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Investigation of the design of creep damage constitutive equations for low stress level

ABSTRACT This paper reports the latest attempt to model the creep deformation and rupture for high Cr (91) alloy under low stress range. While accurately predicting the creep deformation and damage and the ultimate rupture is paramount important, both for the academic research and industrial applications, the current situation is unsatisfactory, particularly for the lower stress range. This paper systematically investigates the formulation as well as tentatively underlines the scientific reasons, to improve and accurately predict the creep deformation, creep damage and the ultimate rupture of P91 in the stress range of 80–125 MPa. This paper contributes to the knowledge on how to accurately model the creep deformation and damage for P91. Furthermore, it contributes to the methodology through the development of a generic model.


Introduction
Creep damage is significant for structural parts working at high temperature, which may lead to fracture failure. Therefore, creep damage plays a significant role in all fields involving high-temperature work, especially for components operation in some dangerous working environments such as high-temperature steam pipes. Creep damage may bring high economic losses and human hazards. For example, Kansas City Power's steam pipe had a creep failure that resulted in a catastrophic accident in 1998 [1]. Therefore, for the safe operation in high temperature environment, accurate prediction of creep damage is essential.
At present, P91 steel has been widely used in thermal power field all over the world, mainly used in superheater and main steam piping system [2]. These structures work for a long time under high temperature and certain stress loading, and so it is essential to predict their fracture life. With the development of society and the improvement of people's standard of living, the requirements for electricity consumption have been increasing. In order to provide energy and improve efficiency of generating units, many countries have been building supercritical thermal power units in recent years [3][4][5][6][7][8][9]. P91 (9Cr-1Mo-V-Nb) is a high chromium (Cr) alloy steel with an excellent creep and oxidation resistance at high temperatures [10] that is more [11] suitable for high temperature steam pipe.
The traditional common pipeline life prediction of thermal power units generally uses isothermal extrapolation method, which is practical and straightforward, and so it is still widely used [12]. Then, a more accurate time-temperature extrapolation method based on the Arrhenius equation appeared, but it was still not accurate enough and easily overestimated the service life of the structure [13].
In order to accurately predict the creep life, it is necessary to understand the creep mechanism and microstructure evolution correctly in the process of materials service according to the physical mechanism of creep, and to use the correct prediction method.
With constant stress loading, there are three different creep deformation stages; these are as follows: (1) Primary creep stage. In this stage, the initial strain rate is high but decreases with time; it is also called decelerating creep stage. The creep deformation will enter next stage during the strain rate decreases to a value. (2) Steady state creep stage. In this stage, the creep strain increases slowly. Strain rate, which is a function of stress, increases as the stress increases. It can be described by power-law relation. Creep damage is accumulating over time and damage occurs within the material. On a macroscopic level, the damages caused during creep are due to particle coarsening and the growth of voids. As the stress level is reduced from high to low levels, the time to creep failure becomes extended. Continuum damage mechanics (CDM) model is a popular method of material life simulation and is widely used to develop the creep constitutive equation of P91 steel.
CDM-based methods were originally proposed by Kachanov [10], developed from continuum mechanics. In order to describe each damage variable, an equation for each variable was developed and added into a CDM model.
The characteristics of CDM based methods are as follows: (a) It is not necessary to know the damage mechanism of materials in detail from the micro-level. The degree of creep damage can be directly evaluated from the value of each damage parameter [20]. (b) The creep constitutive equation based on CDM can be used to analyse the creep damage behaviour simply by using the finite element programme, which can greatly reduce the research time and cost [14]. (c) By introducing different damage parameters, the effects of different damage on creep behaviour can be conveniently studied [15].
A critical review on the development of creep constitutive equations can be found [16][17][18] where the questions about how to model the creep cavitation damage, the coupling of the creep deformation and creep damage, and multi-axial generalisation were raised. Detailed work on some of the these aspects can be found [19-26,and 27].
More specifically, the current creep constitutive equations are unsatisfactory under a wide stress range. Due to its high cost and long duration, the longterm creep tests were rarely carried out; the accelerated creep test under higher stress are usually used hoping to extrapolate to the lower stress level. Therefore, the creep damage equation developed based on the creep test at high stress may not be applicable for low stress levels.
Recently, Yin and Faulkner [38] proposed a cavitation damage calculation method for 9Cr alloy steel based on McLean and 5,CDM framework. On this basis, Chen [2] developed a set of creep constitutive equations suitable for medium-high stress range, and added some damage parameter such as solute depletion, cavity nucleation and growth and particle coarsening. Yang et al. [37] tried to apply it to the low stress range and found that it did not work, but he did not carry out further research. At present, the research of low stress level and long-term creep behaviour is unsatisfactory; the common prediction methods still have high percentage of error.
The main research objectives of this paper are as follows: (1) To research the applicability of the constitutive equation developed by Chen [2] to assess creep behaviour of P91 steel at 600°C in the low stress range of 125 MPa~80 MPa. The calculated results were compared with the collected experimental data, and analysed to find out the reasons why the constitutive equation could not be applied to the low stress range. In order to improve the understanding of the changes in creep mechanisms under different stress loading, this study will discuss from the creep mechanism perspective. (2) To find a method which can improve the applicability of the equation, and then apply it on the Chen's equation [12] by means of comparison with experimental data to verify whether the proposed method is applicable. (3) To find a modified constitutive equation in low stress, based on the Chen's equation [12].

Parameters in creep damage mechanics
In order to accurately describe the creep damage behaviour, it is necessary to add the microscopic damage parameters of the materials in the creep process into the CDM model. During the long-term operation at high temperature, the microstructure and structure of P91 steel will gradually change; the damage parameters related solid solution depletion, particle coarsening and cavitation are defined as Ds, Dp, Dn, respectively.

Solid solution depletion
The decrease of creep resistance is caused by Molybdenum (Mo) depletion in sub grain matrix under the action of long-term high temperature and stress [28]. D s can be calculated by the equation below [29]: C o and C t , respectively, are the initial concentration of solid solution and the average concentration at time t. In Dyson's approach, the rate of D s can be calculated by Wert-Zener equation [30]: Material constant Ks can be calculated by the below equation: D is the diffusion coefficient of Mo in the matrix. After measuring C 0 and, it can be concluded that C 0 = 0.56 mol% and C e =0.33 mol% [31].

Particle coarsening
The coarsening of precipitates in P91 steel has an important influence on the creep resistance of materials [28]. The Chen's creep damage constitutive equation [12] mainly considers the laves phase coarsening. According to Dyson's approach [29], D p can defined as where is the particle diameter at the initial creep stage of material, P t is the particle size at time t.
It is supposed that the coarsening of the particles can be calculated by Livshitz-Wagner equation [29].
where K is a constant related to interface energy, equilibrium solute concentration and diffusion coefficient, r and r 0 are the radians of the particle.
In addition, the formula for calculating D p change rate is defined as where is a constant of rate. The value of K p is the third power of the initial particle size.

Cavity nucleation and crack formation
The creep behavior of P91 steel is controlled by different mechanisms. One of which is the most important one is the void nucleation and cavity formation [28].
The equation of the rate of D n is shown below [29]: The value of A and B depends on temperature [29]. A 0 =AB, B 0 =B-1. The value of D n might be over 1 when A 0 and B 0 at large strain and high strain rates. It will lead to a divergence in the result of calculation. So, the value of D n should be larger than 0 and less than 1.

Dimensionless parameter of strain hardening (H)
The primary creep model is suggested by Ion et al. [9]. H is a dimensionless parameter defined as The rate of H is The value of H is larger than 0 and less than H � is the maximum value of Harding in creep damage behaviour. Constant = EФ. Ф is the volume fraction, E is the Young's modulus.

Kachanov-Rabotnov equation
Kachanov developed the prototype of the Kachanov-Rabotnov equation in 1958. Then it was improved by Rabotnov [17] added the mathematical description of tertiary creep.
Kachanov-Rabotnov equation is a nonlinear first order differential equation, used a power-law creep rate function which is defined as The _ ε is the creep rate when creep is occurring, σ is the loaded stress on the materials during the creep behaviour. _ ε 0 and _ D 0 are the initial state. n and v are material constants called stress exponent.
2. According to change the value of n, this equation can be used on different stress and temperatures easily [33].

Dyson's equation
Kachanov-Rabotnov equation uses D to describe many microdefects like precipitate coarsening, solid solution. It is too simple to describe creep damage behaviour accuracy. Dyson [5] improved the CDM model, combined some damage parameter into CDM model and used sine-law to replace the power law. The Dyson's creep damage constitutive equation is defined as

Chen Yunxiang and Yang Ke's equation
Based on Kachanov-Rabotnov's CDM model, in order to improve the accuracy of description of creep damage behaviour, some damage parameters were added into CDM model by Keller et al. [2]. By only considering the effect of the coarsening of Laves phase, the new equation is defined as: The advantage of equation: By analysing the microstructure changes of materials in the creep process and adding damage parameters in a targeted way, the simulation of creep life is more accurate after experimental verification [2].
Chen's equation was based on the initial CDM model, and various damage parameters were added to facilitate the analysis of the effect of different damage parameters on the creep behaviour.
The disadvantages of Chen's equation: Since Chen's equation is a power-law function, it may lead an imprecise prediction in the primary and steady state creep stage at low stress level; the creep strain is usually lower than experimental data.

The current situation of application to wide stress level
At present, there is no creep model that can be used to predict creep life accurately at all temperatures and stress applied. With the change of the working environment of material, the parameter in creep model may also change.
Yang et al. [37] tried to use the constitutive equation by Chen et al. [2] to predict creep life at 600°C and low stress level; the result was not working. But he did not propose to improve method or assumption. Basirat et al. [1] developed a creep model which can be used at 80 MPa~200 MPa. In his research, the value of damage parameter A will change with applied stress. Basirat tried to establish a mathematical relationship between the constant A and the stress level, but he only optimised A value in terms of macroscopic phenomena; there is no clear trend in the constant A when the stress level is changing from low to high. Therefore, it is difficult to use it in prediction with confidence. Yin and Faulkner [38] stated that the constant A is a function related to temperature, which is used to describe the void nucleation and crack formation in different stress and temperature, and no relationship with applied stress.
From the conclusions of Chen [2] and Rouse et al. [19], in the power-law system, with the change of the stress range, the value of n is not fixed and needs to be changed with the change of the stress level.
Christopher [3] improved a creep model based on Dyson and McLean, developed a model of MX precipitates and formation and the growth of Z-phase with time. The simulated result in his research has a good agreement with experimental data at 50MPa~100MPa. In his research, the value of damage parameter K s and K p is within a specific range; the specific values were calculated by numerical optimisation. The relationship between damage parameters and stress levels needs to be further investigated. Basirat et al. [1] also stated that the value of parameter K s and K p is not fixed while applied stress change.

The method of calculation
From the constitutive equation above, it can be observed that almost all equations are ordinary differential equations of the first order. As generally mathematic differential knowledge, Runge-Kutta method and Euler method are usually used to solve the ordinary differential equations of the first order. The results by Runge-Kutta method are more accurate than Euler method.
In this study, the usability of the improvement of constitutive equation is focused. The accuracy of creep life simulation is not key point. Therefore, in order to simplify calculation, the Euler method has been chosen.

Methodology
(1) The creep test parameters of multiple groups of P91 steel at 600°C for a long time with low stress were collected. The total stress range was 125 ~ 80 MPa, and the creep life range was 4000 ~ 10,000 hours. Then, all the parameters were collected which were used in the creep modelling by Chen [2].
(2) Based on the equation by Chen [2], the creep life is calculated and compared with the collected experimental data. We try to adjust A to improve the applicability of equation at low stress level; if it does not work, then adjust n. (3) Compare the modified results and experimental data again, draw the curves of creep damage parameters and creep life curves on the same figure, analyse the reason from the perspective of creep mechanism, consider that the accuracy of K s and K p could affect on simulated results. Adjusting A, n, K s , K p . (4) According to the final result, to assess the applicability of this improved method for creep constitutive equation at low stress level.

Calculation method and verification
All the parameters used in the equation are calculated by Chen [2], according to the data he collected in different literatures, the material constants and damage parameters used in calculation are shown in Table 2.
Excel software can plot out the curve of relationship between creep strain and time or the creep rate and time when calculation is done. The result is shown below: By comparing the calculation results with those posted in the literature by Chen [2], we found the two results are consistent, which can be proved a point that there is no error in the calculation, and this calculation method can be used in the next research.

Material and experiment
Due to the long time and high cost of low-stress creep test, there are relatively less experimental data that can be collected. In this study, four groups of lowstress creep experimental data of P91 steel for a long time were used. The total stress range was 125~80 MPa, and the creep life range was 4000~10000 hours. The stress decreases gradually in the four sets of data, which is helpful to analyse the trend of each damage parameters in different stress.  [38].
The collected experimental data are shown as Figure 2.

Compare with the experimental data Adjusting the value of A
According to Yin's assumption, coefficient A does not change with the stress level. A 0 =AB,B 0 =B-1, as equation 7.1 and 7.5: Firstly, Chen's equation was used for low stress level according to this assumption, and the results are as follows: Compared with the simulation results and experimental data, there was a big difference between them. Chen's equation cannot be directly used in low stress, this conclusion is consistent with 37.
Through data comparison, the results simulated by Chen's equation have a short life and the strain variable increases sharply, which cannot be consistent with the experimental data.
According to the creep damage mechanism, the sharp increase of strain variable is directly related to the cavity damage, and the coefficient A is directly related to the cavity damage D n . And according to Basirat's results, when the stress is small, the coefficient A decreases with the decrease of stress. Optimisation of A-values is beneficial for understanding creep behaviour at low stress    After adjusting the value of A, the simulated creep life has a positive agreement with the experimental data, but the strain is still not consistent. When the simulated results reach the life, the strain is extremely large and has no engineering significance.      Adjusting the value of A and n ccording to the conclusions of Chen [2] and Rouse et al. [19], the value of n can be changed with the change of the stress level; so, it would be recommended to continue to adjust the parameters A and n at the same time and then observe the comparison results.
By comparing the result, the simulated result could have good agreement with experimental data by adjusting the parameter A and n at 600°C, 125 MPa, 120 MPa.
However, at 100 MPa and 80 MPa, the creep life over 60,000 and 100,000 hours, respectively. The simulated result and experimental data cannot be consistent although the A is adjusted to 1E-30, n is adjusted to 10,000. And through calculation, it can be concluded that when n is at large value and A is at small value, continuously increasing n value and decreasing A value will not bring significant increase in creep life, as shown in the following table: According to Chen' equations [12] 7.1 and 7.5: When the value of n is tending to be extremely large, and A is extremely small, the value of damage D n will also become extremely small, resulting in no physical meaning. At this time, the damage parameters that dominate the creep behaviour may change. So, the damage parameters and creep strain are plotted out together for observation. By comparing the curves drawn with 125 MPa, 120 MPa in 600°C, it can be observed that by adjusting the value of A and n to fit the modelling results with experimental data, during a significant increase in creep strain, the major damage parameter is D n while D s and D p is still increasing.
Under 100 MPa and 80 MPa, 600°C, the value of D n is small until the failure of material occurs. The D s and D p are reaching saturation before the failure of material. It appears that D s and D p are dominating the creep strain in modelling result, but by research, the cavity damage D n should have a jumping when the material failure [39].

Adjusting the value of A, n, K s , K p
According to Basirat et al. [1] and Christopher [3], the material constants K s and K p should be in a certain range. The values of K s and K p calculated by Chen are not fitting well in low-stress range. In this paper, will attempt to adjust the values of K s and K p to observe the modelling result after lessening the D s and D p .

By
adjusting the values of K s and K p , the modelling result is reaching a better agreement with experimental data, as shown in Figure 7. Therefore, the value of K s and K p have a considerable effect to simulate result. Further investigation is required.

Continue to optimise the model
According to the simulated curves in Figure 7, the creep life has been consistently agreed with experimental data. However, the simulation results are always lower than the experimental data at the primary creep stage and at steady state creep stage. Due to a defect in the power function, it is unable to describe more exactly the creep damage in the early stages at long-term low stress level.
To improve this situation, we change the Chen's equation to a hyperbolic sine-law with the damage variable held constant. The modified equation is shown below and simulated result is shown as

Discussion
At present, most of the common creep damage constitutive equations are only applicable to a specific stress range. When the stress level decreases from high stress to low stress, the damage mechanism and creep deformation mechanism may change due to the different stress level. Therefore, the parameters of the creep constitutive equation may also change.
Based on the data used in this study, no obvious rule was found for the value of A and n, which means it is not robust. The adjustment of A value alone has no engineering significance at the long-term of thousands to tens of thousands of hours of creep behaviour; the stress exponent n needs to be adjusted simultaneously to make the simulation results have good agreement with the experimental data.
Through a simulation, it is found that in a certain range of low stress level, a simple adjustment of coefficients A and n can still make the simulation results to have a good agreement with experimental data. However, when the creep life reaches tens of thousands or even hundreds of thousands of hours, the adjustment of coefficients A and n has a small effect.
Therefore, this paper proposes a hypothesis from the micro-level, which may be because the creep damage mechanism under low stress is not well understood at present, and the values of K s and K p of damaged D s and D p are not accurate enough. In the long-term creep behaviour, D s and D p , which are not dominant in the short-term creep behaviour, might have the chance to reach saturation. It is resulting in the predicted life being smaller than the experimental data. If the values of K s and K p are accurate, the simulation results may have a good agreement with experimental data.
By adjusting the constants K s and K p , the simulation creep life was successfully in agreement with the experimental data. Therefore, this hypothesis is applicable in the long-term low-stress creep behaviour. It was shown that phenomenology theory has limitations when applied to the prediction of lowstress and long-term creep. An accurate description of materials at the micro level may be a more effective method to predict the creep behaviour of low-stress and long-term. The value of the constant K s , K p decreases as the stress is reduced. K s , K p could be a function of stress loading.
Hyperbolic sine-law function showed better reliability, but still did not provide a complete description of primary creep stage in this simulation, which might be due to the inaccuracy of some damage variables.
The future work will see the implementation of the improved method to the large stress range and of different creep constitutive equation to verified reliability.

Conclusion
(1) Reported in this paper is a proposed method to improve the accuracy of predicted at low-stress and long-term creep behaviour for P91 steel at 600°C; the following results were obtained: 1. Adjusting the value of A has no obvious advantage on increase the applicability of creep model at low stress level. 2. Adjusting the value of A and n can make the simulated result have good agreement with experimental data in some case, but it is still not working at tens of thousands of hours creep behaviour. 3. Adjusting the value of A, n, K s , K p has significant advantage on increase the applicability of creep model at low stress level. 4. By the results, adjust damage rate constant is an effective method to improve the applicability of constitutive equation. 5. There is a trend in the damage constants K s and K p with the stress levels.
The results show that this method is applicable. This improves method links mechanical analysis with materials science.
(1) The above results showed that the constitutive equation based on phenomenology theory has its limitations in describing the creep behaviour of low stress in a long term. It is necessary to describe the creep mechanisms more accurately from the perspective of microstructures. Each damage constant has a significant influence on the creep life. It is necessary to calibrate them more accurately in the low stress level, and to establish the relationship between the constants of different variables and temperature or stress levels. For example, clarify the relationship between the constant K s , K p and stress level.
(2) The current power law minimum creep strain rate and stress relationship is not suitable to be extended to lower stress level. Compared to the power-law function, the hyperbolic sine-law function provides a more accurate description of damage in the primary and steady phase of creep. The modified sine law should be considered in future work.