Abstract

1. Introduction

Titanium and its alloys are widely explored in the manufacturing industry in a variety of products for aerospace and automotive applications¹1 Ishfaq K, Abdullah M, Mahmood MA. A state-of-the-art direct metal laser sintering of Ti6Al4V and AlSi10Mg alloys: surface roughness, tensile strength, fatigue strength and microstructure. Opt Laser Technol. 2021;143:107366.. In this context, properties such as creep, fatigue, and degradation resistance during service at high temperatures are imperative²2 Gardner A. Tight times for titanium. Metal Bulletin Monthly. 1994;286:34-5. . Thus, the development of materials able to maintain their structural properties under severe environments and conditions for long periods is an industrial necessity since creep failures are considered the major challenge of this segment³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁴4 Betten J. Creep mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg; 2002.. Laboratory creep characterization tests often demand investments and a long time to obtain reliable and reproducible results. In this sense, mathematical and simulation models to predict the creep life of materials become allies in the development of aerospace technologies and projects. Furthermore, this assessment method provides great product reliability, decreases project costs, minimizes study time, and dispenses with the use of specimens and complementary characterization tests. As observed, this study field is highly promising, being able to disseminate national technology and create advances in the aerospace industry. These studies involve the elaboration of constitutive equations with the aim to simulate or predict the creep behavior under different stress-temperature regimes.

Therefore, in this study, a critical analysis is carried out based on the constitutive equations called θ-Projection⁵5 Evans RW, Wilshire B. Introduction to creep. London: Institute of Materials; 1993. 115 p. and υ Concept³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9..

The θ-Projection method is based on an equation that considers the combination of decelerating primary and accelerating tertiary creep stages, enabling its prediction by the interpolation and extrapolation of experimental data⁷7 Wilshire B, Owen DRJ. Recent advances in creep and fracture of engineering materials and structures. Swansea: Pineridge Press; 1982.

8 Guguloth K, Roy N. Creep life estimation of reformer alloy using θ-projection method: a neuro fuzzy approach. Int J Press Vessels Piping. 2023;203:104938.-⁹9 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC. Creep strain modeling for alloy 690 SG tube material based on modified theta projection method. Nucl Eng Technol. 2022;54(5):1570-8.. The θ-Projection method and its modifications have been applied to predict creep curves for 16MND5 steel (used in reactor pressure vessels)¹⁰10 Yu P, Ma W. A modified theta projection model for creep behavior of RPV steel 16MND5. J Mater Sci Technol. 2020;47:231-42., K465 and DZ125 superalloys¹¹11 Fu C, Chen Y, Yuan X, Tin S, Antonov S, Yagi K, et al. A modified θ projection model for constant load creep curves-I. Introduction of the model. J Mater Sci Technol. 2019;35(1):223-30., γ-Titanium Aluminide Ti-45Al-2Mn-2Nb¹²12 Harrison W, Abdallah Z, Whittaker M. A model for creep and creep damage in the γ-Titanium aluminide Ti-45Al-2Mn-2Nb. Materials (Basel). 2014;7(3):2194-209., alloy 690 (a material used in steam generator tubes)¹³13 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC, et al. Modified θ projection model-based constant-stress creep curve for alloy 690 steam generator tube material. Nucl Eng Technol. 2022;54(3):917-25., among others, in which the assessment of the service safety of structural materials were predicted with high agreement. The ν Concept is a mathematical model to express strain-time relations based on the Continuum Damage Mechanics proposed by Barboza³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. and Barboza et al.⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9. and inspired by Ion et al.¹⁴14 Ion JC, Barbosa A, Ashby MF, Dyson BF, McLean M. The modelling of creep for engineering design. I. Report DMA A115. Teddington: National Technical Reports Library; 1986. and Kachanov–Rabotnov formalism¹⁵15 Rabotnov YN. Creep problems in structural members. Amsterdam: North-Holland; 1969.,¹⁶16 Kachanov LM. Introduction to continuum damage mechanics. The Netherlands: Martinus Nijhoff; 1986.. Creep stages are characterized by the νi operational parameters (i = 1-4) extracted from experimental curves. A preliminary study conducted on Ti–6Al–4V showed great agreement with experimental curves at different stress values at 500-600 °C for short-term tests³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9.. The main goal of this study involved predicting a creep curve for the Ti-6Al-4V alloy at 465 MPa and 500 °C, using a database generated by determining θ and ν values from six experimental creep curves at different applied stress values. The interpolation capacity of the constitutive equations and the agreement between the predicted strain-time curves and experimental data has been evaluated.

2. Materials and methods

2.1. The θ-Projection

The θ-Projection method is based on Equation 1, in which the four parameters (θ_i, i = 1-4) are obtained from an experimental creep curve. The expression may be divided into two parts: i) the primary creep rate ${\theta }_1\left(1-e^{-{\theta }_{2}t}\right)$, in which θ₁ represents the primary strain magnitude and θ₂, the decay rate; and ii) the accelerating creep rate ${\theta }_3\left(e^{{\theta }_{4}t}-1\right)$, in which θ₃ refers to the scaling of the tertiary creep strain and θ₄, to its rate¹²12 Harrison W, Abdallah Z, Whittaker M. A model for creep and creep damage in the γ-Titanium aluminide Ti-45Al-2Mn-2Nb. Materials (Basel). 2014;7(3):2194-209.. Thus, θ₁ and θ₃ are denominated scale parameters, whereas θ₂ and θ₄, rate parameters⁵5 Evans RW, Wilshire B. Introduction to creep. London: Institute of Materials; 1993. 115 p..

2.2. ν Concept

The υ Concept considers the Continuous Damage Mechanics framework (CDM) (Equation 2). The υ parameters of this constitutive equation can be independently extracted from the creep curves, as reported in³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9.. To predict the creep curve, the parameters ν_i (i = 1-4) and the fracture time (t_f) are considered. However, in this study, the values of υ and θ will be determined by a computational technique using six experimental curves³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9..

2.3. Time to rupture

Time to rupture must be calculated to predict creep curves. This parameter was used in both the θ-Projection and the ν Concept. The dependence of fracture time on applied stress is frequently described by an exponential fit curve (Equation 3) in which A and m are constants and σ, the applied stress.

2.4. Experimental data

To determine the four θ and ν values and further apply the θ-Projection and ν Concept methods, the experimental data of six creep curves were considered (Figure 1). These curves were previously obtained by Barboza³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. and Barboza et al.⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9. using Ti-6Al-4V specimens with the following chemical composition: 6.61 Al – 4.23 V – 1.18 Fe – 0.13 O – 0.026 C – 0.011 N – 0.003 H – Ti. The microstructure is composed of an ɑ-phase with an average grain size of 10 μm and a β-phase in the grain boundaries of α. The experiments were performed on a dead-weight-creep-rupture machine at six applied stress values (312 to 520 MPa) at 500 °C. Experimental data from a test performed at 465 MPa under the same conditions (not included in the estimation of θ and ν values) were used to compare the results obtained by the θ-Projection and ν Concept methods.

Figure 1
Ti-6Al-4V experimental creep curves (312 to 520 MPa) at 500 °C used to determine the θ and ν parameters³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9..

Creep tests results (Figure 1) are summarized in Table 1, which shows rupture time (t_f), strain at fracture (ε_f) and the applied stress (σ)³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9..

2.5. The Software

A software was developed using the Python programming language to obtain creep curve data considering the θ-Projection and the ν Concept. This computational technique allowed obtaining the θ and ν parameters for a set of Ti-6Al-4V alloy curves, shown in Figure 1. The linear equations were obtained by plotting the experimental θ_1-4 and υ_1-4 values as a function of the applied stress and were further treated by the software Origin ® 2018 fit curve tool. The fracture time to predict the curve at 465 MPa was determined by entering the database (Equation 3) into the software. The equations 4-11 allowed interpolation to determine θ and ν νalues at 465 MPa. These values were implemented in Equations 1 and 2 and the time steps were varied by 0.1 h, enabling the prediction of the strain-time curve at 465 MPa. A flowchart summarizing the steps of this study is shown in Figure 2.

Figure 2
Flowchart describing the steps performed to predict creep curves following the θ-projection and ν Concept methodologies.

3. Results and Discussion

3.1. Short-term creep curve prediction by applying the θ-Projection and ν Concept

Based on creep curve data (Figure 1), θ and ν values were determined. Data are shown in Table 2 and plotted in Figure 3.

Figure 3
Dashed lines represent the linear regression performed in each point set: a) θ-Projection and (b) ν Concept.

A linear regression was performed and represented by Equations 4-11, that enabled the extraction of parameters θ and ν at 465 MPa. These values are shown in Table 3.

Based on Table 1 data, time to rupture as a function of the applied stress (Figure 4) curve was plotted. Based on Equation 3 and standard regression techniques, A = 3.756 × 10²⁶ and an m = −9.89 has been determined. At 465 MPa, rupture time corresponds to 1.6 h.

Figure 4
Dependence of time to rupture (h) on applied stress (MPa)³3 Barboza MJR. Estudo e modelagem sob condições de fluência da liga Ti6Al4V [thesis]. São José dos Campos: Instituto Tecnológico de Aeronáutica; 2001. ,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9..

3.2. Use of the θ-Projection and ν Concept to predict creep curves at 465 MPa

The introduction the θ_i and ν_i values of Tables 2 and 3, the rupture time calculated by Equation 3 for 1.6h at 465 MPa, and the 0.1h time step in software, enabled creep curve prediction by the θ-Projection (Equation 1) and ν Concept (Equation 2) models. Figure 5 shows the software-derived data in comparison with an experimental curve at 465 MPa and a time to rupture of 1.92h. The percentage difference between times to rupture calculated by Equation 3 and the experimental value corresponds to 17%. The creep ductility of the experimental curve corresponds to 0.129. However, values obtained by the θ-Projection and ν Concept methods correspond to 0.103 and 0.137, respectively. Based on Figure 5, the tertiary stage is well defined by the ν Concept (Equation 2) with the typical curvature obtained for Ti-6Al-4V in tests under a constant load.

Figure 5
Creep curves calculated by Equations 1 and 2 and experimental data at 465 MPa.

In general, creep curves have been described by few parameters, such as lifetime, rupture strain, and creep stationary rates. In this case, other information available from a creep curve is ignored. A complete description requires constitutive equations that can describe the normal creep curve under certain conditions of stress and temperature. The ν Concept and θ-Projection methodologies offer a description based on four ν_i and θ_i parameters, respectively. The ν₁, ν₂, θ₁, and θ₂ parameters represent the first and second creep stages; whereas the ν₃, ν₄, θ₃, and θ₄ parameters, the tertiary creep stage. These constitutive equations are based on strain hardening, recovery mechanisms, and creep damage⁵5 Evans RW, Wilshire B. Introduction to creep. London: Institute of Materials; 1993. 115 p.,⁶6 Barboza MJR, Moura Neto C, Silva CRM. Creep mechanisms and physical modeling for Ti-6Al-4V. Mater Sci Eng A. 2004;369(1-2):201-9.

This study obtained ν_i and θ_i parameters by computational techniques for uniaxial Ti-6Al-4V alloy creep test specimens. Such parameters depend on microstructure, applied stress and temperature. The technique is based on creating a database for a given material that allows the extrapolation or interpolation of creep curves⁵5 Evans RW, Wilshire B. Introduction to creep. London: Institute of Materials; 1993. 115 p.,¹³13 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC, et al. Modified θ projection model-based constant-stress creep curve for alloy 690 steam generator tube material. Nucl Eng Technol. 2022;54(3):917-25.,¹⁷17 Liu H, Peng F, Zhang Y, Li Y, An K, Yang Y, et al. A new modified theta projection model for creep property at high temperature. J Mater Eng Perform. 2020;29(7):2020-4779.. In this case, the database presents six curves obtained at 500 ºC at 312 to 520 MPa for Ti-6Al-4V (Figure 1). Based on these six curves, θ_i and ν_i parameters were determined by the software developed in this work based on flowchart shown in Figure 2. The values are shown in Table 2 and Figure 3. These values enabled the prediction of the creep curve at 465 MPa. Regardless of the different strain and damage mechanisms, the linear configuration of ν_i and θ_i as a function of applied stress favors the interpolation process. Both methods have been shown to predict primary, secondary, and tertiary creep behavior for Ti-6Al-4V with acceptable accuracy (Figure 5). However, results indicated that the ν Concept produced a creep curve with better agreement with the experimental data.

As reported in¹³13 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC, et al. Modified θ projection model-based constant-stress creep curve for alloy 690 steam generator tube material. Nucl Eng Technol. 2022;54(3):917-25., the θ-Projection has been widely used for modeling creep behavior under constant stress. Creep behavior under a constant load cannot be accurately describe, as observed in references¹³13 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC, et al. Modified θ projection model-based constant-stress creep curve for alloy 690 steam generator tube material. Nucl Eng Technol. 2022;54(3):917-25.,¹⁷17 Liu H, Peng F, Zhang Y, Li Y, An K, Yang Y, et al. A new modified theta projection model for creep property at high temperature. J Mater Eng Perform. 2020;29(7):2020-4779. and in this work for Ti-Al-4V. In this context, other studies have been presented in the scientific literature to minimize the deviation of the predicted creep curves under constant load¹³13 Moon S, Kim JM, Kwon JY, Lee BS, Choi KJ, Kim MC, et al. Modified θ projection model-based constant-stress creep curve for alloy 690 steam generator tube material. Nucl Eng Technol. 2022;54(3):917-25.,¹⁷17 Liu H, Peng F, Zhang Y, Li Y, An K, Yang Y, et al. A new modified theta projection model for creep property at high temperature. J Mater Eng Perform. 2020;29(7):2020-4779.. However, results indicated that the ν Concept produced a creep curve with better agreement with the experimental data. Such a fact is related to the combination of mathematical functions for the primary and tertiary regimes of the ν Concept, which result in better agreement for tests under constant load conditions.

4. Conclusions

This study developed a software for the direct application of the θ-Projection and ν Concept mathematical models to predict creep behavior. Results indicated that the ν Concept and θ-Projection models showed good agreement with the experimental data. Results indicated that the ν Concept produced a creep curve with greater agreement with the experimental data. Note that the ν Concept better defined the tertiary stage. Furthermore, more experimental data are needed to improve prediction agreement. Databases with ν and θ must be designed for other Ti-6Al-4V microstructures under different stress-temperature regimes. The technique used to obtain parameters is promising and should be improved for future studies.

5. Acknowledgments

The authors thank FINEP, FAPESP, CAPES and CNPq for their financial support.

6. References

Publication Dates

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