A Novel Viscosity-Based Model for Low Cycle Fatigue–Creep Life Prediction of High-Temperature Structures

Abstract

Damage evolution during low cycle fatigue, creep, and their interaction behavior is actually a ductility exhaustion process in response to cyclic and static creep. In this article, a novel viscosity-based model for low cycle fatigue–creep life prediction is presented in an attempt to condition viscosity-based approaches for general use in isothermal and thermo-mechanical loading. In this model, it was assumed that only plastic and creep strains caused by tensile stress lead to ductility consumption under stress-controlled loading. Moreover, with its simple expression, the mechanisms of the loading waveform, temperature, and mean stress effects are taken into account within a low cycle fatigue–creep regime. Predicted fatigue lives using the proposed model were found to be in good agreement with reported experimental data from literature. Compared with the generalized strain energy damage function method, the mean strain rate, Smith–Watson–Topper and Goswami’s ductility models, the proposed model is widely applicable and more precise in the prediction of low cycle fatigue–creep life.

Keywords

creep high-temperature structure life prediction low cycle fatigue viscosity

INTRODUCTION

Currently, a number of mechanical components are used at severe conditions such as high temperature or high pressure for their high efficiency. These components operating in aviation, power generating, petrochemical, and other industries are subjected to cyclically varying loads and often fail within a limited number of load cycles, e.g., below about 10⁵ cycles (Zhang, 2010). The high reliability and safety of these components are largely due to a combination of improved materials, improved life prediction capabilities, and highly conservative design. Thus, a better understanding of the loading conditions to which today’s high-temperature structures are subjected as well as developing better analytical tools of fatigue damage is needed. The motivation behind this is that if the predictive capability of fatigue damage at high temperature can be improved, then statistically, a more efficient design should be feasible.

The most common failure mode for these structures is low cycle fatigue (LCF) at high temperature which is an interactive mechanism of different processes such as time-independent plastic strain, time-dependent creep, and environmental corrosion, oxidation and the complex interaction between them. These damage mechanisms challenge fatigue design methods and may induce dangerously inaccurate life predictions. Thus, even though researchers have explored interactions between fatigue and time-dependent damage mechanisms for many years, studies on the complicated behaviors of materials at high temperature, the designs of such components considering effects of different time-dependent damage mechanisms, and the remaining life to failure are recognized to be essential nowadays.

Increasing attention has been paid to the study of fatigue and creep interaction in either isothermal or thermal–mechanical fatigue conditions. The deformation and failure mechanism of low cycle fatigue–creep (LCF–C) at high temperature are very complex. They may act independently or in combination depending on various test and material parameters, such as temperature, strain rate, hold time, and time-dependent damage processes such as creep, dynamic strain aging, and environmental attack. Some typical methods for LCF–C life prediction have been developed such as linear damage summation (Zhang, 2010), frequency modified Manson–Coffin equation (Coffin, 1976), frequency separation (FS) technique (Coffin, 1974), strain range partition (Manson et al., 1971), strain energy damage function (SEDF) model (Ostergren, 1967), and ductility exhaustion (DE) approach (Goswami, 1995, 1997). In practical application of these models, obtaining various parameters in these equations can be very difficult. Hence, highly precise methods of life prediction are needed to make the design effective.

Robustness of a life prediction method is a crucial key point; there is renewed interest in approaches based on strain energy (Lee et al., 2008; Payten et al., 2010; Zhu et al., 2011a, b). Recently, energy-based approaches have been used to predict damage for fatigue and fatigue–creep cycling based on the hysteresis loops area (Koh, 2002; Lee et al., 2008; Zhu et al., 2011a). In order to account for creep and mean strain or stress effects on the LCF life, a ductility-based model has been previously derived (Zhu et al., 2011a) and applied to a number of LCF tests on a GH4133 Superalloy. Using the mean strain rate (MSR) as the main factor associated with the fracture life, Fan et al. (2007) developed a MSR model based on the investigation of fatigue–creep interaction behavior in stress control mode for 1.25Cr0.5Mo steel. With the assumption that creep damage is measured by the absorbed internal energy density, Payten et al. (2010) put forward a strain energy density exhaustion approach, which is derived from considerations of mechanistic cavity growth. All these models have their own capabilities, shortcomings and have shown their validity on a limited number of alloys and loading conditions. However, the following problems still need to be carefully solved to accurately predict the LCF–C life: quantification of fatigue–creep interaction, description of temperature, loading waveform, mean stress effects, and appropriate damage accumulation rule.

The authors’ previous work (Zhu and Huang, 2010) has clearly showed that it is possible: (1) to correlate the fatigue–creep damage and the life with a viscosity-based parameter E_p; (2) to reflect the effects of time-dependent damaging mechanisms on LCF–C life; (3) to identify the main influential factor of LCF–C life, the maximum stress and stress range at minimum stress σ_min ≤ 0, and mean stress at minimum stress σ_min > 0. Further development and modifications to these issues are in progress which will make the estimation/prediction of LCF–C life via DE theory with high accuracy, simplicity, and wide application scope possible.

The objectives of this article lead to a novel viscosity method for LCF–C life prediction based on the correlation between failure/fatigue and energy. The proposed approach uses the fatigue–creep toughness as the main indicator of damage defining the dynamic viscosity with additional terms to account for the effects of time-dependent damaging mechanisms and mean stress. This model has to comply with industrial requirements for high prediction capability, ease of use, physical observations-based and simplicity. Following this, the prediction results of the proposed method are compared with the MSR (Fan et al., 2007), the generalized strain energy damage function (GSEDF) (Zhu and Huang, 2010), Smith–Watson–Topper (SWT) (Smith et al., 1970) and Goswami’s ductility model (Goswami, 1997). A comparison between the prediction and the experimental results was conducted. It is showed that the proposed model is capable of providing accurate fatigue life prediction. The capabilities of this method are then further discussed.

PREVIOUS RESEARCH SCOPE

According to the interaction mechanisms of failure at high temperature, a good life prediction method must consider not only the effects of stress/strain level, loading history, impurity content, but also creep factors such as hold time, strain rate, and temperature. The LCF life is dependent on test parameters (Zhu et al., 2011c). Though several energy-based methods for predicting LCF life have been developed (Zhu and Huang, 2010), a GSEDF model with the capability to easily incorporate the effects of time-dependent damaging mechanisms on LCF life is essential for the future development of general prediction criterion addressing LCF–C at high temperature. In this study, a trapezoid load diagram was used to analyze the conditions of most alloys under high temperature, pressure, and cyclic loading. The load diagram of fatigue–creep interaction is plotted in Figure 1.

With regard to the stress cycle shown in Figure 1, T_du, T_dl, T′, and T″ represent the tensile hold time, compressive hold time, tension-going time, and compression-going time, respectively, in one loading cycle when σ_max > 0 and σ_min > 0. Here, T_dl is the tensile hold-time when σ_min > 0, T₀ and T the total time period, and the period time not including the hold time where T = T′ + T″.

Considerable effort has been extended in defining suitable damage parameter which correlates the life to failure (Voyiadjis and Kattan, 2009). Similar to the energy criterion proposed in the study of Stowell (1966), the energy parameter accumulated per cycle under fatigue–creep interaction can be described by the stress area under loading waveforms, and above the zero-stress line, as shown in Figure 2.

The parameter E_p per cycle with the shadows, as shown in Figure 2 can be calculated by the function given below

E_{p} = T_{du} σ_{max} + (T_{dl} + T) σ_{min} H (σ_{min}) + \frac{T}{2} f (σ_{max}, σ_{min})

(1)

and

f (σ_{max}, σ_{min}) = {\begin{matrix} Δ σ, σ_{min} >0 \frac{σ_{max}^{2}}{Δ σ}, σ_{min} \leq 0 \end{matrix}

(2)

where

H (σ_{min})

is the unit step function of σ_min and

f (σ_{max}, σ_{min})

called the ‘stress conversion function’ determined by the maximum and minimum stresses and material properties. All materials are sensitive to the tensile stress and tensile dwell. As the dwell time in the tension direction is increased, the resulting life decreased. Compared with tensile dwell effects, the effects of compressive stress and compressive dwell on the fatigue–creep life are complex and vary by materials (Goswami and Hanninen, 2001a, b; Zhang, 2010). Using the assumption that only tensile stress can induce fatigue–creep damage (Chrzanowski, 1976), the function

H (σ_{min})

can be defined as follows

H (σ_{min}) = {\begin{matrix} 1, σ_{min} \geq 0 0, σ_{min} <0 \end{matrix}

(3)

Power law index is established mainly based on the strain energy damage function model and analysis of large amounts of test data. In order to reduce the difference between the approximate and real strain energy absorbed during the damage process and get a higher precision, the strain energy and fatigue life can be written using the power law index

Δ ɛ_{in} {(E_{p})}^{φ} N_{f}^{α} = C

(4)

where

Δ W_{t} = Δ ɛ_{in} {(E_{p})}^{φ}

is the GSEDF and ϕ the stress damage exponent related to environment conditions.

Considering that without hold-time in the loading waveform, T_du = T_dl = 0, the model reduces to an expression similar to the SEDF model when $φ = \frac{1}{2}$ . So, it is called a GSEDF model. The expression of LCF life without hold-time can be written as follows

N_{f} = {\begin{matrix} C {(Δ ɛ_{in} σ_{max} \sqrt{T / 2 Δ σ})}^{β}, σ_{min} \leq 0 C {(Δ ɛ_{in} \sqrt{T σ_{m}})}^{β}, σ_{min} >0 \end{matrix}

(5)

Under different loading waveforms, fatigue–creep life prediction can be determined using Equations (1)–(5). The comparison between the GSEDF, FS, and SEFS methods shows that the GSEDF model provides a higher precision of life prediction than the FS and SEFS methods (Zhu and Huang, 2010). In order to minimize the difference between the approximate and real ductility exhausted during the fatigue process, the accuracy of the proposed parameter E_p will be further investigated based on DE theory in the following section.

A NOVEL VISCOSITY-BASED LIFE PREDICTION MODEL FOR LCF–C

Fatigue is a damage accumulation process in which material property deteriorates continuously under cyclic loading. Experimental results have shown that the toughness of a material, a mechanical property parameter which combines both strength and plasticity of a material, is sensitive to the fatigue–creep damage process. A certain quantity of energy (material toughness), actually the ductility is gradually exhausted during the damage process of a material under LCF–C. Under cyclic loading, the continuous reduction of the material ductility indicates the progressive exhaustion of the ability to absorb energy, which is directly associated with the irreversible process of energy dissipation during fatigue failure. A value of energy dissipated in a material during one cycle of loading or during all the cycles up to the failure, is usually calculated from a history of the changes in cyclic strain and stress combined with the number of cycles. The more damage that is accumulated, the more ductility is exhausted. Once a critical toughness threshold is reached, fracture occurs. According to this relationship, the exhausted ductility can be used to indicate the damage accumulation under LCF–C loading. Several failure criteria and the corresponding life assessment methods have been developed based on this idea (Goswami, 1997; Fan et al., 2007; Zhu and Huang, 2010; Payten et al., 2010; Zhu et al., 2011b).

Under cyclic loading, the interactive behavior between stress and strain during deformation can be represented by hysteresis loops. Recent research suggests that a viscosity-based approach can be used to quantify this interaction (Goswami, 1997, 2004). In each cycle (points 1–5 in Figure 1), the parameter E_p was defined based on the loading waveform. The corresponding hysteresis loop is shown in Figure 3. In this article, the parameter E_p is associated with the stress, strain, and the material toughness based on the DE theory.

Ductility exhaustion theory assumes that, during tension, fatigue, and creep fracture processes, damage evolution usually can be associated with the continuous DE of the material. And failure occurs once accumulated strain reaches a critical ductility. Goswami (1995, 1997) developed a ductility model based on the assumption that deformation for LCF at high temperature can be represented in terms of viscous behavior. The dynamic viscosity should account for the strain range effects and can be presented based on the fundamental viscosity concept. Enlightened by this, dynamic viscosity υ_d is defined as (Goswami, 2004)

υ_{d} = Δ σ \cdot (Δ ɛ_{t} / \overset{\cdot}{ɛ})

(6)

where υ_d, Δσ, Δε_t, and

\overset{\cdot}{ɛ}

are the dynamic viscosity, the stress range, the total strain range, and the strain rate, respectively.

The ability of a material to accommodate permanent deformation was defined in terms of material toughness. Failure criterion was defined in terms of dynamic viscosity, which was equal to the material toughness and could be expressed as

\sum f (υ_{d}) = T_{m}

(7)

where T_m is the total energy dissipated by the material cumulatively up to failure, called the fatigue toughness. The material toughness is a product of ductility and cyclic strength (Goswami, 2004). Using the Edmund and White equation, ductility can be determined as follows

Ductility = Δ ɛ_{p} N_{f}

(8)

Then, material toughness can be obtained by

T_{m} = Δ ɛ_{p} \cdot N_{f} \cdot σ_{sat}

(9)

where σ_sat is the saturated tension stress at half-life (

N_{f} / 2

Since creep damage is sensitive to the tensile hold time instead of compressive hold time (Zhang, 2010). The key factor leading to the failure of high-temperature structures in certain environments is the fatigue–creep interaction, which comes from thermally induced stresses and strains. Meanwhile, the effects of time-dependent test parameters, such as hold time, loading waveform and temperature, showed that the interactions between fatigue and creep are complex interactions with environmental factors. These interactions are very difficult to incorporate into a life prediction model. Hence, defining a suitable damage model to account for the interaction of fatigue and time-dependent damage caused by creep, which can account for temperature and mean stress effects, is considered to be very important when the results of LCF–C testing are applied to simulate thermal fatigue conditions.

According to the physical significance of the parameter E_p in Equation (1) and the dynamic viscosity υ_d in Equation (6), it should be noted that the latter is included in the former. And the essential difference between them is that the former includes the tensile elastic energy input which causes no damage compared with the latter. Based on the above description, the parameter E_p is actually a viscosity-based parameter. In order to reduce the difference between the approximate and real ductility exhausted during the fatigue process, a new definition of dynamic viscosity is presented using the parameter E_p and the tensile elastic energy input ΔW_FL per cycle which causes no damage

υ_{d} = E_{p} - T_{0} \cdot Δ W_{FL}

(10)

where the tensile elastic energy input which causes no damage, ΔW_FL, was determined by

Δ W_{FL} = σ_{lim}^{2} / 2 E

(11)

where σ_lim is the fatigue limit of material.

Substituting Equations (1) and (11) into Equation (10) results in the following equation set

υ_{d} = {\begin{matrix} T_{du} σ_{max} + (T_{dl} + T) σ_{min} + \frac{T}{2} Δ σ - T_{0} \frac{σ_{lim}^{2}}{2 E}, σ_{min} >0 T_{du} σ_{max} + \frac{T}{2} \frac{σ_{max}^{2}}{Δ σ} - T_{0} \frac{σ_{lim}^{2}}{2 E}, σ_{min} \leq 0 \end{matrix}

(12)

For simplicity, it is assumed that the stress cycle period and fatigue life follow a power law index relationship. Thus, the dynamic viscosity during the fatigue process can be expressed as given below

\sum f (υ_{d}) = k {(E_{p} - T_{0} \cdot Δ W_{FL})}^{γ}

(13)

where k and γ are material constants related to the environment conditions, such as temperature.

Fatigue toughness which describes both strength and plasticity of a material is a more sensitive mechanical property parameter to the fatigue damage process than others. Considering that fatigue crack grows only in the tension stage, Ostergren (1967) proposed the strain energy damage function model (Viswanathan, 1995). This model assumes that only tensile inelastic strain energy can induce the crack opening and propagation. The strain energy damage function ΔW_str is approximately expressed by multiplication of the inelastic strain range Δε_in and the maximum tension stress σ_max. The relationship between strain energy and fatigue life is expressed by the power exponent function, i.e.

Δ W_{str} N_{f}^{ν} = Δ ɛ_{in} σ_{max} N_{f}^{ν} = C

(14)

where ΔW_str is the strain energy, Δε_in the inelastic strain range of the stable hysteresis loop or half-life which is replaced by the plastic strain range Δε_p under pure fatigue loading.

It is interesting to note that Equations (9) and (14) have the similar form even though they are derived from different theoretical backgrounds. Comparing two equations based on the Ostergren’s model, the fatigue toughness T_m can be given as

T_{m} = Δ ɛ_{in} σ_{max} N_{f}^{ν}

(15)

As discussed earlier, a new LCF–C life prediction equation is derived equating these two terms, the dynamic viscosity Equation (13) and fatigue toughness Equation (15), as shown in Equation (7). Therefore, a new life prediction equation is derived by rearranging the following equation

k {(E_{p} - T_{0} \cdot Δ W_{FL})}^{γ} = Δ ɛ_{in} σ_{max} N_{f}^{ν}

(16)

where k is a material parameter which balances the units of this equation.

Rearranging various terms in Equation (16), the number of cycles to failure can be calculated from the following viscosity-based approach

N_{f} = k {(E_{p} - T_{0} \cdot Δ W_{FL})}^{p} {(Δ ɛ_{in} σ_{max})}^{q}

(17)

Based on Equation (17), it is worth noting that this empirical model correlates failure/fatigue with energy using DE theory, which includes the factors influencing fatigue and creep lives. It is very easy to use this equation to predict the LCF–C life at high temperatures using only three material parameters (k, p, and q) in Equation (17), which can also be fitted from test data.

The ratcheting behavior of materials is a complex phenomenon and depends on a number of factors, including mean stress σ_m, stress amplitude σ_a, frequency (total time period for one cycle T₀), loading history and micro-structural characteristics (Xia et al., 1996). It is worth noting that Equation (17) incorporates most of these factors. Similar to the parameter developed in the study of Xia et al. (1996) and the SWT parameter (Smith et al., 1970), the effect of ratcheting on the fatigue life was characterized by the inelastic strain range Δε_in, which consists of plastic strain range Δε_p and creep strain range Δε_c. The strains of points 1–5 in Figure 1 were obtained by an extensometer and the inelastic strain range per cycle is expressed as

Δ ɛ_{in} = ɛ_{f} - ɛ_{0}

(18)

where ε₀ and ε_f are the initial and final strains, respectively, in a cycle.

Moreover, these three methods considered the effect of mean stress on the predicted life through the maximum stress, where $σ_{max} = σ_{a} + σ_{m}$ . For the proposed viscosity-based model, both mean stress and ratcheting effects have been taken into account in the fatigue damage process.

For LCF–C interaction with hold time at high temperature, it follows from the discussions above that the fatigue life predicted in terms of the fatigue DE actually represents the degradation of both plasticity and strength of a material. Compared with other methods (Goswami, 1995, 1997; Ye and Wang, 2001), the development of this new model considers not only the loading waveform effects on LCF–C life, but also the effects of mean stress. Additionally, it provides a new way to determine the real dynamic viscosity per cycle and extends the application of DE theory to stress-controlled LCF–C interaction conditions. For actual components, the prediction accuracy of the proposed model was evaluated and was verified by analyzes of LCF–C life prediction in the following section.

VALIDATION OF THE NEW LIFE PREDICTION MODEL FOR LCF–C

To verify the feasibility and prediction capability of the viscosity-based life prediction model for LCF–C at high temperature, the proposed model was evaluated using experimental results of 1.25Cr0.5Mo steel and turbine disk material GH4133 under different temperatures from published sources (Wang, 2006; Fan et al., 2007).

Material and Test Conditions

The applicability of the new life prediction model was evaluated using LCF–C test results from the study of Wang (2006) and Fan et al. (2007). The materials used in these experiments are pearlitic heat resistant steel 1.25Cr0.5Mo and Ni-base superalloy GH4133. The tests for the 1.25Cr0.5Mo steel were conducted using a trapezoid waveform (stress control) with a hold period of 5 s duration at σ_max and σ_min, respectively, where T_du = T_dl = 5s and each cycle took 20 s (T₀ = 20s). The material was machined into cylindrical specimens with 10 mm diameter and 32 mm gauge length. Under different stress ratios and mean stresses $σ_{m}$ , the fatigue–creep tests were performed under the following conditions: two temperatures, 540°C and 520°C, stress amplitudes ranging from 25 to 190 MPa and four defined maximum stresses 200, 210, 220, and 230 MPa. These stress controlled fatigue–creep interaction tests were conducted using an electro-hydraulic servo fatigue testing machine EHF-EG250-40 L equipped with a heating furnace. The combined test conditions and the number of tests are summarized in Table A1.

In order to verify the effects of strain rate and the loading waveform on the LCF life at high temperature, the proposed model will be evaluated using experimental results of GH4133. The heat treatment conditions of this alloy are as follows: austenitization (8 h at 1353.15 K, air-cooled) and tempering (16 h at 1023.15 K, air-cooled). The tests for the GH4133 were performed under axial total strain control with a triangular fully reversed waveform, using an axial extensometer placed on the specimen. Numerous tests were carried out with the following various conditions: mechanical strain range of 0.5–1.4% for isothermal LCF at temperature 500°C and 400°C under strain ratio R_ε = -1, respectively. For further inquiries regarding detailed mechanical properties of the materials, test procedures and specimen specifications refer to Wang (2006), Fan et al. (2007), Chen et al. (2007), and Beijing Institute of Aeronautical Materials (1996).

Results and Discussion

The damage induced in the material due to the loading described in the previous section is to be estimated using each of the following methods: MSR, GSEDF, SWT, Goswami’s ductility model, and the current model stated in Equation (17). The response of the material at half-life ( $N_{f} / 2$ ) is considered to be stable and representative of the life. For 1.25Cr0.5Mo steel at 520°C and 540°C, the Young’s modulus E was 1.77 × 10⁵MPa. Combining the experimental results and the parameters listed in Table A1, for different maximum stresses and stress ratios, the fitted life prediction model for 1.25Cr0.5Mo steel at 540°C according to Equation (17) is given by

N_{f} = 4.89057 \times 10^{14} {(E_{p} - 20 \cdot Δ W_{FL})}^{- 0.837803} {(Δ ɛ_{in} σ_{max})}^{- 0.907999}

(19)

Similarly at 520°C, the fatigue life is approximately fitted as

N_{f} = 1.52045 \times 10^{7} {(E_{p} - 20 \cdot Δ W_{FL})}^{- 0.0101023} {(Δ ɛ_{in} σ_{max})}^{- 0.938895}

(20)

For GH4133, the Young’s modulus E was 1.992 × 10⁵MPa. Under different total strain controls and strain ratios R_ε = -1, the fitted life prediction model for GH4133 at 500°C is given by

N_{f} = 4.08565 \times 10^{39} {(E_{p} - 1.7138 \cdot Δ W_{FL})}^{- 4.0446} {(Δ ɛ_{in} σ_{max})}^{- 0.268971}

(21)

Similarly at 400°C, the fatigue life prediction for GH4133 under strain ratio R_ε = -1 by Equation (17) is expressed as

N_{f} = 1.5365 \times 10^{34} {(E_{p} - 1.7138 \cdot Δ W_{FL})}^{- 3.3111} {(Δ ɛ_{in} σ_{max})}^{- 0.38978}

(22)

Good correlations between the experimental results and the theoretical predictions under different temperatures for these two materials are observed, as shown in Figures 4 and 5. The dashed line in the graph corresponds to the ±1.5 factor indictors, the dot line for the ±1.25 factor indictors, and the solid line for the ±2 factor indictors. From Figures 4 and 5 and Table A1, all the predicted lives are within a factor of ±2, and about 33 out of 34 cyclic lives for 1.25Cr0.5Mo steel, 47 out of 55 cyclic lives for GH4133 are predicted within a factor of ±1.5. Obviously, the predicted results are in good agreement with the test ones.

To reflect the capability of the proposed model and evaluate its applicability under the effects of mean stress and creep, four other methods, the MSR (Fan et al., 2007), the GSEDF (Zhu and Huang, 2010), the Goswami’s (1997) ductility model, and the SWT model (Smith et al., 1970), were employed for comparison purposes, respectively. The test data for 1.25Cr0.5Mo steel was assessed by three methods and is listed in Table A1. This table shows clearly that the life predictions using these methods are in accordance with experimental results. The correlations between the experimental and the predicted lives by these four models for two materials are shown in Figures 6 and 7.

It is found that nearly all the predicted cyclic lives by the GSEDF, the MSR, and SWT models fall into a range within a scatter band of ±2, while about 30 out of 34, 26 out of 33, and 43 out of 55 cyclic lives predicted by the GSEDF, MSR, and the SWT models are within a factor of ±1.5, respectively. From Figure 7, the results show that only about 34 out of 55 cyclic lives are predicted within a factor of ±1.5 to the test ones by the Goswami’s ductility model. Comparing the scatter band and the standard deviation of these methods, results indicate that the proposed model has a better predictability than others. The life prediction method using Equation (17) can predict LCF–C behavior well at a certain temperature, but whether it can be consolidated into one equation for a certain temperature interval or not will be assessed in the following section.

In practical engineering, the failure of high-temperature structures comes from thermally induced stresses and strains. Hence, developing a suitable life prediction model, which can account for temperature effects, is considered to be very important when the results of LCF testing are applied to the simulation of thermal fatigue conditions. If the effect of temperature on fatigue life can be properly described, life prediction at specific elevated temperatures is available in addition to the reference fatigue data normally obtained at room temperature. In general, thermal fatigue test is very difficult to perform and also takes a long time because temperature variation rate is much slow. Therefore, many researchers analyzed thermal fatigue with high-temperature isothermal fatigue data. Using Equation (17), the LCF–C life for 1.25Cr0.5Mo steel under different temperatures is predicted by the following equation

N_{f} = 4.07253 \times 10^{11} {(E_{p} - 20 \cdot Δ W_{FL})}^{- 0.495795} {(Δ ɛ_{in} σ_{max})}^{- 0.92377}

(23)

Similarly for GH4133 under different temperatures, the fatigue life is approximately fitted as

N_{f} = 4.99462 \times 10^{31} {(E_{p} - 1.7138 \cdot Δ W_{FL})}^{- 3.0382} {(Δ ɛ_{in} σ_{max})}^{- 0.370252}

(24)

Life obtained by Equations (23) and (24) at different temperatures are compared to the experimental life data, as shown in Figures 8 and 9, respectively.

The correlation of predicted and experimental life results is satisfactory for different temperature loading conditions. The fatigue life correction factor range is nearly equal to ±1.5 or better, with which 34 out of 36 cyclic lives for 1.25Cr0.5Mo steel and 46 out of 55 cyclic lives for GH4133 are predicted within a factor of ±1.5 by the proposed viscosity-based model. It should be noted that Equation (17) can be used to predict lives under thermal fatigue conditions. Through the DE theory, this model can transform the complex correlation between N_f and mechanisms of loading waveforms and mean stress into a rational relation. By which life prediction for other conditions can easily be made. Furthermore, the proposed model enables the time dependencies of deformation characteristics to be described.

In summary, based on the same theoretical backgrounds and viscosity parameter E_p, the differences between the experimental and calculated LCF–C life by the proposed method and the GSEDF model are relatively small as both of them consider the effects of loading waveform and mean stress. Moreover, the new model incorporates the effect of temperature where the GSEDF model does not. Thus, it can be better used for life evaluation of high-temperature structures under thermal fatigue conditions. According to the theoretical derivation of the new model and the comparisons of the prediction results by these methods from Table A1 and Figures 4, 5, 8, and 9, it is obvious that the viscosity-based model proposed in this article has a better prediction capability than others.

According to the applicable conditions of the proposed model, it is valid for most metallic materials under uniaxial loading, such as carbon steels, cast irons, and alloy steels. Furthermore, it can reflect the fundamentals of fatigue–creep damage under stress or strain control. It has the advantages including few parameters, considering mean stress, temperature, and loading waveform effects, and has a higher life prediction precision when compared to other approaches. However, most engineering components and structures are subjected to complex loading conditions at which stress–strain cycles fluctuate with time. This leads to a cyclic creep–fatigue interaction. Thus, the application of this model under multiaxial loading, different hold times/temperatures and materials need to be further evaluated.

CONCLUSIONS

Based on the DE theory and the GSEDF method, a new viscosity-based model has been proposed for LCF–C life prediction. The feasibility and validity of this proposed model was checked with the fatigue–creep interaction test data from literature. Some conclusions are drawn from the present investigation:

The viscosity-based parameter E_p correlates and describes the fatigue–creep damage with the mechanisms of loading waveform and mean stress. This leads to a great robust assessment method for high-temperature structures in practical engineering applications.

The proposed model is applicable for both the strain-controlled tests and stress-controlled tests under uniaxial loading. On the basis of DE theory, this model is able to describe the damaging processes during the interaction of LCF and creep as a dependence on loading parameters, σ_max, $σ_{m}$ , σ_a, Δε_p, strain rate and hold time within a certain temperature interval.

Compared with the GSEDF and the MSR methods, the proposed viscosity-based model is more suitable for predicting LCF–C life for high-temperature structures. It is worth noting that all the test data were within a factor of ±2 and nearly 97% of the test data for 1.25Cr0.5Mo steel, 85% of the test data for GH4133 were within a factor of ±1.5 of the predicted results. This strongly suggests that the proposed model is believed to be physically feasible and accepted as a general expression. Moreover, it has higher prediction accuracy and the capability to estimate thermal fatigue lifetime using calibration on isothermal fatigue tests only, leading to a great potential cost saving for future test programs given similar materials and loading conditions.

The main characteristics of this model compared with some existing methods for LCF–C life prediction based on energy (ductility) is that it takes the effects of creep, temperature, and mean stress on fatigue life into account. Accordingly, it is better suited to describe the dynamic deterioration process of various high-temperature structural materials and hot section components even under thermal–mechanical fatigue.

Footnotes

Figures 1– appear in color online: ijd.sagepub.com

NOMENCLATURE

ACKNOWLEDGMENTS

This research was partially supported by the National Natural Science Foundation of China under the contract number 51075061 and the National Programs for High Technology Research and Development of China under the contract number 2007AA04Z403. The authors thank Gary Paradee of University of Maryland for his helpful discussions and corrections of the English manuscript of this article.

APPENDIX

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