Unification of the most commonly used time–temperature creep parameters

Materials Science and Engineering A 528 (2011) 2804–2811

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

Unification of the most commonly used time–temperature creep parameters ˇ Domen Seruga, Marko Nagode ∗ University of Ljubljana, Faculty of Mechanical Engineering, Aˇskerˇceva 6, SI-1000 Ljubljana, Slovenia

a r t i c l e

i n f o

Article history: Received 28 September 2010 Received in revised form 9 December 2010 Accepted 9 December 2010 Available online 15 December 2010 Keywords: Creep Larson Miller Manson Haferd Orr Sherby Dorn Time–temperature parameter Time to rupture Master curve

a b s t r a c t The determination of the time to rupture at low stresses and temperatures despite existing methods still remains a challenge. A new unifying time–temperature parameter is proposed. It represents a modification of the efficient, but numerically unstable Manson–Brown parameter. It incorporates the most commonly used time–temperature parameters such as Larson–Miller, Manson–Haferd and Orr–Sherby–Dorn as special cases for the calculation of the master curves. The procedure chooses the most appropriate master curve for the test data and calculates the corresponding coefficients. The inputs for the determination of the master curves are minimised to the least possible amount of test data. The proposed time–temperature parameter is included in a fast creep damage calculation procedure which is a part of a method for evaluating the total damage of the thermomechanically loaded components. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The lifetime of automotive components, gas and steam turbines, power plants and other products that operate at high temperature depends on thermal and mechanical loading (TML). Typical examples of TML are start-up, full load, partial load and shut-down [1]. Damage mechanisms arising from TML are mainly mechanical fatigue, creep and oxidation. Fatigue and oxidation influence the fatigue damage [2] while the creep damage becomes significant when temperatures exceed the creep temperature typically determined as 40% of the melting temperature of the material [3–5]. The aim of the presented procedure therefore deals with the creep properties as an additional part in damage estimation of the TML components, where creep does not represent a dominant part of the damage, and not as an independent calculation which needs detailed knowledge of the creep phenomenon and skilful treatment of the creep rupture data. Thus the principal advantages of our procedure are simplicity, speed and robustness. When the creep damage of TML components is calculated, the creep rupture data on the whole stress–temperature working area of the component must be known. This also includes

∗ Corresponding author. Tel.: +386 1 4771 507; fax: +386 1 2518 567. E-mail address: [email protected] (M. Nagode). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.12.034

creep rupture data at low temperatures and stresses where the time to rupture can exceed 105 h, for example [4]. Due to the enormous amount of time and expenses spent on achieving creep rupture data at low temperatures and stresses they are gained rather using the time–temperature parameters. The basic idea of every time–temperature parameter is to predict long-term creep behaviour by performing tests at high temperature and stress. Time–temperature parameters enable the extrapolation of creep behaviour to lower temperatures and stresses, thus reducing test time and effort. They can be based on some physical or purely mathematical model to link stress, temperature and time to rupture together. Since the introduction of time–temperature parameters, first by Larson and Miller [6], Manson and Haferd [7] and later by Orr, Sherby and Dorn [8], they have become a common method in representing creep rupture data [9–11]. Each of them is based on a different representation of the family of the master curves, involving time to rupture tr , temperature T and stress . Although all of them are reasonably accurate in the test time range, they differ in their representation of the behaviour in the extrapolated range. After numerous proposals on alternative parameters [10–16], the Larson–Miller (LM), the Manson–Haferd (MH) and the Orr–Sherby–Dorn (OSD) parameters remain the most commonly used in determining creep rupture data [12,13,17–21]. There have also been many attempts to generalise parameters (Murry, Grounes, Manson and Ensign, Goldhoff and Sherby, Manson and Succop. . . [10,13]) but it has not yet been concluded which of the

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Fig. 1. Master curves, relation between stress and time to rupture over the temperature range.

existing time–temperature parameters is the best for extrapolation purposes [10,22]. There also exists a variety of recommendations for the assessment of creep rupture data which are based on a comprehensive review of existing procedures and an extensive evaluation of their effectiveness [14,22,23]. ISO 6303 method [24] and DESA Version 2.2 method [25] (and also graphical averaging and cross-plotting methods, the SIMR method, the SPERA method and the minimum commitment method) are the most commonly used state-of-theart procedures representing highly flexible and efficient tools for applying time–temperature parametric equations to the assessment of the creep rupture data [14,22]. Despite this it is not practical to recommend a single procedure for use. Implementation of the recommendations still requires significant additional effort. Without pre-assessment of the test data, repeated main assessments of the times to rupture and post assessment tests, the uncertainty, particularly that associated with extrapolated data, is unacceptably high [22]. The procedures either use common time–temperature parameter forms or exclusively fit the test data. The proposed parameter combines these properties. It can either use the most frequently used time–temperature parameters or select one that optimally fits the test data. Simple input of the test data and robust determination of the time to rupture are its additional advantages. Using time–temperature parameters a master curve (Fig. 1) can be obtained representing the relation between logarithmic stresses and logarithmic time to rupture for a single temperature. An important challenge is the modelling of the master curves in the case of very limited test data. At least 27 tests are recommended [22] while industry tends towards satisfying results with the least possible number of tests. Especially in the lifetime prediction of the TML components, where creep does not contribute to the major part of the total damage, usually tests at three different temperatures and three different stress levels are performed [26–28]. It is also not always straightforward to determine the coefficients for the existing time–temperature parameters due to nonlinear regression problems. The Manson–Brown (MB) parameter [29] in particular, given by MB =

log tr − log ta (T − Ta )q

,

(1)

appears to be very efficient [10] but numerically unstable due to the coefficients Ta and q. They make the set of equations for determination of the coefficients nonlinear and not trivially solvable. Instead of the globally optimal, often only locally optimal sets of

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Fig. 2. Determination of RMB coefficients q and Ta .

coefficients are obtained, depending on the initial guesses in the numerical algorithms. 2. Restrained Manson–Brown parameter The restrained Manson–Brown (RMB) parameter is proposed as RMB =

log tr − log ta · T |q|−1 (T − Ta · q)q

,

(2)

where log ta , Ta and q are the coefficients obtained by the least squares method. The value of q is q if q > 0 and 0 if q ≤ 0. The MB parameter (Eq. (1)) is hard to solve numerically due to the unrestrained coefficients q and Ta . Therefore the RMB parameter is introduced as a substitute resembling all the properties of the MB parameter. It is numerically friendlier to solve because the coefficients can be gained by solving a set of linear equations and not nonlinear, as in the case of the MB parameter. The limit states of the RMB parameter represent the LM, OSD and MH parameter (Fig. 2). If q = 1, the RMB parameter represents the MH parameter, given by MH =

log tr − log ta , T − Ta

(3)

if q = 0, the RMB parameter represents the OSD parameter, given by OSD = log tr −

log ta , T

(4)

and if q = −1, the RMB parameter represents the LM parameter, given by LM = T (log tr − log ta ).

(5)

If q = / − 1, 0 or 1, the RMB parameter examines whether or not it is possible to describe the test data more accurately than with the LM, OSD and MH parameters. Every time–temperature parameter is also a function of the stress, RMB = LM = OSD = MH = f (),

(6)

and usually a second degree polynomial is sufficient to describe the stress function [30], RMB = LM = OSD = MH = a0 + a1 · log  + a2 · log2 , especially in the case of limited test data.

(7)

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Considering only the RMB parameter (Eq. (2)), the relation between the time to rupture, the temperature and the stress is thus given by log tr − log ta · T |q|−1 (T − Ta · q)q

= a0 + a1 · log  + a2 · log2 .

(8)



This yields

Ta min = min

log tr = log ta · T |q|−1 +(T −Ta · q)q (a0 + a1 ·log  + a2 · log2 ).

(9)

Coefficients log ta , Ta , q, a0 , a1 and a2 are obtained for the existing creep rupture test data by the least squares method,

R2 =

mT n  

i=1

The unknown coefficient Ta multiplied by the other unknown coefficients a0 , a1 and a2 would also give a nonlinear set of equations that is numerically unpleasant to solve since no initial guess for Ta can be found. Therefore new coefficients Ta a0 , Ta a1 and Ta a2 are introduced. After solving the linear set of equations (Eq. (12)), Ta can be limited by

|q|−1

(log trij − log ta · Tj

q

2

2

− (Tj − Ta · q) (a0 + a1 · log ij + a2 · log ij )) ,

j=1

(10)

⎡

1  ⎢ (T − T ) a j ⎣ (Tj − Ta )log ij i=1 j=1 (Tj − Ta )log2 ij n

=

mT

n mT  

log trij



,

Ta max = max

Ta a0 Ta a1 Ta a2 , , a0 a1 a2



.

(13)

The optimal Ta and the corresponding set of coefficients can be chosen by searching the minimum of the function 2 Rm =

mT n  

2

2

(14)

(log trij − log ta − (Tj − Ta )(a0 + a1 · log ij + a2 · log ij )) .

j=1

2 is calculated, T runs from T When the minimum Rm a a min to Ta max in distinct steps and the value q = 1 remains (Fig. 4), 2 2 2 2 ∂Rm ∂Rm ∂Rm ∂Rm = = = = 0. ∂ log ta ∂a0 ∂a1 ∂a2

(15)

Eqs. (14) and (15) can be rewritten in the matrix form (Eq. (16)) and the coefficients are calculated numerically by LU decomposition [31].

(Tj − Ta ) (Tj − Ta )2 (Tj − Ta )2 log ij (Tj − Ta )2 log2 ij

(Tj − Ta )log trij



2.2. Searching for the optimal coefficient Ta

i=1

where the index i represents the test stress level and the index j the test temperature, n is the number of test stress levels for a single temperature Tj and mT is the number of test temperatures. The complete procedure of assessing the unknown RMB coefficients is depicted in Figs. 2 and 3.

Ta a0 Ta a1 Ta a2 , , a0 a1 a2

(Tj − Ta )log ij (Tj − Ta )2 log ij (Tj − Ta )2 log2 ij (Tj − Ta )2 log3 ij

(Tj − Ta )log ij log trij

⎤⎧

⎫

(Tj − Ta )log2 ij ⎪ ⎨ log ta ⎪ ⎬ a0 (Tj − Ta )2 log2 ij ⎥ ⎦ (Tj − Ta )2 log3 ij ⎪ ⎩ a1 ⎪ ⎭ 2 4 a2 (Tj − Ta ) log ij

(Tj − Ta )log2 ij log trij

T

(16)

i=1 j=1

After solving the linear set of equations, coefficients log ta , Ta , a0 , 2 are gained (Fig. 3, step 2). This a1 and a2 for q = 1 and minimum Rm state of the RMB parameter still corresponds to the MH parameter. It still has to be checked whether or not a different set of coefficients gives the minimum R2 by changing q.

2.1. Preparing the limit values As the coefficient q makes Eq. (9) nonlinear, it is first set to 1. The RMB parameter in this case resembles the MH parameter. In order to find the other RMB coefficients for q = 1 that will give the minimum R2 , all partial derivatives from Eq. (10) have to be calculated and set to zero. ∂R2 ∂R2 ∂R2 ∂R2 ∂R2 ∂R2 ∂R2 = = = = = = =0 ∂ log ta ∂a0 ∂Ta a0 ∂a1 ∂Ta a1 ∂a2 ∂Ta a2

(11)

Eqs. (10) and (11) can be rewritten into the matrix form (Eq. (12)) and the coefficients are calculated numerically by LU decomposition [31].

⎡

1 ⎢ Tj ⎢ n mT ⎢   ⎢ −1 ⎢ Tj log ij ⎢ i=1 j=1 ⎢ −log ij ⎣ T log2  j ij −log2 ij n  

Tj Tj 2 Tj Tj 2 log ij −Tj log ij Tj 2 log2 ij −Tj log2 ij

−1 −Tj 1 −Tj log ij log ij −Tj log2 ij log2 ij

Tj log ij Tj 2 log ij Tj log ij Tj 2 log2 ij −Tj log2 ij Tj 2 log3 ij −Tj log3 ij

−log ij −Tj log ij log ij −Tj log2 ij log2 ij −Tj 2 log3 ij log3 ij

Tj log2 ij Tj 2 log2 ij −Tj log2 ij Tj 2 log3 ij −Tj log3 ij Tj 2 log4 ij −Tj log4 ij

⎤

⎧ ⎫ −log2 ij ⎪ log ta ⎪ ⎪ ⎪ ⎪ −Tj log2 ij ⎥ ⎪ ⎪ a0 ⎪ ⎥⎪ ⎪ ⎨ Ta a0 ⎪ ⎬ log2 ij ⎥ ⎪ ⎥ 3 a1 −Tj log ij ⎥ ⎥ ⎪ Ta a ⎪ 1 ⎪ log3 ij ⎥ ⎪ ⎪ ⎪ ⎪ a2 ⎪ ⎪ ⎦⎪ 2 ⎪ −Tj log ij ⎪ ⎩ ⎭ 4 T a a 2 log  ij

mT

=

log trij

Tj log trij

−log trij

Tj log ij log trij

−log ij log trij

i=1 j=1

After solving the linear set of equations, coefficients q = 1, log ta , a0 , Ta a0 , a1 , Ta a1 , a2 and Ta a2 are gained (Fig. 3, step 1).

Tj log2 ij log trij

−log2 ij log trij

T

(12)

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Fig. 4. Graphical determination of the coefficient Ta .

Fig. 3. The assessment of the RMB coefficients.

2.3. Searching for the optimal coefficient q The coefficient q is limited by qmax = 1 ≥ q ≥ qmin = − 1 and can exert any value within this interval (Fig. 2). Optimal q and the corresponding set of coefficients coincide with the minimum of Eq. (10), where q runs from qmax to qmin in distinct steps. Ta is now set to the optimal value, as defined by Eq. (15) for q = 1, and is linearly driven by the term q. When q reaches the value 0 or less, the term Ta q vanishes. For q = 1 the RMB parameter corresponds to the MH parameter, for q = 0 to the OSD parameter and for q = −1 to the LM parameter. If q exerts any value between qmax and qmin except 1, 0, −1, the RMB parameter corresponds to the MB parameter (Fig. 2). ∂R2 ∂R2 ∂R2 ∂R2 = = = =0 ∂ log ta ∂a0 ∂a1 ∂a2

(17)

Eqs. (10) and (17) can be rewritten in the matrix form (Eq. (18)) and the coefficients are calculated numerically by LU decomposition [31],

⎡ n mT  ⎢ ⎣ i=1 j=1

=

Tj 2(|q|−1) (Tj − Ta q)q Tj |q|−1 (Tj − Ta q)q Tj |q|−1 log ij (Tj − Ta q)q Tj |q|−1 log 2 ij

n mT  

Tj |q|−1 log trij

(Tj − Ta q)q Tj |q|−1 (Tj − Ta q)2q (Tj − Ta q)2q log ij (Tj − Ta q)2q log2 ij

(Tj − Ta q)q log trij

Fig. 5. Determining master curves with the RMB parameter.

temperature Ti expected in the creep damage calculation for i = 1, . . . kT , where kT represents the number of discrete temperature divisions. Values of the polynomial coefficients b0i , b1i and b2i are stored before the creep damage calculation, preserving a high computational speed (Fig. 3, step 4). log tr (Ti ) = b0i (Ti ) + b1i (Ti ) · log  + b2i (Ti )log2 

(Tj − Ta q)q Tj |q|−1 log ij (Tj − Ta q)2q log ij (Tj − Ta q)2q log2 ij (Tj − Ta q)2q log3 ij

(Tj − Ta q)q log ij log trij

⎤ ⎧

(19)

⎫

(Tj − Ta q)q Tj |q|−1 log2 ij ⎪ ⎨ log ta ⎪ ⎬ a0 (Tj − Ta q)2q log2 ij ⎥ · ⎦ 2q 3 (Tj − Ta q) log ij ⎪ ⎩ a1 ⎪ ⎭ a2 (Tj − Ta q)2q log4 ij

(Tj − Ta q)q log2 ij log trij

T

(18)

i=1 j=1

Finally, the optimal RMB coefficients log ta , Ta , a0 , a1 , a2 and q are known (Fig. 3, step 3). Although the system of equations seems quite tough to solve, the solution is straightforward, robust and fast. The RMB parameter always assures the relevant forms of the master curves even in the case of input polynomials for the known temperatures having an unexpected form. However, the calculated master curves for the known temperatures always deviate from the input master curves since the RMB parameter tries to find the optimal description of the test data over the entire stress and temperature range (Fig. 5). 2.4. Transformation for a faster creep damage calculation For a faster creep damage calculation and less computer memory consumption the RMB coefficients are transferred to the second degree polynomial (Eqs. (19)–(22)) representing the relation between the time to rupture and the stress for every distinct

b0i (Ti ) = log ta · Ti |q|−1 + (Ti − Ta · q)q · a0

(20)

b1i (Ti ) = (Ti − Ta · q)q · a1

(21)

b2i (Ti ) = (Ti − Ta · q)q · a2

(22)

3. Restrained Manson–Brown parameter in the creep damage calculation Although the creep damage at controlled test conditions is relatively easy to obtain, components rarely operate under constant conditions. The most commonly used approach to creep damage assessment under variable thermomechanical loading is by calculating the time that the component is subjected to some loading. In combination with the fatigue damage calculation, Robinson’s linear accumulation rule is usually used also for the creep damage

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Fig. 7. Determination of the time to rupture for stresses lower than the limit stress.

If  i =  i+1 , the value of the time division tij is evaluated from the preset temperature class width T,

Fig. 6. Creep damage calculation.

tij = calculation [1,4,5,32–35],

Dc =

ni mij   tij (ij , Tij ) i=0 j=0

trij (ij , Tij )

(23)

3.1. Discretisation of the load history

sTi =

Ti+1 − Ti ti

(24)

The class widths  and T depend on the expected ultimate stresses and temperatures and the preset number of classes in the calculation. The value of the time division tij is deducted from the preset stress class width , tij =

 . si

mij =

ti+1 − ti tij

(25)

(27)

In general, a master curve does not have its vertex on the abscissa axis. The limit stress  limi , at which the vertex appears, is calculated by finding the extreme of the master curve equation (Fig. 7). The first derivative of the time to rupture (Eq. (19)) is set to zero and the limit stress is expressed as such (Eq. (28)), d log tr (Ti ) = b1i + 2b2i log  = 0 d log  log limi = −

b1i 2b2i

(28)

b1i

limi = 10− 2b2i The time to rupture log trlimi for loading stresses under the limit stress  limi is then the same as for the limit stress otherwise absurd results may be obtained (Fig. 7), log trlimi (Ti ) = b0i −

Creep damage in the presented procedure is calculated continuously between two successive times ti and ti+1 separated by time increment ti . The time increment ti is further divided into additional time divisions tij for a more accurate calculation (Fig. 6). The size of the time division tij depends on the stress or temperature variation in the time increment ti . First, the slopes of the stress and temperature variation are calculated as i+1 − i ti

(26)

From the known time increment tij , the number of the time divisions mij is determined by

where tij is the actual time under some loading and temperature and trij is the corresponding time to rupture at the same loading and temperature defined by the RMB parameter. Index i represents the number of the time steps in the load history and index j represents additional subdivisions of time step i (Fig. 6). Loading temperature T has to be higher or equal to the creep temperature Tc and loading stress  has to be higher than the temperature dependent elastic limit of the material k(T) otherwise no creep occurs [3–5]. If the loading temperature is lower than the limit temperature for which material data is available, then the material data for the limit temperatures must be taken into account. The damage accumulation in a single time step is independent of the previously accumulated damage. When the sum of individual creep damages reaches the defined limit damage sum, creep rupture occurs.

si =

T . sTi

b21i 4b2i

.

(29)

3.2. Creep under compressive stresses The creep damage is calculated as a simple integration over all the time increments according to Eq. (23) (Fig. 6). However, there are three different creep relations for calculating creep damage due to compressive stresses [5,36,37]. The creep relation can allow either tensile creep only 0 < tr < ∞ if  > k(T ) and T > Tc , otherwise tr = ∞ tensile–compressive creep



0 < tr < ∞ if | |>| k(T ) and T > Tc , otherwise tr = ∞ or compressive healing −∞ < tr < 0 if  < −k(T ) and T > Tc , 0 < tr < ∞ if  > k(T ) and T > Tc , otherwise tr = ∞

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Fig. 8. Influence of creep relation upon creep damage accumulation.

Table 1 Polynomial coefficients b0 , b1 and b2 for the short-term test temperatures. Ti (◦ C)

b0i (s)

b1i (s/MPa)

b2i (s/MPa2 )

500 550 600

−170.599 2.265 1.852

152.082 6.156 6.227

−33.102 −2.639 −2.947

Fig. 9. Test master curves (thick lines) and interpolated master curves (thin lines) using the RMB parameter on the short-term test data [38].

Creep healing is possible only if Dc (t) > 0. Moreover, the tensile–compressive creep is larger than or equal to the tensile creep only and that is larger than or equal to the compressive healing. The appropriate creep relation depends on the material. Some materials tend to heal under compression loading while for other materials creep damage continues to grow regardless of the direction of the loading. The suitable creep relation for each material has to be determined experimentally. The influence of creep relation on creep damage accumulation is shown in Fig. 8. The temperature is supposed to be constant while stress changes from tension to compression at t1 . Before t1 the creep relation does not affect the Dc . However, after t1 the tensile–compressive creep results in the highest Dc , while compressive healing yields the lowest. 4. Examples One isothermal and one non-isothermal sample are considered for creep damage simulation and verification purposes. The RMB parameter is compared to the other time–temperature parameters. In both cases  = 160 MPa. In the first case the temperature equals 550 ◦ C while in the second case the temperature increases linearly from 20 ◦ C to 550 ◦ C. Both cases start at time t = 0 and last 1031.3 h. The first case thus exactly corresponds to one of the test results. The material under investigation is 1.25Cr0.5Mo steel. The existing short-term creep rupture test data [38] were gained at temperatures of 500 ◦ C, 550 ◦ C and 600 ◦ C and at three stress levels with standard creep rupture testing [ASTM E 139-00]. Every test was performed at a single temperature and a single stress level until rupture. There are only 9 test results available in the short-term test data set (Fig. 9).

The creep temperature TC is chosen as 450 ◦ C. This is the temperature where the first significant change in material behaviour occurs [39]. Creep also occurs below this temperature but it can be neglected due to the enormous times to rupture. If one estimates that noticeable creep deformation occurs below this temperature the creep temperature can also be set to a lower number. Polynomial coefficients b0i , b1i and b2i for each test temperature Ti are determined (Table 1). The RMB is compared to the LM, OSD and MH parameters with the R2 value. The resulting coefficients b0i , b1i and b2i for the test temperature 500 ◦ C are given in Table 2. They are calculated according to Eqs. (20)–(22). The R2 value confirms that the RMB parameter optimally describes the test data among the compared parameters although three significant decimal places are the same as for the MH parameter. It is assumed that the RMB parameter optimally describes the extrapolated area, too. Although the coefficients b0 , b1 and b2 for the test temperature of 500 ◦ C are quite different for each parameter, the measured test area is described well by any of the parameters (R2 > 0.9). The difference among them occurs when the master curves for lower stresses and lower temperatures are described. Master curves for the test temperature of 500 ◦ C are depicted in Fig. 10. Although the form of the input polynomial is unusual, the forms of the master curves defined by time–temperature parameters keep the anticipated form. The master curves described using the RMB parameter for the unknown temperatures are depicted in Fig. 9 with the temperature step T equal to 10 ◦ C for the temperature range 450–650 ◦ C. The input master curves are also pointed out. The interpolated master curves in Fig. 9 are also depicted for stresses above the temperature dependent tensile strength of the material. However, if stresses higher than the temperature dependent tensile strength appear,

Table 2 R2 value and coefficients at 500 ◦ C. Parameter RMBP LMP OSDP MHP input

q (−) 0.024 −1.0 0.0 1.0 /

Ta (◦ C) 438.236 0.0 0.0 438.236 /

R2 (–)

b0 (s)

b1 (s MPa−1 )

b2 (s MPa−2 )

0.972 0.955 0.960 0.972 1.000

10.973 18.320 9.757 5.047 −170.599

0.246 −5.995 1.573 4.878 152.082

−1.372 −0.047 −1.713 −2.273 −33.102

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D. Sˇ eruga, M. Nagode / Materials Science and Engineering A 528 (2011) 2804–2811 Table 3 Creep damage calculation.

Fig. 10. Short-term test data [38] and master curves for test temperature 500 ◦ C calculated with different time–temperature parameters.

static failure is presumed and the creep damage calculation terminates. The interpolated master curves are spread equidistantly in the temperature area but the test master curves are slightly modified. Although only a couple of short-term test data at high temperatures are available due to the accelerated creep tests, the RMB parameter ensures a trustworthy prediction in the extrapolated area. This is usually the case in that engineering praxis where the creep takes only a supplementary share in the damage prediction of the TML components. The applicability of the RMB parameter is verified on the existing long-term creep rupture test data for 1.25Cr0.5Mo steel [39]. In Fig. 11, both the short-term and the long-term test data are compared with master curves described using the RMB parameter calculated from the short-term test data. It can be seen in the figure that the long and short-term test data do not coincide and that the number of data points for the long-term test data is larger. However, it can be seen that the long-term test observations do coincide with the predicted times even though the predicted rupture times have been estimated from a completely different data set (i.e. the short term data set [38]). For the temperature of 650 ◦ C the difference between observed long-term test data and the predicted value of the RMB parameter is slightly bigger. The RMB parameter is determined from the short-term test data at temperatures of 500 ◦ C, 550 ◦ C and 600 ◦ C. Values of these

Damage

Constant temperature

Linearly increasing temperature

RMBP LMP OSDP MHP

1.018 1.036 0.998 1.010

0.0216 0.0215 0.0219 0.0207

short-term test data deviate from the average value of long-term test data at temperatures of 500 ◦ C, 550 ◦ C and 600 ◦ C (Fig. 11), thus also the prediction using the RMB parameter at 650 ◦ C can deviate from long-term test data. The calculated creep damage values for the isothermal and nonisothermal case using different time–temperature parameters for determining master curves are given in Table 3. If fatigue only is taken into consideration, no damage is obtained for the applied load history at all. Creep damage calculation is performed according to Eq. (23). In spite of the simplicity of the example, some differences between defining master curves using different time–temperature parameters are observed. High temperature plays a crucial role in creep damage estimation and constant temperature is about 50 times more damaging than linearly increasing temperature. This is due to the fact that the creep damage Dc equals 0 until the load temperature reaches the creep temperature. The isothermal case corresponds exactly to one of the test points. Thus it can refer to Dc = 1. The predicted creep damages deviate from this value depending on the parameter used for the master curve description. The damage predicted using the RMB parameter stays stable regardless of the load history. It firmly remains the midmost apart from the inconvenient alterations in the damage prediction of other parameters. 5. Summary and conclusions The RMB parameter is presented as a successful attempt to choose the optimal description of the master curves resembling the properties of the efficient, but numerically unpleasant MB parameter. If creep does not occur as an independent phenomenon but is considered an additional occurrence in thermomechanical fatigue, the RMB parameter seems to be a promising choice for modelling the master curves. Due to the robustness of the algorithm to determine the RMB coefficients, a detailed pre-assessment of the test data and the form of the test master curves do not need to be treated carefully. The number of tests can be reduced and the RMB parameter still ensures a trustful prediction in the extrapolated area. Instead of the LU decomposition for solving sets of linear algebraic equations any other numerical tool can be used. The introduced procedure is integrated in the engineering software LMS Virtual.Lab. Acknowledgements The authors thank to the Institute for the Promotion of Innovation by Science and Technology in Flanders IWT and to the LMS Deutschland GmbH, Kaiserslautern for the financial support to carry out this investigation through sponsored Project No. IWT 060532. References

Fig. 11. Comparison of the predicted results using RMBP with existing long-term creep rupture test data (empty symbols) [39] and existing short-term creep rupture test data (solid symbols) [38].

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