Creep Constitutive Model Research of High Temperature Reactor

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Structure Using Artificial Neural Network
Fan Zhanga, Shoutong Daib
a. China Institute of Atomic Energy, Box 73, Box 275, Beijing, zhangfan@163.com

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b. Corresponding author, China Institute of Atomic Energy, Box 73, Box 275, Beijing,
daishoutong@163.com

Abstract ： The new types of nuclear reactors require higher design temperatures and longer

service life than before. Consequently, the investigation of the long-term creep in metallic

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components utilized in high-temperature environments has gained greater importance. The analysis

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of creep necessitates precise constitutive models which need a substantial quantity of long-term

experimental data. However, the acquisition of such data through experiments incurs considerable
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costs, thus the need is highlighted to develop a dependable model that can forecast the long-term
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creep behavior of materials based on short-term creep test data. At present, the amalgamation of

artificial neural network (ANN) techniques in the creep of reactor design lacks any established

precedent. This academic article presents an ANN model that characterizes the creep deformation
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properties of materials and predicts the long-term creep deformation behavior. This is achieved
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through the denoising of limited experimental data, identification of the cut-off point between the

primary and secondary creep, and the incorporation of ANN techniques with Norton Bailey's power

law for segmented regression and reverse parameter calibration. The analysis indicates that the
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optimized constitutive model can describe the creep behavior of materials more accurate than either

of the creep theory or neural network training model . Overall, the model predicts the long-term
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creep features with an improved accuracy.

Key words ： Reactor structure mechanics; Reactor structure; Creep; Steel; Artificial neural
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network; Norton-Bailey’s power law; Creep constitutive model; Creep cut-off point; Long-term

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 creep

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Creep is defined as a time-dependent plastic deformation that occurs at a certain temperature and

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load [1]. Extreme working conditions of the reactor system, such as high temperature, high radiation,

and strong corrosion, make metal components highly susceptible to creep [2,3]. Creep rupture is

one of the most commonly-encountered failure modes of engineering materials and structures [4].

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To ensure the secure operation of the reactor, it is necessary to understand the high-temperature

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mechanical properties of the materials, especially the long-term creep properties [5].

For reactor components or materials serving in high-temperature conditions [6], the design life is
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getting longer and longer. However, the implementation of long-term creep testing presents
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challenges and incurs significant experimental expenses. To accurately analyze creep behavior, it is

necessary to investigate a more precise constitutive model that can also extrapolate long-term creep

performance from short-term creep test data [7,8,9]. The intricate nature of high-temperature creep,
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its sensitivity to numerous items, and the divergence of experimental data, all above are obstacles
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to obtaining an accurate creep model. Inaccurate models lead to overly conservative evaluation

methods, posing numerous difficulties for engineering evaluation of high-temperature creep.

Therefore, it is necessary to conduct in-depth research on creep analysis methods for high-
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temperature reactor structures to obtain more accurate and practical creep constitutive.

Artificial Neural Network (ANN) is an information processing system based on imitating brain
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function, which can quickly and accurately realize the simulation and prediction of data information

[10].
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ANN has already played an active role in creep research. Gui and Chen [11] confirmed the

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 feasibility of using artificial neural network methods to establish the creep constitutive model of

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materials. Li and Peng [12] further applied ANN to design guidance, using Backpropagation (BP)

neural networks to predict the influence of material composition on the creep rupture life. Wang

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and Sun et al. [13] used BP neural networks and back propagation learning algorithms to back-

calibrate the constitutive parameters of a θ-projection method, which combines artificial neural

networks and existing creep models. However, the θ-projection method recognized that the creep

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process of metal materials consists of strain hardening in the primary creep and strain softening

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process in the tertiary creep, without the existence of secondary [14]. This method is more suitable

for materials with short secondary creep, but not for materials used in reactor design [15].
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Currently, there is no integrating the Artificial Neural Network (ANN) approaches with reactor
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creep analysis. Based on the creep experimental data of an austenitic stainless steel for 10000 hours,

a model, which amalgamates the ANN technique with Norton-Bailey's power law, was obtained

using data within 4000 hours. Using the model to predict creep deformation for 4000 to 10000 hours,
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and verify the reliability of the model by comparing the predicted data with experimental data. This
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study ultimately resulted in a creep constitutive model that can be used to predict long-term creep

using short-term data, while overcoming the limitations of solely employing the BP neural network

or Norton Bailey's power law in constructing creep models, which amalgamates the ANN technique
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with Norton-Bailey's power law. Owing to its strong nonlinear processing features, ANN could

output nonlinear relationships of many complicated scientific problems [16,17,18], and the existing
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creep theory can guide neural networks to make more accurate predictions of long-term creep in the

absence of sufficient data samples.

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1. Experiment

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 The creep test of the austenitic stainless steel was carried out using a high-temperature creep

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specimen with a diameter of 10 mm( Figure 1).

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Fig. 1. Sample for Creep Test.

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The high-temperature creep test was conducted on the austenitic stainless steel using RD2-3 and

RC-1130A creep endurance testing machines to obtain the total strain. The displacement sensor
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resolution of the device is 0.001mm.
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Fig. 2. The process of sample sampling and processing. (a) Marking of steel stamps. (b）Wire cutting sampling. (c)

Brinell hardness test. (d) Mark transfer and processing.

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This experiment selects 5 levels of constant creep stress at 500℃, 550℃, 600℃, 650℃ and 700℃,

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 and determines the relationship between creep deformation and time under the specified temperature

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and stress of the sample.

2. BP neural network creep model

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The signal and error in the BP neural network are propagated in opposite directions, i.e., the signal

is propagated forward and the error is propagated in the reverse direction [19] . If it matches, the

learning will be complete . Otherwise the backpropagation will be performed. According to the

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original communication path, the reverse calculation is carried out between the actual output and

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the ideal output, and the neuron weight and displacement of each layer are adjusted by regression

method or other algorithms to reduce the signal error [20,21]. The whole process of solving is
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depicted as follows: Sample normalization processing; Set threshold and weight values for the first
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time based on experimental parameters; Set network structure and parameters; Calculate system

error; Correction weight; If the error reaches the minimum or acceptable level, the result will be

output. Otherwise, the threshold and weight values will be adjusted before next calculation [22,23] .
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2.1 Training programs

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To prevent any convergence issue or model overfitting [24], the gradient descent method is used to

construct the neural network. To eliminate the impact of large differences in magnitude, the data is

normalized so that the input and output data can be between 0 and 1. The normalized formula is as
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follows.

Xn = (Xi−Xmin)/(Xmax−Xmin) (1)
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Based on the experimental data of the austenitic stainless steel at 500℃,220MPa, a three-layer BP

neural network was established using MATLAB toolbox. The appropriate training results are
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ultimately obtained by constantly adjusting the network parameters [25] such as the number of

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 hidden neurons, learning rate, number of iterations, training function, etc. The research found that

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while the trained neural networks have good regression performance, there are significant errors in

the primary creep, especially at the beginning of creep. Therefore, the model cannot predict the

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long-term creep behavior.

The aforementioned issue may be attributed to the inadequate number of experimental data

samples for the austenitic stainless steel at 550℃, 220MPa, as well as insignificant creep at 550 ℃.

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Consequently, these data points were substituted with creep experimental data at 700℃, 28MPa,

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which possessed a larger sample size for analysis. A comprehensive total of 407 sets of creep data

were collected for the austenitic stainless steel at 700℃, 28 MPa, with the experiment being
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conducted over a duration of 3992.3 hours. The neural network model, trained using this data,
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showed an enhanced regression performance. However, it still does not provide accurate

extrapolations.

The above issues may be caused by the functionality of the function itself, the following efforts
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have been made to solve the problem. Replacing the built-in Matlab toolbox with“Newff”, By
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using the trainlm training function, the training objective and learning rate are adjusted continuously,

an extrapolable neural network model is obtained. It is worth mentioning that the “newff” is a

function of code, while the “trainlm” is a training function for training neural networks.
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2.2 Training results

Using the above scheme, it is shown in Figure 3 of the neural network correlation results
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obtained from the training of experimental data at 550 ℃, 220 MPa and 700 ℃, 28 MPa. The

dataset is divided into a training set, a testing set, and a validation set, accounting for 70%, 15%,
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and 15% respectively [26]. The correlation image consists of four small images, representing the

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 training set, validation set, test set and the overall regression image respectively. The target is the

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actual value of the input sample (the normalized value obtained from the experiment), and the output

is the predicted value of the obtained neural network. The point falling on the diagonal means that

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the predicted value is completely consistent with the true value. A good correlation model means

that the R-value approaches 1, the fit line coincides with the diagonal, and the sample points can be

evenly distributed on the fit line.

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Fig. 3. Regression of the training model. (a) Regression of 550, 220 MPa model. (b) Regression of 700, 28 MPa
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model.
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As shown in Figure 3, it can be seen that 700℃, 28 MPa model with a larger sample has significantly

better regression than the model of 550℃, 220 MPa.

The fitting images of the resulting model are shown in Figure 4.

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Fig. 4. Fitting results of the training model. (a) 550℃, 220 MPa model fitting result. (b) 700℃, 28MPa model

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 fitting result.

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At 550 ℃, 500MPa, the initial error of the model in the first stage is relatively large (the orange

part represents the error). As can be seen in Figure 4, after replacing the input data with a larger

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number of samples, the resulting neural network model not only improves the regression

performance significantly, but also fits better, while the error becomes significantly smaller. From

these two figures, it can also be seen that temperature and stress levels have a significant impact on

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the creep behavior of materials.

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As depicted in Figure 4, the substitution of input data with a greater quantity of samples yields a

notable enhancement in the regression performance of the neural network model, leading to
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improved fitting and a substantial reduction in error. The aforementioned figures also demonstrate
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the significant influence of temperature and stress levels on the creep behavior of materials.

However, although the model improves its accuracy, it still lacks the ability to extrapolate long-term

creep performance based on short-term creep data.

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Subsequent to the adjustment of the training function, the resultant model exhibits the capability to
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extrapolate predictions, which are presented in Table 1. The model for creep deformation prediction

reveals superior performance in 4000 hours, followed by a subsequent decline in prediction

accuracy over time. Notably, an error exceeding 25% at 8000 hours signifies inadequate prediction
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accuracy.

Table 1. 700℃, 28MPa model prediction results

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Time(h) Predictive value(%) True value(%) relative error

3000 0.0765129070 0.0761363640 0.492130099%

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4000 0.0856556584 0.0863636360 0.826539171%

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 5000 0.0905434329 0.0960227270 6.051564311%

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6000 0.0924609807 0.1034090910 11.840789727%

7000 0.0929619079 0.1079545450 16.127720989%

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8000 0.0930388416 0.1170454550 25.802786183%

2.3 Analysis of training results

The results of the above multiple sets of training can be analyzed as follows:

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(1) The accuracy of a neural network is contingent upon the quantity of samples available. A greater

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number of samples enables the neural network to depict the creep behavior of the material more

precisely and make more accurate predictions. Nonetheless, the establishment of reliable and precise
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neural networks is easily hindered by the inadequate density of experimental data. Additionally, the
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limited duration of creep testing poses challenges in predicting creep behavior. Consequently,

relying solely on neural networks for machine learning of samples is insufficient to achieve accurate

predictions.
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(2) As the experiment failed to collect points with sufficient density in the primary creep, a large
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amount of data in the secondary, in turn, made the neural network to shave off some data in the

primary as noise, and the neural network had difficulty in accurately describing the primary creep.

The strain accumulation in the first stage of creep cannot be ignored in many cases [27] , the creep
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constitutive model should have the ability to accurately describe the primary creep characteristics.

(3) In the case of insufficient experimental samples, considering the significant shortcomings of
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directly trained neural network models in describing the primary creep and predicting the long-term

creep, a new neural network model is considered to be constructed. It is proposed for a method of
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constructing creep constitutive model by combining ANN and Norton-Bailey’s power law.

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 3. Norton-Bailey’s power law

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3.1 Norton-Bailey equation

Primary creep by aging theory creep strain can be described as:

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εc = Aσntm (2)

where the parameters A, n, and m depend on the temperature and are determined by the creep

experimental data.

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The secondary creep can be described when the stress is constant, i.e., as a constant velocity theory.

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It degenerates to the Norton-Bailey’s power law by taking m=1 [28] :

εc = εf + A0σn0(t−tfp)
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3.2 Creep constitutive model based on Norton-Bailey’s power law

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Among many creep analysis methods, the Norton creep equation is the most widely used and is

suitable for the special case of uniaxial stress. The Norton-Bailey’s power law has the greatest

advantage over other models in that the stress equations have the same form regardless of the value
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of the stress, making the stress analysis much more convenient [29]. The RCC-MRx specification
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and the internationally emerging advanced high-temperature design method, the linear matching

method, use this model to describe the creep behavior of materials and are built into the commercial

finite element software ANSYS and ABAQUS. The method [30] use this model to describe the
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creep behavior of materials and are built into the commercial finite element software ANSYS and

ABAQUS.
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In the usual research, the fitting methods are used to obtain creep curves, such as a fitting calibration

method based on the creep intrinsic parameters of Norton-Bailey model, and some results of which
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are shown in Figure 5.

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Fig. 5. Comparison of creep curves of 316H stainless steel at 500°C

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The Norton-Bailey model, although possessing a straightforward framework and practical

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applicability, exhibits inadequate efficacy in accurately representing primary creep, as evident from

Figure 5. To rectify the primary creep discrepancy, precautionary actions must be implemented to
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ensure safety, such as advancing the cut-off point's location and augmenting the strain rate of

secondary creep. However, due to the protracted duration of the secondary creep in reactor
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component materials, this approach engenders an excessively conservative creep constitutive

equation, thereby imposing substantial strain on the evaluation of structural creep.

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4. Neural network combine with Norton-Bailey’s power law

This paper presents a novel approach for acquiring a creep model by integrating the Norton-Bailey's
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power law with neural networks, taking into account the characteristics and limitations of the

aforementioned methods. In order to address the challenge of limited availability of long-term creep
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data, the Norton-Bailey's power law is employed in the secondary stage of the constant velocity

theory to enhance the predictive capability of the model. Additionally, the utilization of neural
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networks enables effective handling of nonlinear problems and provides a more accurate depiction

of the primary creep behavior compared to the Norton-Bailey's power law.

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4.1 Optimization of training programs

In the second stage of creep, m=1, logarithmically solving for the Norton-Bailey’s power law:

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 𝑙𝑛𝜀 = 𝑙𝑛𝐴 + 𝑛𝑙𝑛𝜎 + 𝑙𝑛𝑡 (4)

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included among these 𝜀 = 𝜀𝑐 + 𝜀𝑓

The experimental data of 700 ℃, 28 MPA is treated by logarithm method , which is showed in

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Figure 6, the key to this study is to find the cut-off point of primary and secondary creep. At the

beginning of the primary stage, the process time was too short to describe.

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Fig. 6. Creep curve after taking logarithmic value. After calculating the logarithm, this part was amplified, which
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is more helpful in finding the cut-off point.

Since the small number of experimental data samples and the significant impact of noise, Gaussian
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Noise Reduction is applied to the samples before searching for the cut-off point. It is more

convenient to find the cut-off point by adjusting the window length to obtain a smooth curve.
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Set the threshold N for the first screening, it is used of the criterion that the strain rate derivative is

less than the threshold to screen out the portion of the slope that is too large due to the small
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deformation at the beginning of the experiment.

The data are more concentrated after the first screening, and a larger window length is used to further

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 prevent the influence of noise perturbations on the slope judgment.

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Bin [31] developed a creep strain rate curve and proposed that the first stage ends when the

derivative of the creep strain rate equals zero. From the formula (4), in the primary and secondary

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creep demarcation the value of the strain rate tends to 1 after the logarithm, which is taken as a

criterion to finally find the demarcation point of the primary and secondary creep.

However, the above procedure does not find the cut-off point. Linear regression is performed on the

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part with stable slope, and it is found that even at the stable slope on the graph, the slope does not

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exceed 0.5. This means that at this temperature and stress level, the austenitic stainless steel used in

the experiment does not enter the second stage of creep.

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For higher stresses, such as 700℃, 40MPa, use the above method to find the cut-off point between
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primary creep and secondary creep. Normalize the data from the primary and secondary creep for

training, and use a linear neural network for training in the secondary. The returned w and b values

are the desired values. Linear neural networks can effectively avoid the impact of noisy data and
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prevent data dispersion from disrupting the results.

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Assemble the obtained models for the primary and secondary creep to obtain the overall creep

constitutive model. The method flow is shown in Figure 7.

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Fig. 7. Creep modeling flow proposed in this paper.
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4.2 Optimization of training results

Two screening techniques are able to successfully identify the cut-off point between primary and

secondary creep utilizing data with a stress level of 700℃,40MPa. It is observed that an increase in
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stress significantly hastened the material creep process.

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The creep data of the first and second stages are analyzed independently, which leads to the

development of a corresponding creep model as depicted in Figure 8. Following 10,000 iterations,

the secondary creep linear neural network model yields a w value of 1.04671230 and a b value of -
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0.01604907.

𝑙𝑛𝜀 = 1.04671230𝑙𝑛𝑡−0.01604907 (5)

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Fig. 8. Model creep curves obtained by segmentation. (a) Primary creep model. (b) Secondary creep model.
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After establishing cut-off point, the primary and secondary creep models demonstrate a strong fit to

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 the data. However, the creep model derived from direct training exhibits inadequate description of

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the primary stage and unsatisfactory regression performance. It is evident in Figure 8, which depicts

the regression comparison between the direct training model and the optimized model for primary

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creep.

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Fig. 9. Model regression. (a) Regression of direct training model. (b) Regression of primary
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Notably, the optimized model exhibits significantly superior regression results compared to the

model obtained through direct training. By integrating the primary and secondary creep models, the
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de-normalized outcome yields a comprehensive creep model that combines Norton-Bailey’s power

law with neural networks.

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Fig. 10. Creep curve of the model obtained from training. (a) Direct training of the resulting model. (b)
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Optimization model.

The optimized model exhibits a closer resemblance to the experimental data and surpasses the

outcomes achieved through direct training.

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Table 2 presents the predictions of both models. Notably, when forecasting creep behavior for a

duration of 10000 hours, the optimized model demonstrates a relative error that is merely half of
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that observed in the original model. Evidently, in contrast to directly trained models, the optimized

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 creep model excels in forecasting long-term creep features by utilizing short-term data, yielding

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reduced errors and more dependable outcomes.

Table 2 Prediction results of the two models

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Optimization model Optimization model Original model Original model
Time(h)
prediction(%) relative error prediction(%) relative error

3000 0.1435863437 1.084785961% 0.1444551816 1.696447835%

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4000 0.1757897885 0.836529549% 0.1748146886 1.386585917%

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5000 0.2056643705 2.957294339% 0.2015935714 4.878100377%

6000 0.2338052515 4.079896819%

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7000 0.2605816820 6.403314206% 0.2453353133 11.879560954%

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8000 0.2862429299 6.532920837% 0.2629084265 14.152350535%

9000 0.3109674122 8.477818480% 0.2780515852 18.165419734%

10000 0.3348884625 11.234383436% 0.2910785803 22.846641356%

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4.3 Analysis of training results

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（1）The austenitic stainless steel at 700℃, 28MPa, 3000 hours is still in the primary creep, it can

be seen that the first creep stage of this material takes a long time.
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（2）When training directly with neural networks, due to the insufficient number of samples, a

large number of secondary creep data will cause the machine learning ignore the primary with a

small number of samples, i.e., the model obtained from the direct training is not good at regressing
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the primary creep. The problem can be well solved by finding the cut-off point between the first and

second stages of creep and processing the experimental data in segments.

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（3）The forecasting results of the model obtained using neural networks in combination with the

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 Norton-Bailey’s equation for long-time creep features are better than that of the model obtained by

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direct training.

（4）The combination of neural networks with the Norton-Bailey’s equations yields a more

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accurate creep constitutive model.

5 Conclusion

（1）For materials exhibiting prolonged primary creep, an elevation in stress level at a constant

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temperature considerably expedites the transition from the primary to the secondary creep. Judging

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whether the creep enters the second stage with the naked eye is not accurate, and in comparison, the

machine-based assessment is more reliable.

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（2）A proposed methodology is presented for identifying the cut-off point between the primary
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and the secondary creep. The data is processed by the Gaussian noise reduction and the multiple

screenings are conducted based on the creep characteristics. By segmenting the data multiple times

according to the different criteria, when the error is less than the set threshold, the cut-off point is
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determined . The feasibility of this method has been verified through experiment data.
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（3）The accuracy of the neural network is contingent upon the density of the samples. In practical

terms, it is difficult to acquire a substantial quantity of long-term creep data with adequate density.

Insufficient samples prevent the models that rely entirely on neural networks from achieving the
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expected functionality. Because of the scarcity of primary creep samples coupled with an abundance

of secondary creep data, it does not result in machine training results that capture the primary
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characteristics inadequately. If the experimental program can be adjusted based on the above issues,

which has great positive significance for designing a program for data collection in creep
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experiments.

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 （4）In this study, a novel approach is introduced that integrates the creep theory and the neural

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network method for creep model optimization. The proposed method successfully captures the

behavior of the first and second stages of creep, while also demonstrates notable capability for

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predicting the long-term creep performance. The optimized model outperforms the direct training

model by a significant margin, which yields more precise and dependable results than the latter.

（5）The optimization model derived in this study has great significance both in engineering

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application and theoretical research for the data acquisition of the creep experiment of the high-

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temperature reactor , as well as for the analysis and evaluation of creep life in such structures.

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[3] Mohsin S ,A.R. O ,S. K , et al.Limitations on the Computational Analysis of Creep Failure
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Atmosphere[J]. ACS omega,2020,5(11).

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 [8] Jianxing M,Zhixing X,Dianyin H, et al. A Concise Binomial Model for Nonlinear Creep-Fatigue

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