Development of a P91 uniaxial creep model for a wide stress range with an artificial neural network

ABSTRACT A uniaxial creep model that describes creep over a wide stress range was developed for P91 steel using an artificial neural network (ANN). The training dataset was based on measurements from uniaxial creep tests and information derived from a combination of the logistic creep strain prediction and the Wilshire models. The ANN model reproduces the training dataset with high accuracy (R2 = 0.975; RMSE (Root Mean Square Error) = 0.19). The model can be easily implemented in finite element analysis (FEA) codes since it provides an analytical expression of the true creep rate as a function of temperature, true stress and true creep strain. In FEA simulations under the same conditions as the training dataset, the model provides times to rupture and minimum creep rates very close to those in the training dataset. The model can be adapted for heats with different properties from the average behaviour of the training dataset by means of a stress-scaling factor.


Introduction
Structures that operate at high temperatures are affected by creep, which can threaten the integrity of a component and the entire system.In this context, it is crucial to understand and accurately predict the creep effects on structures during both design and operational stages to ensure the component's longterm structural integrity.Most engineering creep models focus on only one or two stages of creep as described in Holdsworth's overview [1], and there are only a few models that cover the entire creep phenomenon, which includes a primary, a secondary and a tertiary creep stage.One such model is the modified Garofalo model, which was extended to include tertiary creep [2].Another model is the modified omega model that extends secondary and tertiary creep to include primary creep [3].More detailed models were developed that consider various deformation and damage mechanisms, but those models require additional microstructural data [4][5][6].Therefore, those models are not easily implemented in finite element (FE) codes, e.g.Abaqus, since the knowledge of the material microstructure is required.The aim of this work is to developan analytical model that can be employed in FE codes for numerical simulations of relevant creep phenomena to investigate the creep behaviour from the initial loading to component failure, including plasticity effects, for a wide stress range.
In recent years, we developed a true stress-strain constitutive model for the plastic hardening and models for true creep rate as a function of true creep strain, true stress and temperature for austenitic stainless steel 316 L(N) [7,8].It must be noted that true creep strain, true stress and temperature are variables that are available in finite element analysis (FEA) codes; therefore, the model can be easily implemented in those codes.
In this paper, the focus is on P91, which is another relevant material for nuclear installations, for the power industry in general and for petrochemical plants.In the study by Baraldi et al. [7], three different modelling approaches were developed: the first creep model (phenomenological model) was based on a parametrised equation describing the creep rates with a predefined form that accounts from primary to tertiary creep.The second model was based on artificial neural networks (ANNs) and the third was based on a symbolic regression method.Since the ANN model was proven to be the most accurate for 316 L(N) [7], we selected that approach for P91 in this work.
Beyond the conventional approaches to creep modelling, more recently, ANN models were developed for the prediction of creep behaviour.Abendroth [9] used ANN and creep small punch test data to derive the values of the parameters for the P91 creep strain rate expressed as a combination of two Norton laws.Ghatak and Robi applied ANN for the prediction of creep curves for HP40Nb micro-alloyed steel where the creep strain is a function of stress, temperature and time [10].He and Sandström developed constrained ANNs for creep rupture predictions of two austenitic steels (Sanicro 25 and Super304H) [11].Ma et al. developed a back-propagation neural network model to predict the creep curves of MarM247LC superalloy under different conditions [12].

Methodology and models
The research methodology was already applied to 316 L(N) [7], and it is based on several stages (Figure 1).The experimental tensile properties were used to develop a true stress/strain model for the elastic-plastic properties, while the available creep uniaxial test data for different temperatures and stresses were instrumental in developing the engineering strain/time curves by means of the logistic creep strain prediction (LCSP) model [13][14][15] and Wilshire model [16].In the study by Baraldi et al. [7], the required data for the 316 L(N) models were extracted from the RCC-MRx [17].However, that possibility was not available for P91 because the amount of available data and information in the RCC-MRx is much smaller for P91 than for 316 L(N).It was therefore necessary to rely on other sources for P91 [18,19].It must be highlighted that the experimental data were generated with P91 from different heats [18,19].The synthetic engineering data (developed with the LCSP and the Wilshire model and based on the experimental data) represent an average of the properties of the multiple heats in the dataset.
With the engineering strain/time curve and the true stress/strain, curves of creep true strain versus true stress and versus creep strain were generated for a wide range of engineering stress (40-200 MPa) and for three temperatures (T = 600°C, 625°C and 650°C).Those curves were used to train an ANN and develop a new model for the true creep rate as a function of temperature, true stress and true creep strain.Finally, Abaqus/Zmat [20,21] calculations were performed with the new model to simulate uniaxial creep tests.

Elastic-plastic constitutive model
The yield and the ultimate tensile strength at the relevant temperatures were found in the literature [18].The constitutive model for the true stress/ strain is based on a Ramsberg-Osgood exponential law [22] until the true value corresponding to the ultimate tensile strength.After that point, the assumption of a linearly growing true stress was made.The plastic hardening model is derived by fitting the true stress σ and the true plastic strain ε p with an analytical expression that includes a constant term, a linear hardening term and one nonlinear hardening term: R 0 is the elastic limit, H linear is the plastic hardening and Q 1 and b 1 are related to the amplitude and rate of nonlinear hardening.The resulting constitutive law are shown in Figure 2 for temperatures of 600°C, 625°C and 650°C, and the corresponding parameters for Equation 7 are given in Table 1.

Synthetic engineering data -LCSP model
The three creep stages (primary, secondary and tertiary) are described by the LCSP model [13][14][15].The LSCP creep model is based on the expressions: where t e is the time to the given engineering strain e, t u is the time to rupture and x 0 and p are fitting parameters.C is a Larson-Miller-type parameter, which identifies the minimum time that can be predicted by the model as 10 −C hours.Apart from defining the shortest time of the model applicability, the C parameter has a limited effect on the LCSP fit of the data.The p and x 0 parameters require optimisation either by fitting each condition of time to rupture, temperature and engineering stress separately or by nonlinear regression on 'curve family' strain data, as it has been done for this work: The time to rupture is derived by the Wilshire model (WE) [16]: where s/s UTS is the ratio of engineering stress to ultimate tensile strength, Q c is the creep activation energy determined at constant s/s UTS , R* is the universal gas constant, T is absolute temperature and k and u are fitting constants.
The p and x 0 parameters for Equation 3 and the parameters k and u for Equation 7 are derived by fitting the available time to strain and time to failure experimental data [18,19].
The modelling result of the LCSP and failure assessment are illustrated in Figure 3.The scatter factor Z defined by ECCC (European Creep Collaborative Committee) [23] for all levels of time to strain is Z = 5.23 (2.5 standard deviations) when the t u in Equation 2is predicted by the WE model and Z = 3.13 when the actual measured t u is used.The WE model for time to failure has a scatter factor Z = 4.45, indicating that the predicted time to strain using the LCSP is not exceedingly increasing the scatter in predictions over the whole range of strains.The analysis for three specific cases (AP2-140 MPa, AP3-180 MPa and MC06-155 MPa) is included in the 'Results and discussion' paragraph.
Table 1.Young modulus, ultimate tensile strength and parameters for the expressions of the true stress vs. true plastic strain at T = 600°C, 625°C and 650°C.2).The curve families are times to strain for 0.5%, 1%, 2% and 5% (for AP3, the 0.5% data were not available in the measurements and it is not included in the figure).UB = upper bound and LB = lower bound (defined by the scatter factor Z). The extracted times to strain for the example curves (MC06-155 MPa, AP3-180 MPa and AP2-140 MPa) are discussed in the "Results and discussion" paragraph.

Synthetic true data
The true values of the creep strain rate as a function of the temperature, the true stress and the true creep strain are derived from the synthetic engineering strains and times to the strain that are generated with the LCSP model and Wilshire model as described in the above paragraph.The detailed procedure to extract the true creep rate from the engineering strain and time to strain is described in [7,8].
It should be emphasised that the following main assumptions were made: • the engineering creep strain is the total engineering strain under constant load, which includes both the creep strain and the rise in elastic and plastic strain with rising true stress.
• the strain and the strain rate can be separated in an elastic component, a plastic component and a creep component: • the elastic-plastic constitutive model is not affected by the accumulated creep strain.

ANN true creep rate model
The true creep rate data as a function of temperature, true stress and true creep strain were used to develop an ANN model in JMP [24].In order to prevent overfitting, the synthetic dataset (6,048 data points) has been randomly split into a training set (60% of the data), a validation set (30% of the data) and a test set (10% of the data).
Modelling directly the creep rate over a very wide range, which spans several orders of magnitude, can cause negative values in the region with very small creep rates (closer to zero).The issue was addressed by modelling the logarithm of the creep rates instead of the creep rate.Among the several ANN models that were developed with different number of nodes, we selected the model with the minimum number of nodes, which was able to achieve an R 2 equal to 0.999.The selected ANN has a single hidden layer with n = 8 nodes, as shown in Figure 4.The structure of the selected ANN model with one single hidden layer structure and eight nodes/neurons is consistent with the work by Zhong et al. who found that such relatively simple structure was sufficient for long-term creep behaviour predictions in the engineering area although for a different type of material [25].It was shown that a relatively simple ANN structure was also suitable for the creep prediction of 316 L(N) [7].
The general expression of the true creep strain rate as a function of the temperature T, true stress σ t and true creep strain ε f is: where k i are model parameters and H i are the activation functions based on the hyper-tangent: The k i and h i parameters for Equation 6 and Equation 7 are given in Table 2.

Results and discussion
The statistical indicators of the ANN model performance in Table 3 show the high level of accuracy of the model of the logarithm of the true creep rate compared to the LCSP true data.The statistical performance indicators have similar values for the three datasets (training, validation and test) and that is an indication that the ANN model does not overfit.The statistical performance indicators for the complete dataset of the creep model (R 2 = 0.975, RMSE = 0.19) are a bit worse than for the logarithmic model (from Table 3: R 2 = 0.999, RMSE = 0.0587), but the accuracy is still high as shown by the comparison between predicted and actual creep rates in Figure 5.The synthetic true creep strain rates in function of the true stress and the true creep strain for T = 600°C are described as circles in Figure 6.The comparison between the predicted values and the synthetic values for the true creep strain rate is illustrated in Figure 6 for T = 600°C.A similar agreement was achieved for the other two temperatures (625°C and 650°C), and the figures are not shown because of the limited length of the paper.It must be emphasised that the initial loading phase and the elastic contribution are not included in the LCSP engineering data.Because of that, at the end of the loading stage (1 second = 2.7e-4 hour), in the FEA simulations, the strains/times are completely different from the LSCP engineering data (Figure 7).After the end of the loading stage, there is a transition phase for the simulation to converge to the LCSP data.The lower the load is, the higher the time to rupture is, the larger the time difference between the end of the loading phase in the simulation and in the LCSP data is, and the longer the transition phase is (ranging from about 1 h for the 220 MPa case to 10 6 hfor the 40 MPa case).However, the end of the transition phase occurs at very low strains, ranging from about 10 −3 for the 40 MPa case to 10 −2 for the 220 MPa case.
Apart from the initial loading stage, the agreement between the simulations and the LCSP is satisfactory, and the model predictions for times to rupture can be considered as satisfactory.A further confirmation of the   agreement between the simulation results and the LCSP data is shown in Figure 8 both for the time to rupture and for the minimum strain rate for different stresses.
In Figure 9, a comparison between experimental data of uniaxial creep tests and simulation results for the engineering strain as a function of time is shown for three experiments (engineering stress = 140 MPa (AP02), 155 MPa (MC06) and 180 MPa (AP03); T = 600°C.Those three tests were not included in the initial datasets, which were used to develop the LCSP synthetic engineering data.Moreover, the three tests were performed with different material heats in EU co-funded projects: MATTER (MATerials TEsting and Rules) and MATISSE (Materials Innovation for a Safe and Sustainable nuclear in Europe) [26,27].It must be noted that those three tests were the only tests (from different batches from those in the initial dataset) that were available to the authors with the full strain/time curves [28].
It is well known that significant differences can exist in the properties of different batches/heats of the same materials.If the training datasets does not include the properties of a batch/heat with different behaviours from that which is consider in the model development, the model cannot be expected to capture with high accuracy the relevant material properties.In Figure 3, the discrepancy between the three tests for the time to strain predictions is illustrated by the distance of the three tests from the unity line, which represents the perfect agreement between the experiments and the predicted values with the WE model.For the MC06 and AP3, the predicted times to strain (from 0.5% to 5% strain) are longer compared to the experiment (overprediction), while for the AP2 they are shorter (underprediction).Consistent with that is the difference between the experimental data (circles) and the LCSP-WE engineering data (dotted lines) in Figure 9. Since the development of the ANN model was based on the LCSP-WE data, the results of the FEA simulations with the ANN model follow closely the LCSP curves in Figure 9.
In the Monkman-Grant graph in Figure 10 [29], the experimental minimum creep rates versus times to rupture for NIMS (National Institute for Materials Science, Tsukuba, Ibaraki, Japan) [18]    experiments in the initial dataset [18] and for the three additional tests are compared.The three additional tests are not aligned with the dotted line, which represents the Monkman-Grant relationship for the experimental NIMS data.The largest discrepancy occurs for the 140 MPa case (AP2) with a much smaller minimum creep rate compared to the NIMS experiments at the same stress although with a similar time to rupture.The 140 MPa case exhibits a very anomalous behaviour in the Monkman-Grant graph and that could be an indication of potential issues during the experiment.Therefore, the 140 MPa cases was not considered in the following application of the stressscaling factor.
One way to adapt the average ANN model to a specific material batch/heat with creep properties outside the range of the dataset that was used for the model development is to consider a scaling factor that accounts for the difference between the specific batch and the average material behaviour.The dashed lines in Figure 9 represent the simulation results with the ANN model when a stress-scaling factor (SF = 1/0.96= 1.042) is applied to the model for the 180 MPa and 155 MPa cases.The stress-scaling factor improves significantly the agreement of the model results with the experimental curve.
As shown in Figure 7 in the FEA numerical simulations, the strain can reach values as high as 100% and that is not consistent with the experiments in Figure 9.In the tests, the rupture is most likely caused by crack formation, and the lack of a damage mechanism in the ANN model will results in much larger strains at times to rupture in the FEA simulations compared to experiments.It must be noted that the time to rupture in the simulations is mainly determined by the tertiary creep regime in the ANN creep model.Although the damage model and the subsequent crack formation are not included in the simulations, the effect of crack formation on the time to rupture is partly taken into account in the simulations by including the tertiary creep regime in the ANN creep model (Figure 6).However, in FEA simulations, strains will grow to unphysical larger values compared to the tests until the calculation is not able to cope with that situation from the numerical point of view, and it does not converge anymore.
It must be emphasised that ANNs can be more accurate than other approaches within the condition range of the training data, but they have a major drawback when they are applied outside that range.ANN models are highly nonlinear models whose accuracy can decrease significantly when applied in regions outside the range of the training data, and therefore, in those regions, their output cannot be considered reliable.Extrapolation is always an issue with any model, but with some of the other models, e.g.linear models or phenomenological models, the model performance outside the validation range is more predictable than with ANN models.For example, the creep rate in the tertiary region increases with the true stress and the true strain as shown in Figure 6.For larger true stress, outside the range of the training dataset, the creep rate will keep increasing (as expected from the qualitative point of view) in a linear model or in a phenomenological model, while in an ANN model, it might significantly decrease or strongly fluctuate.Another disadvantage of ANN models is that it is often not possible to associate its terms with physical mechanisms or phenomenological observations due to the model complexity.

Figure 1 .
Figure 1.Layout of the methodology.

Figure 3 .
Figure 3. Predicted vs. measured time to strain (and failure) for the LCSP model based on WE-predicted failure times (t u in Equation.2).The curve families are times to strain for 0.5%, 1%, 2% and 5% (for AP3, the 0.5% data were not available in the measurements and it is not included in the figure).UB = upper bound and LB = lower bound (defined by the scatter factor Z). The extracted times to strain for the example curves (MC06-155 MPa, AP3-180 MPa and AP2-140 MPa) are discussed in the "Results and discussion" paragraph.

Figure 4 .
Figure 4. Diagram of the artificial neural network with one hidden layer and n = 8 nodes.

Figure 5 .
Figure 5. Actual creep rate (LCSP model) vs. predicted creep rate (ANN model) for the complete dataset and for all temperatures (600°C, 625°C and 650°C).

Figure 7 .
Figure 7.Comparison of FEA results (lines) and LCSP synthetic engineering data (circles) for the strain versus time at T = 600°C.

Figure 8 .
Figure 8.Comparison between of FEA results (red squares) and LCSP synthetic data (blue circles) for the times to rupture and the minimum strain rate at T = 600°C.

Figure 9 .
Figure9.Comparison of experimental data and FEA simulation results for strain vs. time at T = 600°C for an engineering stress of 180, 155 and 140 MPa.Note that the 140 MPa test data show anomalous behaviour, with times to strain differences of a factor of 2 in comparison to the general creep strain behaviour (see Figure3).

Figure 10 .
Figure 10.Experimental values for minimum creep rate versus time to rupture of NIMS experiments in the initial dataset (cyan colour) and for the three additional tests (orange colour).

Table 2 .
Parameters (k i and h i ) for the ANN model as defined in Equation 6 and Equation7for the prediction of log 10 (_ ε f ).

Table 3 .
Statistical indicators for the training, validation and test datasets for the logarithm of the creep rate.