In engineering, many structural components operate under high-temperature conditions for a long time, such as steam turbine, boiler and main steam pipeline in thermal power generation equipment, high-temperature and high-pressure reaction vessel and pipeline in petrochemical system.

In addition to normal working stress, they also need to bear other additional stress, cyclic stress and rapid temperature fluctuation in a large range.

Therefore, their life is often affected by creep fatigue and creep fatigue interaction.

Fatigue creep interaction is the main mechanism of equipment failure under cyclic load under high temperature environment.

Its life prediction is of great significance to the selection, design and safety evaluation of high temperature equipment.

It has always been a concern of the engineering and academic circles. Many scholars have proposed corresponding life prediction models.

This post briefly introduces the common life estimation methods.

Table of Contents

**Life-time fraction method**

Linear cumulative damage method, also called life-time fraction method, is mainly used to estimate the life of fatigue creep interaction.

The life-time fraction method considers that the damage caused by the interaction of fatigue and creep is the linear accumulation of fatigue damage and creep damage, as shown in the following formula:

Where N_{f} is the fatigue life, n_{i} is the fatigue cycle times, t_{r} is the creep failure time, and t is the creep retention time.

This method simply adds the calculated fatigue damage and creep damage to obtain the total damage.

The calculation is very simple, but it needs to obtain the test data of pure creep and pure fatigue under the corresponding temperature environment.

Because the interaction between fatigue and creep is not considered in this method, its calculation results and accuracy are poor.

In order to overcome the shortcomings and improve the calculation accuracy, researchers have proposed a variety of improved forms.

For example, Xie’s correction formula is as follows:

The amendment proposed by Lagneborg is as follows:

In the above formula, n is the interaction creep damage index, 1 / N is the interaction fatigue damage index, and a and B are the interaction coefficients.

The interaction term is added to the two modified expressions, which can be used to adjust the error between the prediction results of the cumulative damage method and the experimental results, and greatly improve the reliability of the prediction results.

## Frequency correction method (FM method) and frequency separation method (FS method)

At present, most of the fatigue creep life estimation methods widely used in engineering are based on the strain control mode.

The frequency correction method was proposed by coffin.

It is believed that the main damage in low cycle fatigue is caused by plastic strain.

Eckel proposed the following formula on this basis:

Where: t_{f} is the failure time, K is the temperature dependent material constant, ϑ is the frequency, ∆ε_{p} is the range of plastic strain.

By substituting the above formula into Manson coffin formula, the expression considering frequency correction can be obtained as follows:

The frequency separation method is another improvement on the basis of the frequency correction method.

The method assumes that the fatigue damage is caused by inelastic strain and considers the influence of the holding time on the life under high temperature.

The tensile holding frequency and the compression holding frequency are introduced, and the fatigue life is expressed in the exponential form of inelastic strain and holding frequency, so that the influence of the loading frequency on the fatigue life is more significant.

As follows:

Wherein, ϑ_{C} is the frequency of the compression carrier, ϑ_{t} is the frequency of tensile load retaining, ∆ε_{in} is the inelastic strain.

Both the frequency correction method and the frequency separation method use the fatigue life estimation model, but they successfully use the loading frequency to introduce the creep factor into the fatigue life estimation model, making the new model suitable for the life estimation of fatigue creep interaction.

**Strain range division (SRP) and strain energy division (SEP)**

The strain range division method was proposed by Manson.

The basic idea is that for the two types of strain that are time dependent and time independent, even if the amount of strain is the same, the damage caused is not the same.

Considering the interaction between creep and fatigue, the inelastic strain range in a stress-strain cycle is divided into pure mechanical strain range components and time-dependent strain range components according to different quality, and then the damage caused by each part is determined and the total damage is obtained.

Which has the following expression, Cij, β_{i}_{j} is the material constant.

The strain range division method is widely used, but it requires different types of cyclic test data.

The strain energy division method is established on the basis of the strain range division method, and the relationship between the strain energy of each strain and the life is established:

Where, C_{ij} ,β_{ij} is the material constant determined by the test;

∆U_{ij} is the strain energy;

α_{ij} is tensile strain energy and rectangular area σ_{max}∆ε_{P}.

According to the linear cumulative damage method, the following life estimation formula is obtained, and F_{ij} is the weight coefficient.

Dong Zhaoqin and he Jinrui used the frequency separation method to modify the relationship between strain energy and life, called SEFS method, and obtained the following expression, where C, β, m, K is a constant.

The strain range division method and the strain energy division method need a large number of reliable test data as the basis, and many material parameters and mechanical variables need to be considered.

Therefore, it is a long-term work to use this method for life estimation.

**Stress relaxation range method**

In the strain control mode, the interaction of creep and fatigue for a long time will lead to greater stress relaxation, and the stress relaxation and creep effect are the main reasons for the reduction of creep fatigue life under a long time.

From this point of view, Nam Soo Woo et al. introduced the stress relaxation range into the creep fatigue life prediction model.

According to the relationship between life and holding time and the relationship between holding time and stress relaxation range, the normalized life prediction method is derived as follows:

Among, Φ, f is the material constant.

Since the stress relaxation range is a function of holding time, initial stress, strain level, temperature and other parameters, the above formula can be used to predict the life under different holding times, different waveforms and different strain ranges, and the coffin Manson curve obtained under different conditions can be normalized to obtain a main curve.

The stress relaxation range method is suitable for the life prediction of fatigue creep interaction under strain control mode.

**Ductility loss method**

The estimation method of fatigue creep life of ductile depletion is based on the theory of ductile depletion.

According to the theory of ductility depletion, fatigue and creep cause the damage of components in the form of viscous flow.

Fatigue causes the depletion of intracrystalline ductility, while creep causes the depletion of grain boundary ductility.

They accumulate and stack each other, and finally reach the critical value, resulting in material failure.

Goswamirunf has done a lot of research on the fatigue creep interaction of Cr Mo steel and put forward a new prediction model of ductile dissipation life.

Where ∆σ is the stress range, ∆ε_{P} is the plastic strain range, ∆εt is the total strain range, is the strain rate, ∆σ_{s} is the saturated stress at the half life, and K, A, m and n are the material constants.

This model is established on the basis of the concepts of strain control mode, strain rate and viscous flow.

It is suitable for the life prediction of Cr Mo Steel under the interaction of fatigue and creep under strain control and plastic strain dominated.

In addition to the ductility depletion model, the fatigue creep life estimation method based on the stress control mode also has the energy life estimation model and the average strain rate estimation model.

In contrast, the ductility depletion model is more suitable for the stress control mode, and this method can also comprehensively reflect the influence of stress ratio, loading rate, holding time, average strain rate and other factors on the component life, with high prediction accuracy.

**Metallographic life prediction method**

Nam Soo Woo et al. proposed a new damage parameter based on the nucleation and growth mechanism of creep holes in austenitic stainless steel.

This damage parameter is suitable for describing the damage of materials with grain boundary creep holes.

This method needs to know the micro quantities such as pore area, grain boundary thickness, grain boundary diffusivity and atomic volume of creep.

**Damage mechanics prediction method for fatigue creep life**

The concept of damage mechanics was first proposed by Kachanov, and then Lemaitre et al. applied damage mechanics to predict fatigue creep life.

According to the classical damage theory, the damage variable D represents the extent to which the effective bearing area of the material decreases during the damage process caused by the microcracks and micro voids, that is, due to the formation and expansion of the microcracks and micro voids, the cross-sectional area a of the specimen decreases to the effective bearing area a *, and the decrease of the effective bearing area leads to the increase of the stress.

According to the above definition of damage mechanics, it can be assumed that the damage increment can be expressed by the sum of fatigue damage increment and creep damage increment:

The expression of fatigue damage increment and creep damage increment adopts Lemaitre model, and the specific form of fatigue creep interaction damage increment is as follows:

It can be seen from the above formula that the damage accumulation described by the damage mechanics model is nonlinear, and the fatigue creep interaction is considered.

In addition to Lemaitre damage model, Shang et al. put forward a nonlinear uniaxial fatigue damage accumulation model based on chaboche continuous fatigue damage theory according to the change performance of material toughness in the process of fatigue damage.

This model takes into account the inseparability of fatigue limit, average stress, damage variable and loading parameters, as well as the influence of loading sequence.

Jing et al. proposed a nonlinear continuous damage mechanics model for the creep fatigue life of steam turbine rotor.

In the model, the influence of complex multiaxial stress and the coupling effect of fatigue and creep were considered, and the nonlinear evolution of damage was considered.

**Prediction method of fracture mechanics**

Fracture mechanics divides life prediction into two stages: crack formation and crack propagation.

Since the 1970s, many scholars have proposed to use C * integral to describe the local stress field and strain rate field at the crack apex of an object at any time under creep conditions.

Meanwhile, C * integral is also called creep fracture parameter.

Therefore, the measurement and calculation of C * integral has become an important research direction in the fatigue creep life estimation method.

Chapuliot and curtit et al. gave an experimental method to determine the parameter C * of the surface crack in a plate subjected to bending moment and obtained the calculation formula of C *.

Fookes and Smith have proved through experiments that the total displacement rate can be used to determine the parameters.

Yatomi et al. proposed to determine the parameters by using the creep load line displacement rate calculated numerically.

**A new forecasting method based on multivariate statistics**

The typical representative of multivariate statistical methods is Goswasmi, who proposed a general formula for predicting fatigue creep life of high-temperature materials based on a large number of experimental data.

He also gave the basic formulas for fatigue creep life prediction of Cr Mo steel, stainless steel and alloy steel containing tin, titanium and other materials.

**A new prediction method based on Neural Network**

Neural network (ANN) is an advanced nonlinear analysis tool developed in recent years. It can fully approach any complex nonlinear relationship.

The most outstanding advantage of neural networks is that they can find solutions in uncertain systems and variable relationships.

At present, many scholars have applied the neural network method to predict the fatigue creep life of materials.

For example, venkatech et al. proposed the method of back propagation neural network to predict the fatigue creep life of materials at (0.7 ~ 0.8) melting point;

Srinivasan et al. used neural network method to predict the life of 316L (N) stainless steel under fatigue creep interaction.

In 2013, Wang, N and others proposed to build a new type of adaptive network for creep fracture life prediction.

The network is a four layer structure system, which accurately predicts the creep fracture life of 9-12% chromium ferritic steel.

The results show that this method is more accurate than Larsen Miller parameter method and more effective than BP neural network.

**Prediction model based on energy conservation and momentum conservation**

Most of the above existing fatigue creep interaction life prediction models require a large number of different types of test data, or for the fatigue creep interaction under the strain control mode, it is very inconvenient to apply and cannot be applied to the stress control.

Jiang et al. deduced a new fatigue creep interaction life prediction model based on the energy conservation law and momentum conservation law reflecting the motion of the system, striving to have a better theoretical basis and a simple model expression, and can be applied to the fatigue creep interaction under stress control.

The expression is:

Using the above formula to predict the life of fatigue creep interaction, the physical meaning is clear, and it is applicable to the life prediction of fatigue creep interaction under the strain control and stress control modes.

The required test parameters are easy to obtain and the number is small.

In order to test the accuracy of the model, Jiang et al. carried out stress controlled trapezoidal wave loading tests on smooth specimens of 1.25Cr0.5Mo steel at 540 ℃ and 520 ℃, and used the model to predict the fatigue creep interaction life under the above two temperature environments.

The predicted results are in good agreement with the actual results.

**Service condition endurance strength (SCRI) interference model**

Zhao proposed a service condition creep property interference model (scri model) based on Z parameter to predict the reliability of endurance life of high-temperature materials.

Using the Z-parameter method, the dispersion of the endurance strength of high-temperature materials obeys the normal distribution, and the distribution of service conditions caused by the fluctuation of service temperature and stress can be simulated by Monte Carlo method, so as to realize the reliability analysis of the endurance life of materials under the condition of considering the dispersion of performance data and the fluctuation of service conditions.

**Extrapolation model of creep fracture data based on dynamic process**

Liu, H et al. proposed a model for extrapolating creep fracture data based on dynamic process. The model describes the relationship between stress and fracture time.

The expression parameters are few, the calculation process is relatively simple, and the calculated value is closely consistent with the experimental results.

The expression is:

Where: C is Larsen Miller constant; Q is the activation energy of the creep process and R is the Boltzmann constant.

In addition, the model enhances the preciseness of long-term creep life prediction.

According to the comparison of the test data of 2.25Cr1.0mo steel and Ti Al metal compound, this evaluation method is more accurate than the traditional Larsen Miller parameter (LMP) method.

**Conclusion**

This paper summarizes the research results of fatigue creep life estimation methods in recent decades.

The correction formula of linear cumulative damage considers the interaction of fatigue and creep, which effectively improves the calculation accuracy;

The life prediction methods of damage mechanics and fracture mechanics have relatively mature theoretical basis and can specifically solve the life prediction problems of complex and defective components.

The prediction results of frequency correction method, frequency separation method and strain range division method are ideal, while the prediction results of strain energy division method and strain energy frequency correction method are poor.

Multivariate statistical method and neural network method are new methods for fatigue creep life estimation.

Among them, the three types of materials mentioned in the multivariate statistical method can be directly predicted by using the basic calculation formula;

Neural network method is mainly used to solve complex or unknown life prediction problems.