologija, 2005. -286p. (in Lithuanian).

[4.] Sniuolis, R. Dependence of Low Cycle Fatigue Parameters on Mechanical Characteristics of Structural Materials. Doctoral dissertation.-Kaunas: 1999, -117p. (in Lithuanian).

[5.] Catalog of Data for Structural Materials of Mechanical and Low Cycle Loading Characteristics. -Moscow. 1990.-400p. (in Russian).

[6.] Mishnaevsky, L.L. Methods of theory of complex system in modeling of fracture: a brief review. -Engng. Fract. Mech., 1997, 56 (1), p.47-56.

[7.] Makhutov, N.A. Deformation Criterions and Strength Counting for Construction Elements. -Moscow: Mashinostroenie, 1981.-272p. (in Russian).

[8.] Manson, S.S. Fatigue: a complex subject - some simple approximations. -Experimental Mechanics, 1965, v.5, No7, p.193-276.

M. Daunys *, A. Stulpinaite **

* Kaunas University of Technology, Kqstucio str. 27, 44312 Kaunas, Lithuania, E-mail: Mykolas.Daunys@ktu.lt ** Kaunas University of Technology, Kqstucio str. 27, 44312 Kaunas, Lithuania, E-mail: agnes@erdves.lt

Table 1
Statistical characteristics of low cycling curves parameters
[m.sub.e], [C.sub.e], [m.sub.p], [C.sub.p] at room and elevated
temperatures

 Corrosion and heat-resistant
Parameters steels at room temperature

 [m.sub.p] [C.sub.p] [m.sub.e] [C.sub.e]

Number of materials 42 42 42 40
Mean value 0.791 189 0.152 2.40
Median value 0.811 212 0.137 1.81
Minimum value 0.39 4.44 0.06 0.65
Maximum value 1.12 555 0.30 6.49
Kurtosis coefficient -0.06 0.56 -0.46 0.71
Skewness coefficient -0.63 0.39 0.78 1.25

 Corrosion and heat-resistant
Parameters steels at elevated temperature

 [m.sub.p] [C.sub.p] [m.sub.e] [C.sub.e]

Number of materials 36 36 34 29
Mean value 0.735 154 0.200 2.58
Median value 0.798 193 0.197 2.35
Minimum value 0.36 6.05 0.11 0.83
Maximum value 1.10 347.8 0.34 5.08
Kurtosis coefficient -0.94 -1.10 -0.73 0.30
Skewness coefficient -0.38 -0.36 0.52 0.71

 Weld metals of corrosion and
Parameters heat-resistant steels at room temperature

 [m.sub.p] [C.sub.p] [m.sub.e] [C.sub.e]

Number of materials 25 24 23 22
Mean value 0.608 93.1 0.171 2.3
Median value 0.665 52.7 0.157 1.6
Minimum value 0.24 2.88 0.05 0.5
Maximum value 0.97 348 0.36 6.9
Kurtosis coefficient -0.50 1.16 -0.42 0.7
Skewness coefficient -0.19 1.38 0.47 1.3

Table 2
Correlation analysis of parameters [m.sub.e], [C.sub.e],
[m.sub.p], [C.sub.p] and modified plasticity ([[sigma].sub.u]/
[[sigma].sub.y]) * Z at room and elevated temperatures

 Pearson correlation coefficient

Material [m.sub.p] [C.sub.p] [m.sub.e] [C.sub.e]

 At room temperature

Corrosion and heat- -0.660 -0.461 0.610 0.747
resistant steels
Corrosion and heat- 0.559 0.707 0.545 0.466
resistant steels
weld metals

 At elevated temperature

Corrosion and heat- -0.567 -0.402 0.501 0.436
resistant steels

Table 3
Analytical dependences of low cycle curves parameters on modified
plasticity (aujay) Z at room and elevated temperatures

 At room temperature At elevated temperature

 Corrosion and heat-resistant steels

[m.sub.p] = 1.00 - 0.149 [m.sub.p] = 1.08 - 0.189
([[sigma].sub.u]/[[sigma].sub.y])Z ([[sigma].sub.u]/[[sigma].sub.y])Z
[C.sub.p] = 294 - 73.8 [C.sub.p] = 266 - 60.7
([[sigma].sub.u]/[[sigma].sub.y])Z ([[sigma].sub.u]/[[sigma].sub.y])Z
[m.sub.e] = 0.073 + 0.059 [m.sub.e] = 0.105 + 0.052
([[sigma].sub.u]/[[sigma].sub.y])Z ([[sigma].sub.u]/[[sigma].sub.y])Z
[C.sub.e] = 0.260 + 1.61 [C.sub.e] = 1.44 + 0.645
([[sigma].sub.u]/[[sigma].sub.y])Z ([[sigma].sub.u]/[[sigma].sub.y])Z

 Weld metals of corrosion and heat-resistant
 steels at room te[m.sub.p]erature

[m.sub.p] = 0.363 + 0.244 ([[sigma].sub.u]/[[sigma].sub.y])Z
[C.sub.p] =-79.7 + 181 ([[sigma].sub.u]/[[sigma].sub.y])Z
[m.sub.e] = 0.057 + 0.114 ([[sigma].sub.u]/[[sigma].sub.y])Z
[C.sub.e] = 0.147 + 2.26 ([[sigma].sub.u]/[[sigma].sub.y])Z

Table 4
Comparison of experimental [N.sup.exp.sub.f] and calculated
[N.sup.cal.sub.f] durability for corrosion and heat-resistant steels
and their weld metals at room and elevated temperatures

 Number of specimens, when scatter of
 results between experimental and
 calculated durability is
 Total
 number of fourfold ninefold
Material specimens number % number %

 At room temperature

Corrosion and 449 142 32 212 47
heat-resistant
steels
Weld metals of 228 88 31 145 50
corrosion
and heat-resistant
steels

 At elevated temperature

Corrosion and 362 58 16 100 28
heat-resistant
steels

 Number of specimens, when scatter of
 results between experimental and
 calculated durability is
 Total
 number of sixteenfold
Material specimens number %

 At room temperature

Corrosion and 449 260 58
heat-resistant
steels
Weld metals of 228 173 60
corrosion
and heat-resistant
steels

 At elevated temperature

Corrosion and 362 139 39
heat-resistant
steels