# Correlation Analysis of Established Creep Failure Models through Computational Modelling for SS-304 Material

^{1}

^{2}

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## Abstract

**:**

^{2}value of the sine-hyperbolic model proved the model’s selection for predicting the creep response of stainless steel 304. The method can be applied to select a suitable creep damage model as per the feasibility of the operating conditions.

## 1. Introduction

#### Research Objectives

## 2. Creep Damage Constitutive Models

#### 2.1. Norton Bailey Model

#### 2.2. Omega Model

_{r}is the rupture life, t indicates the current time, Ω is the material creep damage constant, and $\dot{\epsilon}$ is the current creep strain rate. At the point in time when the rupture occurs, t → t

_{r}and t/t

_{r}→ 1, and Equation (5) collapses. Thus, the life fraction rises from zero to (nearly) one (0 ≤ t/t

_{r}< 1). According to Prager [7], the creep strain will have the following relationship, as shown in Equation (6):

_{0}and Ω [6]. Equation (7) can calculate the constant Ω using experimental data. When Ω is high, the material has a low creep strain rate for the bulk of its life before rapidly increasing its creep strain rate before failing. In the tertiary creep regime, a low value of Ω means that an ample amount of time is disbursed in this regime [27].

#### 2.3. Kachanov–Rabotnov Model

#### 2.4. Theta Projection Model

#### 2.5. Sine-Hyperbolic Model

## 3. Methodology

#### 3.1. Creep Strain Analytical Analysis

#### 3.2. Regression Analysis of Creep Plot through Extrapolative Prediction

^{2}value of 0.9803, which was significantly higher than the criterion set by the ASME FFS-1/API-579 standards for this plot. To assess the effects of material degradation at the tertiary stage of creep for the SS-304 material, the tertiary creep damage constant (ω) was derived in the regression equations [22].

#### 3.3. Finite Element Geometry Modelling and Pre-Processing

#### 3.4. Sensitivity Analysis of Established Models Using RSM and ANOVA

^{2}value of more than 80%. Once the appropriate model was selected, the evaluation involved 3D surface plots showing the connection between the design components and the outcomes. These graphs were used to understand how all of the responses behaved and correlated to each other. The optimization criteria for each design parameter were then specified with a suitable significance. The ideal values of the design parameters were established in the following surface plots. Analysis of variance (ANOVA) was also used to assess the differences between two or more means and variables through significance tests. For each model, design matrices with 30 simulation runs were created after assigning low and high values to specified factors. To increase the dependability of the design and analysis, these matrices also included replicas of the core points [43].

## 4. Results and Discussions

#### 4.1. FE Analysis of Dog-Bone Specimen of SS-304

#### 4.2. Model Comparison—Minimum Creep Strain Rate

_{s}regulates the bend in the SH model. Compared to the KR, NB, Omega, and TP models, the SH model better fit the simulation data over a wide range of stresses. When the applied stress range was constrained, the KR model performed well compared to the other models.

#### 4.3. Model Comparison—Creep Deformation and Damage

#### 4.4. Model Comparison—Stress-Rupture

_{t}in the sin-h model. As the rupture duration approached zero, the sin-h rupture predicted bends at the yield strength and achieved a result that was less than but close to the nominal ultimate tensile strength of SS-304. Furthermore, as the rupture period approached zero, the KR rupture prediction did not bend at the yield strength. Instead, it achieved a value that was 1.35× more significant than the ultimate tensile strength. Penny [53] found that the weakness in the KR rupture prediction was caused by the “brittle curve” phenomena and revised the KR model to handle the high-stress to low-stress bend by adding additional components and material constants but kept the flaw that the critical damage is less than unity. As a result, to introduce the bend, two sets of constitutive equations were used. Near rupture, the critical damage was modest (between 0.2 and 0.4), and the damage trajectory was nearly endless. Without these additional issues and limits, the sin-h model replicated the transition from high- to low-stress regions. Figure 10 depicts the stress versus the minimum creep strain rate of the models.

#### 4.5. Creep Experimental Testing

#### 4.6. Validation of Models by Creep Experiment

#### 4.7. Data Optimization by Statistical Modelling

^{2}), modified coefficient of determination (adjusted R

^{2}), and expected coefficient of determination (predicted R

^{2}) for each example were used to verify the applicability of the selected regression models. Table 5 displays the fit statistics for the response strain produced from a central composite design. The relevance of the models can be seen in the R

^{2}values. Furthermore, the modified R

^{2}and anticipated R

^{2}values were close to each other. The term “adequate precision” refers to the comparison between the predicted values, referred to as “signal,” and the average prediction error is referred to as “noise”. The models’ performance is shown by the suitable relationship between the signal and noise. It was found that all the quadratic models were significant for searching the design space [54].

## 5. Conclusions

- The creep strain rate curve modeled by the SH model was better as compared to the KR, NB, Omega, and TP models primarily because of the material constants in its formulation. The model accurately modeled all three creep stages for the SS-304 material while running the simulation and extrapolating to 18,000 h.
- The KR, NB, Omega and TP models could not represent the minimum creep strain rate vs. stress bend accurately. However, the SH model represented the lowest creep strain rate bend precisely.
- The stress rupture predictions of the SH model exhibited a smooth curve for the creep strain and damage evolution as compared to the KR, NB, Omega, and TP models in conditions up to 720 °C and 60 MPa.
- The damage evolution differed between the KR, TP and SH models, whereas the NB and Omega models were incapable of predicting the damage evolution. The NB and Omega models depicted zero damage evolution, whereas the KR and TP models exhibited a conservative damage evolution. The best damage evolution criteria were modelled by the SH model for ω = 0–1.10.
- The combined effects of the design factors on the response SH model’s creep strain rate (ε
_{t}) and contour creep deformation maps from the RSM results were better as compared to the other models. The relative error of the SH model’s ANOVA results was 0.84, which was comparable to the other models, which proves the significance of the model.

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Norton’s power-law constant |

n | Stress exponent |

T | Temperature |

R | Universal gas constant |

Q | Activation energy |

t_{r} | Rupture time |

σ_{1}, σ_{2}, and σ_{3} | Principal stresses |

S_{1} | Stress parameter |

Q_{c} | Norton’s activation energy |

α | Triaxiality parameter |

ω | Omega damage parameter |

δ_{Ω} | Omega parameter |

ε_{0} | Initial creep strain |

Ω | Omega material damage constant |

ε_{t} | Creep strain rate |

Ωm | Omega multi-axial damage parameter |

Ωn | Omega uniaxial damage parameter |

σ_{e} | Effective stress |

Δ^{cd} | The adjustment factor for creep ductility |

εΩ | Accumulated creep strain |

Ωt | Omega material damage constant over time |

A_{Ω} | Stress coefficient |

A_{0} | Stress coefficient |

${\mathsf{\Delta}}_{n}^{cd}$ | Creep rupture life |

n_{BN} | Norton–Bailey coefficient |

Q_{Ω} | Temperature dependence of Ω |

β_{Ω} | Omega parameter to 0.33 |

FEA | Finite element analysis |

FFS | Fitness for service |

T_{refa} | Reference temperature |

API | American Petroleum Institute |

UTS | Ultimate tensile strength |

MPC | Material Properties Council |

ASME | American Society for Mechanical Engineers |

BPVC | Boiler and pressure vessel codes |

UTS | Ultimate tensile strength |

ASTM | American Standards for Testing of Materials |

CDM | Continuum damage mechanics |

KR | Kachanov–Rabotnov model |

NB | Norton–Bailey Model |

ANOVA | Analysis of variance |

TP | Theta Projection model |

SH | Sine-hyperbolic model |

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**Figure 1.**The overall methodology flowchart for acquiring creep data for the FE dog-bone specimen using the Omega–Norton–Bailey regression model [22].

**Figure 2.**The resulting strain rate versus stress curve was generated by a resulting curve-fitting of MPC Omega to the NB, TP, and KR models based on strain hardening [26].

**Figure 3.**(

**a**) Standard dimensions (in mm) of dog-bone specimen, (

**b**) SS-304 specimen with pre-defined boundary conditions, (

**c**) specimen with pre-defined temperature of 720 °C, which was constant throughout the region.

**Figure 5.**Von Mises stress and relaxed stress distribution with Omega–Norton–Bailey Regression model’s visco-elastic–plastic run-time of 18,000 h.

**Figure 6.**(

**a**) Induced Von Mises stress in the specimen after running simulation; (

**b**) creep strain (CEEQ) for the applied stresses, and (

**c**) plastic strain (PEEQ) at 720 °C and 60 MPa.

**Figure 7.**(

**a**) Comparison of natural logs of creep strain rates of established models at 720 °C and 60 MPa. (

**b**) Comparison of creep strain rates of established models at 720 °C and 60 MPa.

**Figure 14.**(

**a**) Stress and stress exponent interaction effect on NB model’s creep strain rate; (

**b**) stress and stress exponent interaction effect on TP model’s creep strain rate; (

**c**) stress and stress exponent interaction effect on KR model’s creep strain rate, and (

**d**) stress and stress exponent interaction effect on SH model’s creep strain rate.

**Figure 15.**(

**a**) The combined impact of design factors on the response NB model’s creep strain rate (ε

_{t}); (

**b**) contour creep deformation map for NB model; (

**c**) the combined impact of design factors on the response TP model’s creep strain rate (ε

_{t}); (

**d**) contour creep deformation map for TP model; (e) the combined effect of design factors on the response KR model’s creep strain rate (ε

_{t}); (

**f**) contour creep deformation map for KR model; (

**g**) the combined effect of design factors on the response SH creep strain rate (ε

_{t}); (

**h**) contour creep deformation map for SH model.

**Figure 16.**(

**a**) Actual vs. predicted values for NB model’s creep strain rate (ε

_{t}); (

**b**) actual vs. predicted values for TP model’s creep strain rate (ε

_{t}); (

**c**) actual vs. predicted values for KR model’s creep strain rate (ε

_{t}); (

**d**) actual vs. predicted values for SH model’s creep strain rate (ε

_{t}).

**Table 1.**Material coefficients for the SS-304 material type from MPC Omega (MPa, °C) [35].

$\mathbf{Parameter}\mathbf{Strain}\mathbf{Rate}\u2014\left({\mathit{\epsilon}}_{\mathit{c}0}\right)$ | Parameter Omega—(Ω) | |||
---|---|---|---|---|

Type-SS 304 | A_{0} | −19.17 | B_{0} | −3.40 |

A_{1} | 37,917.40 | B_{1} | 10,521.29 | |

A_{2} | −12,389.36 | B_{2} | −7444.83 | |

A_{3} | 4112.12 | B_{3} | 3266.58 | |

A_{4} | −936.22 | B_{4} | −552.00 |

**Table 2.**Material and physical properties of SS-304 material [34].

Material Model | Elastic–Perfectly Plastic |
---|---|

Young’s modulus | (201,000–17,100) MPa @ −25 °C to 720 °C |

Poisson’s ratio | 0.31 |

Density | 8000 kg/m^{3} |

Thermal expansion coefficient | 17.3 × 10^{−6} °C^{−1} |

Thermal conductivity | 16.2 W m^{−1} °C^{−1} |

Yield stress | (207–126) MPa |

Plastic strain | (0–0.015) |

Norton–Bailey Model | Kachanov–Rabotnov Model | Sin-h Model | Theta Projection Model | Temperature (°C) | |
---|---|---|---|---|---|

Creep parameters | 1.93 × ${10}^{-18}$ | 2.10 × ${10}^{-18}$ | 4.71 × ${10}^{-15}$ | 2.47 × ${10}^{-15}$ | 680 |

4.71 × ${10}^{-18}$ | 5.15 × ${10}^{-18}$ | 1.06 × ${10}^{-14}$ | 8.24 × ${10}^{-15}$ | 690 | |

1.13 × ${10}^{-17}$ | 1.23 × ${10}^{-17}$ | 2.35 × ${10}^{-14}$ | 2.68 × ${10}^{-14}$ | 700 | |

2.67 × ${10}^{-17}$ | 2.90 × ${10}^{-17}$ | 5.13 × ${10}^{-14}$ | 8.51 × ${10}^{-14}$ | 710 | |

6.18 × ${10}^{-17}$ | 6.73 × ${10}^{-17}$ | 1.10 × ${10}^{-13}$ | 2.64 × ${10}^{-13}$ | 720 | |

Stress exponent | 7.10 | 7.08 | 7.11 | 7.91 | 680 |

7.03 | 7.01 | 7.03 | 7.65 | 690 | |

6.69 | 6.94 | 6.96 | 7.40 | 700 | |

6.88 | 6.87 | 6.89 | 7.16 | 710 | |

6.82 | 6.80 | 6.82 | 6.92 | 720 |

Independent Design Factors | Response | ||||||
---|---|---|---|---|---|---|---|

Models | Values | Stress (A) MPa | Stress Exponent (B) ‘n’ | Creep Parameter (C) MPa ^{−n}h^{−1} | Damage Parameter (D) ‘ω’ | Strain Rate 10 ^{−5}/h | |

Norton–Bailey | Low | 3 | 6.82 | 1.93 × ${10}^{-18}$ | 0 | 1.11 × ${10}^{-8}$ | |

High | 81 | 7.16 | 6.18 × ${10}^{-17}$ | 0 | 15.89 | ||

Theta Projection | Low | 3 | 6.72 | 8.24 × ${10}^{-15}$ | 0.05 | 8.71 × ${10}^{-9}$ | |

High | 81 | 8.11 | 2.64 × ${10}^{-13}$ | 0.40 | 17.28 | ||

Kachanov–Rabotnov | Low | 3 | 6.66 | 5.15 × ${10}^{-18}$ | 0.05 | 3.83 × ${10}^{-7}$ | |

High | 81 | 7.08 | 1.23 × ${10}^{-17}$ | 0.40 | 521.65 | ||

Sine-Hyperbolic | Low | 3 | 6.68 | 1.10 × ${10}^{-13}$ | 0.05 | 1.99 × ${10}^{-5}$ | |

High | 81 | 7.25 | 4.71 × ${10}^{-15}$ | 0.40 | 11.74 |

Type of Creep Test | Creep Models | Maximum Deviation up to 5% | |
---|---|---|---|

FEA | Experiment | ||

1000 h | NB Model | 0.1596 | 0.0994 |

KR Model | 0.2282 | 0.0994 | |

TP Model | 0.2878 | 0.0994 | |

SH Model | 0.3332 | 0.0994 |

Fit Statistics for NB Model’s Creep Strain Rate (ε_{t}) | |
---|---|

Statistical Parameters | Values |

R^{2} | 0.78 |

Adjusted R^{2} | 0.62 |

Predicted R^{2} | −0.29 |

Adequate precision | 4.71 |

Fit Statistics for TP Model’s Creep Strain Rate (ε_{t}) | |

R^{2} | 0.84 |

Adjusted R^{2} | 0.74 |

Predicted R^{2} | −0.07 |

Adequate precision | 7.72 |

Fit Statistics for KR Model’s Creep Strain Rate (ε_{t}) | |

R^{2} | 0.82 |

Adjusted R^{2} | 0.74 |

Predicted R^{2} | 0.26 |

Adequate precision | 12.60 |

Fit Statistics for SH Model’s Creep Strain Rate (ε_{t}) | |

R^{2} | 0.84 |

Adjusted R^{2} | 0.73 |

Predicted R^{2} | −0.10 |

Adequate precision | 6.88 |

Response: NB Model’s Creep Strain Rate—Model Summary | ||||||
---|---|---|---|---|---|---|

Source | Std. Dev. | R^{2} | Adjusted R^{2} | Predicted R^{2} | Press | |

Linear | 3.62 | 0.09 | −0.08 | −0.54 | 222.62 | |

2FI | 3.81 | 0.09 | −0.21 | −1.73 | 393.94 | |

Quadratic | 2.12 | 0.78 | 0.62 | −0.29 | 186.96 | Suggested |

Cubic | 1.91 | 0.87 | 0.69 | −4.20 | 750.21 | Aliased |

Response: TP Model’s Creep Strain Rate—Model Summary | ||||||

Linear | 38.04 | 0.02 | −0.16 | −0.81 | 27,007.75 | |

2FI | 40.09 | 0.02 | −0.29 | −1.99 | 44,564.25 | |

Quadratic | 17.93 | 0.84 | 0.74 | −0.07 | 15,998.53 | Suggested |

Cubic | 21.15 | 0.84 | 0.63 | −8.60 | 1.43 × 10^{5} | Aliased |

Response: KR Model’s Creep Strain Rate—Model Summary | ||||||

Linear | 138.51 | 0.69 | 0.65 | 0.58 | 6.73 × 10^{5} | |

2FI | 147.27 | 0.69 | 0.61 | 0.48 | 8.34 × 10^{5} | |

Quadratic | 119.07 | 0.82 | 0.74 | 0.26 | 1.18 × 10^{6} | Suggested |

Cubic | 108.94 | 0.88 | 0.78 | −6.52 | 1.22 × 10^{7} | Aliased |

Response: SH Model’s Creep Strain Rate—Model Summary | ||||||

Linear | 757.0 | 0 | −0.20 | −0.86 | 1.07 × 10^{7} | |

2FI | 797.95 | 0 | −0.33 | −2.08 | 1.76 × 10^{7} | |

Quadratic | 356.15 | 0.84 | 0.73 | −0.10 | 6.31 × 10^{6} | Suggested |

Cubic | 421.41 | 0.84 | 0.62 | −8.91 | 5.68 × 10^{7} | Aliased |

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## Share and Cite

**MDPI and ACS Style**

Sattar, M.; Othman, A.R.; Muzamil, M.; Kamaruddin, S.; Akhtar, M.; Khan, R.
Correlation Analysis of Established Creep Failure Models through Computational Modelling for SS-304 Material. *Metals* **2023**, *13*, 197.
https://doi.org/10.3390/met13020197

**AMA Style**

Sattar M, Othman AR, Muzamil M, Kamaruddin S, Akhtar M, Khan R.
Correlation Analysis of Established Creep Failure Models through Computational Modelling for SS-304 Material. *Metals*. 2023; 13(2):197.
https://doi.org/10.3390/met13020197

**Chicago/Turabian Style**

Sattar, Mohsin, Abdul Rahim Othman, Muhammad Muzamil, Shahrul Kamaruddin, Maaz Akhtar, and Rashid Khan.
2023. "Correlation Analysis of Established Creep Failure Models through Computational Modelling for SS-304 Material" *Metals* 13, no. 2: 197.
https://doi.org/10.3390/met13020197