2021 Volume 62 Issue 2 Pages 198-204
This paper proposes an isothermal fatigue life prediction model to predict the fatigue life of the creep-fatigue damage interaction. The fatigue life is caused by high-frequency mechanical loads and low-frequency temperature loads, i.e. stepped-isothermal fatigue loads. The model is based on continuum damage mechanics (CDM), in which the interaction between creep and fatigue damage is considered nonlinear. To verify the proposed model, the cast aluminum alloy is subjected to fatigue tests at 200–350°C. The results show that the predicted life of the model can reach a good consistency with the experimental data.
With the development of aerospace, energy, and chemical industries, the application of high-temperature equipment has become more and more extensive, and the load it bears has become more and more complex. Among these loads, low cycle fatigue (LCF) and creep-fatigue interaction (C-F) are the most common damage modes.1–3) These types of equipment suffer from creep damage during steady-state operation and suffer from thermal fatigue or thermo-mechanical fatigue (TMF) damage during start-stop or sudden changes in working conditions. The potential danger is enormous. Once an accident occurs, it is often catastrophic. Therefore, the safety and reliability of equipment under C-F have become increasingly prominent with the development of power machinery, chemical machinery, and aerospace. The damage assessment and life prediction of high-temperature components are currently important areas that need to be studied. To study the fatigue behavior of these components, the high temperature and mechanical load during the load cycle are mainly kept constant or cycled at the same time, which are called isothermal high-cycle fatigue (HCF) or TMF respectively.4) However, for certain components, such as engine pistons and cylinder heads, the load in each cycle is a combination of high-frequency mechanical load and low-frequency temperature load. During the entire cycle life of the engine, the working state changes alternately, for example, the engine changes from an idling condition to full speed and full load condition. Under such cyclic conditions, this combined cyclic load is defined as a stepped-isothermal fatigue load.5)
In recent years, many researchers have researched the fatigue problems of aluminum alloy materials widely used in internal combustion engine (ICE) piston and cylinder head structures.6–9) The fatigue problem of aluminum alloy materials is affected by many complicated factors.10) To meet the requirements of fatigue analysis of aluminum alloy engine components, Minichmayr et al.11) tested different life prediction methods. The results showed that the Neu-Sehitoglu life prediction model12,13) based on linear cumulative damage has the best accuracy in describing the life results of the material test. The model fully considers the effects of creep, fatigue, and oxidative damage on the fatigue life of the material, but the biggest disadvantage of the Neu-Sehitoglu model is that it requires too many material parameters and does not consider the nonlinear coupling between the three types of damage. When studying Al–Si alloy pistons, Wang et al.14) proposed a new energy-based LCF and TMF life prediction model, which is based on the hysteresis energy with strain rate modification, while considering fatigue and creep damage.
Under the combined action of cyclic thermal load and mechanical load, the coupled creep-fatigue damage is the decisive factor affecting the fatigue life of materials. Creep damage is a material degradation process in which microvoids are formed and grown between the grain boundaries of material, and then intergranular cracks are formed, leading to creeping fracture. It is closely related to factors such as loading stress and temperature.
Fatigue damage is a deterioration process in which lattice dislocations and slippage occur locally under the action of cyclic loading, which leads to the formation of transgranular cracks in the material.15) The essence of fatigue damage is the failure phenomenon caused by the accumulation of internal damage of the material to the limit value.16) The use of damage mechanics analysis methods to predict and analyze the fatigue life of materials can take into account the nature of material creep-fatigue damage and can be effectively applied to the life prediction analysis of aluminum alloy materials.17) In the research of the fatigue life model based on damage analysis, many scholars18–21) have carried out corresponding research and successfully applied the model to engineering practice. For the thermo-mechanical fatigue problem, Wang et al.22) introduced the Kachanov-Rabotnov creep damage model into the constitutive model of the material to evaluate the creep-fatigue life of the material. Berti et al.23) proposed a life prediction model that can consider the effect of temperature changes on creep-fatigue damage. Many researchers have also proposed other fatigue damage models considering C-F under high-temperature conditions.24–27) The damage accumulation and failure mechanism under stepped-isothermal fatigue load are different from the damage accumulation and failure mechanism under conventional isothermal fatigue or thermo-mechanical load conditions. At the same time, it is obvious that a stepped-isothermal fatigue life prediction model is needed in engineering applications. With this model, the need for a large number of expensive fatigue tests can be greatly reduced.
In the research, to consider the influence of the non-linear ultimate tensile strength σu of the material with the temperature change on the creep-fatigue damage life of the material, the creep-fatigue life model in Ref. 5) is modified. In this model, the creep and fatigue damage evolution model is derived from CDM, and the interaction between creep and fatigue damage is nonlinear. To verify the model, the cast aluminum alloy is subjected to stress-controlled uniaxial fatigue tests at 200–350°C by applying specific isothermal fatigue loads.
The material tested in the stepped-isothermal fatigue test is the cast Al–12Si–CuNiMg alloy for diesel engine pistons.28,29) Table 1 lists the chemical composition of cast aluminum alloys. After the material is proportioned according to the chemical composition, it is smelted in a power frequency furnace for 1 hour, then refined and degassed and slag removed, then calmed for 30 minutes, and poured into a 120 × ϕ12 mm cylindrical bar with a pouring temperature of 790–810°C. The heat treatment process is 7.5 hours of artificial aging treatment at 240°C. Finally, a cylindrical specimen is processed with a gauge length of 27 mm and a diameter of 6.35 mm, as shown in Fig. 1.
The geometry of the creep-fatigue test specimen (dimension in mm).
The stepped-isothermal fatigue load shown in Fig. 1 is used in the test. Servo hydraulic machine MTS-810 is used to control mechanical load and thermal load. Since there is no standard procedure for the stepped-isothermal fatigue test, the fatigue test in this study is conducted concerning the GB/T 15248-2008 (China) standard, which is the standard for the LCF test.
The temperature range of the test is 200–350°C, the mechanical load cycle period is 2 s, the uniaxial stress control of the triangular wave is adopted, the stress ratio is −1, the temperature cycle period is 1 hour, and the trapezoidal wave control is adopted. For the 200–350°C test, the lowest temperature is 200°C and the highest temperature is 350°C. The temperature is kept at 200°C for 30 minutes, and then the temperature is raised to 350°C, kept for another 30 minutes, and then enter the next cycle. Heat all test samples to the test temperature and keep them for 30 minutes to ensure that the temperature of the entire sample is uniform. The temperature along the gauge length is controlled within ±2°C. Cooling is achieved by blowing compressed air onto the surface of the test bar through a nozzle.
Under isothermal fatigue or TMF load, the temperature variable is usually controlled to be constant, or it changes cyclically at the same frequency as the mechanical load, that is, the in-phase (IP) or out-of-phase (OP) TMF problem. This paper mainly studies the creep-fatigue life prediction model of aluminum alloy materials under the coupling action of low-frequency thermal load and high-frequency mechanical load. The creep-fatigue coupling load is shown in Fig. 2. The characteristic of the creep-fatigue load shown in Fig. 2 is that the cycle period of temperature is much longer than that of mechanical load, that is, the frequency of thermal load is very different from the frequency of the mechanical load. Besides, the temperature changes cyclically in the form of a trapezoidal wave, and the mechanical load is simplified to the form of a triangular wave to facilitate model development and parameter verification.
The stepped-isothermal fatigue loading: (a) temperature variation, and (b) loading stress.
In the follow-up of the paper, we will discuss the creep and fatigue damage evolution under thermal load and mechanical load respectively, and then combine these two damages through implicit nonlinear superposition.
3.2 Creep damage evolutionIt is recognized that damage mechanics is proposed by Kachanov when he is studying metal creep in 1958. At that time, he proposed the concepts of continuity factor and effective stress and used effective stress to give the evolution equation of continuity factor.30) For creep damage, it is a process of internal fracture of the material. The damage starts from microcracks or holes in the grain boundary and gradually grows up. Under the action of stress concentration and grain boundary slippage, local voids are formed. Creep cracks are generally intergranular cracks, which are different from fatigue failure in the form of transgranular damage. Kachanov considered that the material damage is the cross-sectional loss caused by the existence of internal micro-holes and micro-cracks, so he proposed the creep damage evolution equation to describe the relationship between initial stress and cumulative damage:
| \begin{equation} dD_{c} = \left[\frac{\sigma}{A(1-D_{c})}\right]^{r}dt \end{equation} | (1) |
| \begin{equation} dD_{c} = \left(\frac{\sigma}{A}\right)^{r}(1+r) \frac{(1-D_{c})^{-m}}{m+1}dt \end{equation} | (2) |
In the above equation, because in the pure creep test, the loading stress is a fixed value. To be suitable for the situation where the load stress changes in a large area, Berti et al.23) proposed an improved creep damage model:
| \begin{equation} dD_{c} = \left(\frac{\sigma(t)}{A}\right)^{r}(1+r)\frac{(1-D_{c})^{-m}}{m+1}dt \end{equation} | (3) |
| \begin{equation} \sigma (t) = \begin{cases} \left(\dfrac{\sigma_{\text{max}}}{t_{1}}\right)t & \text{($0 < t < t_{1}$)}\\ \dfrac{\sigma_{\text{max}}(t-t_{2})}{t_{1} - t_{2}} & \text{($t_{1} < t < t_{2}$)} \end{cases} \end{equation} | (4) |
Stress evaluation in each mechanical cycle.
Finally, combining the above evolution equation of creep damage and the expression of tensile stress in each cycle, the creep damage evolution model under stepped-isothermal fatigue load can be described as:
| \begin{equation} \left\{ \begin{array}{l} dD_{c} = \left(\dfrac{\sigma(t)}{A}\right)^{r} (1+r) \dfrac{(1-D_{c})^{-m}}{m+1}dt\\ \sigma (t) = \begin{cases} \left(\dfrac{\sigma_{\text{max}}}{t_{1}}\right)t & \text{($0 < t < t_{1}$)}\\ \dfrac{\sigma_{\text{max}}(t - t_{2})}{t_{1}-t_{2}} & \text{($t_{1} < t < t_{2}$)} \end{cases} \end{array} \right. \end{equation} | (5) |
The above equation can be rewritten into the following form:
| \begin{equation} dD_{c} = f_{c} (\sigma (t),T,D_{c})dt \end{equation} | (6) |
Assuming that the initial creep damage is N = 0 when Dc = 0, and N = Np when Dc = 1, then the specific period of creep damage can be obtained by integrating eq. (5).
3.3 Fatigue damage evolutionPrevious studies have shown that even under the same maximum temperature and stress amplitude load, the TMF life of the material is still lower than the isothermal fatigue life.33,34) This shows the important influence of temperature cycle changes on the damage evolution of materials. Therefore, when establishing the fatigue damage evolution model of materials, it is necessary to consider the influence of temperature changes on fatigue damage.
Based on the classic fatigue damage model proposed by Lemaitre et al.,35,36) this paper proposes an improved fatigue damage evolution model:
| \begin{equation} \text{d}D_{f} = \frac{(1-D_{f})^{-m}}{m+1}\frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\beta (T)}\text{d}N \end{equation} | (7) |
| \begin{equation} \beta (T) = \gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}} \end{equation} | (8) |
Substituting eq. (8) into eq. (7), the fatigue damage evolution model can be obtained:
| \begin{equation} \text{d}D_{f} = \frac{(1-D_{f})^{-m}}{m+1}\frac{\sigma_{a}}{\sigma_{u}-\sigma_{a}}\left( \frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - {T_{\text{min}}}}{T_{\text{max}}}}\text{d}N \end{equation} | (9) |
The above equation can be rewritten into the following form:37)
| \begin{equation} dD_{f} = f_{f}(\sigma_{a},T,D_{f})dN \end{equation} | (10) |
Assuming that N = 0 when Df = 0, and N = Np when Df = 1, the fatigue damage under cyclic loading can be obtained by integrating eq. (10).
3.4 A creep-fatigue damage interaction modelFrom a mesoscopic point of view, the coupling between creep damage and fatigue damage is very complicated. In a high temperature environment, the local fatigue crack initiation of a material is fatigue damage, and the formation of grain boundary holes in the material causes creep damage. The two types of damage interact on the microscopic mechanism. On the one hand, when the transgranular microcracks caused by fatigue damage meet with the grain boundary pores caused by creep damage, the growth and aggregation speed of fatigue cracks will accelerate, and the creep damage accelerates the development of fatigue damage; on the other hand, the grain boundary pores caused by creep damage will accelerate the nucleation and growth of the pores under the action of the micro cracks caused by fatigue damage, and they will continue to accumulate, eventually leading to accelerated accumulation of creep damage. In short, creep holes can be used as crack sources to promote the initiation and growth of fatigue cracks, and the microcracks of fatigue cracks will also aggravate the nucleation and growth of creep holes.38) Under this interaction, the internal stress concentration of the material becomes more obvious, thereby reducing the creep-fatigue life of the material. When the total damage of the material including creep damage and fatigue damage reaches a limit threshold, it will lead to failure. In this case, the creep damage Dc and fatigue damage Df per cycle can be linked to the total damage D of the material, as shown in the following equation:
| \begin{equation} dD_{c} = f_{c}(\sigma (t),T,D)dt \end{equation} | (11) |
| \begin{equation} dD_{f} = f_{f} (\sigma_{a},T,D)dN \end{equation} | (12) |
Also, considering the coupling effect of creep damage and fatigue damage, after each cycle of loading, there is a cumulative effect between creep damage and fatigue damage, and the total damage can be expressed as:
| \begin{equation} D = D_{c} + D_{f} \end{equation} | (13) |
Therefore, the evolution equation of total damage can be expressed as:
| \begin{align} dD &= dD_{c} + dD_{f} \\ &= f_{c}(\sigma (t),T,D)dt + f_{f} (\sigma_{a},T,D)dN \end{align} | (14) |
In the above equation, creep damage and fatigue damage accumulate in each incremental step of the cycle. As the number of load cycles continues to increase, creep damage and fatigue damage achieve implicit nonlinear accumulation of damage.
Based on the above damage accumulation analysis, combined with the evolution expressions of creep damage and fatigue damage, the total damage expression under the combined effects of creep-fatigue can be obtained:
| \begin{align} dD &= \left(\frac{\sigma(t)}{A}\right)^{r}(1+r)\frac{(1-D)^{-m}}{m+1}dt \\ &\quad + \frac{(1-D)^{-m}}{1+m}\frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}} \left(\frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}}dN \end{align} | (15) |
Based on the above equation, the tensile stress area of the creep damage evolution equation is integrated to obtain:
| \begin{align} \frac{dD}{dN} &= \frac{(1-D)^{-m}}{m+1}(1+r) \int_{0}^{t_{2}} \left(\frac{\sigma (t)}{A}\right)^{r}dt \\ &\quad + \frac{(1-D)^{-m}}{1+m}\frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}} \end{align} | (16) |
| \begin{align} \frac{dD}{dN} &= \frac{(1-D)^{-m}}{m+1} \\ &\quad \times\left[(1+r) \int_{0}^{t_{2}} \left(\frac{\sigma (t)}{A}\right)^{r}dt + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M} \right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}}\right] \end{align} | (17) |
Then integrate both sides of eq. (17) at the same time, and assume that N = 0 when D = 0 and N = Np when D = 1. Finally, the following equation is obtained:
| \begin{align} &\int_{0}^{1}(1+m)(1-D)^{m}dD\\ &\ = \int_{0}^{N_{p}} \left[ (1+r) \int_{0}^{t_{2}} \left(\frac{\sigma (t)}{A} \right)^{r} dt + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}} \right] dN \end{align} | (18) |
| \begin{equation} N_{p} = \left\{(1+r) \int_{0}^{t_{2}} \left(\frac{\sigma (t)}{A} \right)^{r} dt + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}} \right\}^{-1} \end{equation} | (19) |
The creep-fatigue life prediction model established in this paper is shown in eq. (19). In the equation, $\int_{0}^{t_{2}}(\frac{\sigma (t)}{A})^{r} dt$ is directly related to the cycle period of the mechanical load. Each cycle period of the mechanical load in this test is 2 s, the mechanical load waveform is a triangular wave, and the stress ratio is −1. The time of stress action is 1 s, that is, the time t2 = 1 in the integral term. Besides, according to Fig. 3, the tensile stress reaches its maximum value at the time t1, and segmental integration is required for numerical integration. According to the above analysis, the tensile stress area of the integral term $\int_{0}^{t_{2}}(\frac{\sigma (t)}{A})^{r} dt$ in a single cycle is integrated to obtain:
| \begin{equation} \int_{0}^{1} \left(\frac{\sigma (t)}{A} \right)^{r} dt = \int_{0}^{0.5}\left(\frac{\sigma (t)}{A}\right)^{r}dt + \int_{0.5}^{1} \left(\frac{\sigma (t)}{A}\right)^{r}dt \end{equation} | (20) |
Substituting eq. (4) into eq. (20), we can get:
| \begin{align} \int_{0}^{1} \left(\frac{\sigma(t)}{A}\right)^{r} dt & = \int_{0}^{0.5} \left(\frac{\sigma(t)}{A}\right)^{r} dt + \int_{0.5}^{1} \left(\frac{\sigma(t)}{A}\right)^{r} dt\\ & = \frac{1}{A^{r}}\bigg[\int_{0}^{0.5}\left(\frac{\sigma_{\text{max}}}{t_{1}}t\right)^{r}dt \\ &\quad + \int_{0.5}^{1} \left(\frac{\sigma_{\text{max}}}{t_{1} - t_{2}}(t - t_{2})\right)^{r}dt\bigg] \\ & = \frac{\sigma_{\text{max}}^{r}}{A^{\text{r}}(1 + r)} \end{align} | (21) |
Substituting eq. (21) into eq. (19), the expression of creep-fatigue life Np can be obtained as:
| \begin{align} N_{p} & = \left\{\frac{1 + r}{A^{r}}\left[\frac{\sigma_{\text{max}}^{r}}{(1 + r)}\right] + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma\frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}}\right\}^{-1}\\ & = \left\{\frac{\sigma_{\text{max}}^{r}}{A^{r}} + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma\frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}}\right\}^{-1} \end{align} | (22) |
In the aluminum alloy material test in this paper, the maximum value of the mechanical load is equal to the amplitude value. The above equation can be rewritten as:
| \begin{equation} N_{p} = \left\{ \frac{\sigma_{a}^{\text{r}}}{A^{\text{r}}} + \frac{\sigma_{a}}{\sigma_{u} - \sigma_{a}}\left(\frac{\sigma_{a}}{M}\right)^{\gamma \frac{T_{\text{max}} - T_{\text{min}}}{T_{\text{max}}}}\right\}^{-1} \end{equation} | (23) |
In this way, the creep-fatigue life prediction model of the aluminum alloy material is obtained, and the material parameters in the model can be obtained by fitting the test data.
Table 2 lists the ultimate tensile strength σu of aluminum alloy materials with temperature changes. Table 3 lists the test results of the cast aluminum alloy under stepped-isothermal fatigue conditions. The stress range is 80–95 MPa, and 2–3 specimens are selected for the test under each load level, and the test is performed in temperature range. Four sets of fatigue tests with different stress levels are performed.5) The material parameters in the model can be obtained by fitting the experimental data, as shown in Table 4. According to the obtained material parameters, draw the creep-fatigue model curve in the temperature range of 200–350°C, and compare it with the experimental data,39) as shown in Fig. 4. It can be seen from the figure that the creep-fatigue life prediction model has a good fit with the test data. The model can describe the creep-fatigue life data of the material well at 200–350°C.
Model curves and experimental data at 200–350°C.
According to the creep-fatigue life prediction model of aluminum alloy materials, the creep-fatigue life of materials at 200–350°C is predicted, and the corresponding stress amplitudes is 75 MPa. The results show that the life prediction results of the materials under this load are within a scatter band of factor two, as shown in Fig. 5. It shows that the creep-fatigue life prediction model established in this paper based on CDM analysis has a good prediction effect, and can analyze the creep-fatigue problems under the action of high-frequency mechanical load and low-frequency thermal load.
Comparison between predicted and experimental data under stepped-isothermal fatigue condition (with a significance level of 0.05).
In this paper, based on CDM theory, the creep damage and fatigue damage of materials are analyzed by nonlinear coupling accumulation, and a creep-fatigue life prediction model of aluminum alloy materials is established. The model has a clear load pertinence, that is, the coupling load effect of mechanical load with high-frequency cyclic change and thermal load with low-frequency cyclic change. According to the characteristics of this load, the evolution equations of creep damage and fatigue damage are analyzed and deduced, and finally, the nonlinear coupling analysis of the two types of damage is realized. The following conclusions are obtained:
This work is supported by the China Postdoctoral Science Foundation funded project (Grant No: 2019M663443), the Chongqing Research Program of Basic Research and Frontier Technology (Grant No: cstc2017jcyjAX0120), and the youth project of science and technology research program of Chongqing Education Commission of China (Grant No: KJQN201901113).
