Fatigue life prediction of AZ31B magnesium alloy based on fracture fatigue entropy

Abstract

In this paper, changes in entropy generation, entropy flow and damage are discussed under cyclic loading for AZ31 magnesium alloy. Results show that the entropy generation increases sharply at the beginning of the cycle, decreases to a certain extent and remains stable. The evolution law of entropy flow is similar to that of temperature in the magnesium alloy fatigue process. Fatigue damage rises exponentially and increases rapidly after the damage reaches 0.3. The relationship between the fatigue fracture entropy model and cycle times, which can predict the fatigue life well, is established. The relationship between the stable value of entropy generation and cyclic loading is also established. The fatigue limit is 102.25 MPa, and the error is 6.8%.

Keywords

FFE model entropy generation fatigue life fatigue limit

Introduction

The magnesium alloy has low density, high-specific strength and good thermal conductivity. This alloy has good application prospects in the field of rail transit, in which magnesium alloy components often bear dynamic loads [1]. Fatigue fracture is always the main cause of component failure under dynamic loading [2]. After a certain number of cycles, the component appears as small cracks under the response of dynamic load and breaks rapidly after a small number of cycles [3]. Therefore, the study of fatigue process evolution and life prediction is extremely important for structural design and service evaluation.

At present, many methods for fatigue life prediction and evaluation are available. The commonly used fatigue evaluation methods include nominal stress, local stress–strain and fracture mechanics [4,5]. However, most of these methods have their own shortcomings. For components with notches, a large error occurs when using the nominal stress method. The local stress method is applicable to this type, whereas the notch coefficient (K) is difficult to obtain, thereby affecting accuracy. The basis of the fracture mechanics theory is that components have defects. In addition, these methods all rely on destructive tests [6].

With the rapid development of sensitive components, an infrared thermal imager is used to detect the temperature change in the fatigue process [7]. Therefore, the method of predicting fatigue life based on temperature has been studied. The typical temperature change in the fatigue process is divided into three stages [8]. Amiri and Khonsari [9] proposed a method based on the initial temperature rise slope for predicting fatigue life. Jiang et al. [10] found that the relationship between the fatigue life and temperature rise of specimens under uniaxial fatigue loading can be established. Fargione et al. [11] found that the integral area of fatigue life and temperature rise are constant through a large number of experiments. Huang [12] believes that for high-ductility metal materials, the initial temperature rise slope in the third stage is related to fatigue life.

Temperature can predict the fatigue life and fatigue limit well but cannot show the damage in the fatigue process. In the framework of thermodynamics, fatigue damage essentially belongs to an irreversible dissipation process. According to the second law of thermodynamics, the index of thermodynamic entropy, which connects energy and temperature, is introduced into the study of fatigue. This index can measure the energy conversion direction of the irreversible dissipation process [13] and the thermodynamic index of the damage degree of the system (metal specimen). The disorder caused by irreversible processes increases with time until failure occurs. Entropy increases monotonically with the disorder. A large entropy results in uniform system, poor organisation, high degree of disorder, and low energy state. In the process of fatigue, the total entropy of the system is the sum of entropy flow across the system boundary (change in system entropy caused by thermal convection and radiation) and entropy generation (change in entropy caused by irreversible damage in the system).

Khonsari's team introduced thermodynamic entropy into fatigue damage for the first time. Bryant et al. [14,15] correlated the system degradation of the irreversible dissipation process with the entropy generation of the dissipation process by theoretical deduction and proposed the degradation–entropy generation theory. On this basis, Nader [16] studied the fatigue damage degree under different loading conditions in accordance with thermodynamic entropy. Wei [17] used fracture fatigue entropy (FFE) to evaluate the fatigue reliability of Q460 welded joints during high-cycle fatigue. The Morrow model based on plastic energy density and the temperature model of the fatigue process proposed by Salimi [18] are used for entropy generation, in which two different types of plastic property changes are considered. However, the change trend of plastic strain energy in the process of fatigue is determined by the nature of materials. Therefore, a general method to represent plastic strain energy should be found. In this paper, a plastic strain is introduced to establish a general model of FFE. At the same time, the experimental evidence of the whole process is not sufficient, the existing research is based on steel, and the material is single. This study has tried to use magnesium alloys to prove the superiority of thermodynamic entropy in the fatigue life determination process.

In this work, a general FFE model for predicting fatigue life is proposed. The changes in entropy generation, entropy flow, and damage degree of magnesium alloy during fatigue are studied.

Experiments

The specimens are commercial AZ31 magnesium alloy plates with a thickness of 5 mm and extruded by casting at 350–380°C compression moulding. The specimen is an aluminium–zinc magnesium alloy, and its chemical composition is shown in Table 1. The specimen size is shown in Figure 1.

Table 1

Chemical composition of AZ31 (wt-%).

Element	Al	Zn	Mn	Fe	Cu	Ni	Si	Be	Mg
Content	3.22	1.15	0.4	0.0019	0.0011	0.00051	0.0133	0.00006	Bal

The specimens of AZ31 magnesium are evaluated on a fatigue testing apparatus by using the DS100 electrohydraulic servo-controlled fatigue testing machine with 100 kN capacity. Fatigue tests are conducted under the sinusoidal waveform stress controlled mode, stress ratio of 0.1, and frequency of 10 Hz. At the same time, an infrared thermal imager is used to monitor the fatigue process.

Before the fatigue test, the tensile test is carried out to obtain the yield strength for the selection of the loading amplitude required for the fatigue test. Figure 2 shows the tensile curve of the magnesium alloy obtained with the equipment, and the yield limit is 156 MPa. With decreasing units of 10 MPa from 150 MPa to 70 MPa, these stress values are selected as the maximum stress loaded in fatigue tests. During the fatigue process, the number of cycles and the surface temperature of the sample are recorded for each σ _max by an infrared thermal imager, and deformation data are recorded by the sensor. The specimen is considered to not break if cycles exceed 10⁷. In this paper, the derivation of the equation and the demonstration of the test results are predominantly carried out with 140 MPa as an example.

Theoretical background of entropy evolution

This study is based on the thermodynamic framework, and the required thermodynamic assumptions and premise definitions must be assumed in advance. Figure 3 shows the fatigue system schematic. For fatigue research, the fatigue characteristics of the material rather than the fixture are the focus. Thus, the thermodynamic system defined in this study is the specimen gauge section. No material conversion, only heat exchange, is observed between the system and the environment. The system is considered as a closed system. The heat exchange between the thermodynamic system and the environment set occurs through radiation and convection. Heat conduction only occurs at the holding end. Given that the holding part of this experiment is far from the heating section, the heat loss through heat conduction is considered small. The concept of local equilibrium in thermodynamics is applicable to this paper. The total volume of the system is V, and the system boundary (Ω) is calibrated (Figure 3). Part of the system is exposed to the environment, and the system is fixed on the fatigue testing machine. The cyclic load f(t) is applied to the system externally.

During the fatigue process, the external cyclic load does work (W) to the sample, part of the energy is dissipated in the form of heat (Q), and the remaining energy exists in the form of internal energy. The first law of thermodynamics and strain are introduced, and the expression [19] can be stated as: (1) where ρ is the material density, u is the specific internal energy, t is the time, σ is the stress tensor, D is the deformation rate tensor, and is the heat flux. According to the Fourier law of heat conduction [20], is linearly related to the temperature gradient vector ∇T. For solid materials with small deformation [21], the expression can be changed into the following equation (2) where is the strain rate tensor, and k is the material thermal conductivity rate. The relationship among u, T, and s is established using the Helmholtz free energy [15] as follows: (3) where is the Helmholtz free energy, and s is the thermodynamic entropy. The two sides of Equation (3) are derived as follows: (4) Equations (2) and (4) are combined to obtain the following equation: (5) Equation (5) defines the entropy balance of the system. The entropy equilibrium equation can be written as a sum of the entropy generation and the entropy flow, where the local balance assumption holds: (6) where is entropy generation, and indicates the total entropy flow per unit time through the unit area. The second law of thermodynamics states that the entropy generation is 0 for a reversible process and that the entropy generation is nonnegative when the process is irreversible. (7) The total entropy flow can be simplified as the heat flux through the unit area per unit time, and Equation (6) can be organised into Equation (8), which is the Clausius–Duhem inequality [22]. (8) is the plastic strain rate tensor, and V_K is any internal variable such as work hardening and damage. A _K is the thermodynamic forces related to internal variables. For metals, the entropy produced by energy dissipation relates to the evolution of the internal variable A_KV_K , which only accounts for 5–10% of the entropy produced by plastic dissipation. Therefore, the first term on the right side of Equation (8) can be considered only when plastic deformation dominates the system's entropy generation under cyclic loading [15]. (9) During the fatigue process, . f is the frequency of fatigue loading, and is the strain energy density in one cycle. Then Equation (9) can be written as follows. (10) Entropy generation from initiation to fracture is FFE: (11) According to the Morrow theory [23], the strain energy density of each cycle is given by Equation (12) in the steady-state hysteresis hoop. Equation (14) is a simplified form of calculating the hysteresis curve, which can be used to calculate the density of fatigue plastic energy, especially low-cycle fatigue. It is sensitive to the work hardening index. (12) where is the stress amplitude, represents the steady plastic strain amplitude, and is the work hardening index.

Entropy generation from initiation to fracture is FFE: (13)

Results and discussion

Proposed prediction model of FFE

At present, Equation (13) is a general equation. However, a big problem is that different materials produce different plastic energy densities due to their properties, causing Δw to be different in Equation (13). Yan's study showed that the plastic strain change law of the material is the same during the fatigue process [24]. Figure 4(a) shows the stress–strain curves of different cycles (σ _max = 140 MPa). Irreversible plastic deformation occurs under tension–tension cyclic loading, which has a hysteresis effect. With the increasing number of cycles, plastic hoops move to the right, indicating an increase in the plastic strain in the material. Subsequent shaping hoops coincide, indicating that plastic hoops are basically stable. The plastic hoops have stabilised about the cycles number at 3000 times. In the last few cycles, the value of plastic strain reaches the maximum. In accordance with the stress–strain curve of each cycle, a horizontal line is drawn at the position of the average stress to intersect the plastic hoop, and the difference value is the plastic strain, as shown in Figure 4(b).

The plastic strain predominantly experiences three stages, and the initial stage strain increases rapidly due to a large amount of plastic deformation in stage I. Stage II is a steady stage in which the strain remains constant. Stage III is the shortest and is characterised by a sharp increase in plastic strain. The third stage occurs at the moment of fracture, which may sometimes not exist and depend on the magnitude of the loading stress amplitude. In the fatigue process, stages I and II play a major role, especially the latter. The plastic curves under different cyclic loadings exhibit the same trend [25]. The law of plastic strain can be expressed by the following equation. (14) where a, b, and d are all constants, related to material properties, stress levels and frequency, respectively, and t is time. Substituting Equation (14) into Equation (12), the plastic work density is obtained as follows: (15) By substituting Equation (15) into Equation (2), the first law of thermodynamics can be expressed as follows [26]: (16) According to a large number of experiments and previous studies [27,28], the maximum temperature is distributed at the centre of the scale length. This view is also confirmed by the temperature distribution of different cycles (σ _max = 140 MPa), as shown in Figure 5(b) (using the centreline as shown in Figure 5(a) for temperature measurement). Therefore, X defaults to the central point temperature in the model, and only the partial derivative of time can be obtained to simplify the calculation and ensure safety. The boundary condition is defined as follows. Considering that test equipment clamps have a larger volume than the end of the sample, they act as radiators. Thus, the two ends are considered in equilibrium with room temperature [18]. Finally, the temperature expression can be obtained using Equation (17) by importing the boundary conditions and assumptions into Equation (16). The comparison between the measured and model temperature curves is shown in Figure 6, and the error is within the allowable range. (17) The actual surface temperature change of the AZ31B magnesium alloy can be divided into three stages during the fatigue process (Figure 6). In the first stage, the surface temperature of the magnesium alloy increases approximately linearly. At initial cycles, plastic deformation occurs locally and a large amount of heat is generated. The heat generation rate is higher than the heat conduction rate, leading to a temperature increase. The surface temperature of the magnesium alloy reaches the peak value after finite plastic deformation. The temperature enters stage II and drops rapidly because of the work hardening of magnesium. Magnesium alloys are HCP structures with few slip systems [29]. With increasing cycles, the heat generation and heat dissipation in the sample reach a balance, and the surface temperature of the sample fluctuates in a small range and enters stage II. The duration of this stage is almost 80% of the total fatigue life.

Substituting Equations (15) and (17) into Equation (13), FFE can be established as follows:

(18)

Influencing factors of the FFE model

FFE is predominantly affected by temperature and plastic energy density. Next, their change laws are analysed during the test. For the AZ31B magnesium alloy, the temperature curve changes under different σ _max values (Figure 7). From the law of fatigue temperature evolution, the temperature evolution of magnesium alloys under high cyclic loadings follows the law of four stages. Temperature is almost horizontal and straight under low cyclic loadings [30].

The variation in the plastic strain curve for different cyclic loadings is plotted in Figure 8. Plastic strain results show that the fatigue failure of magnesium alloy is an irreversible damage accumulation process, and the change in cyclic strain follows the same law. With increased cyclic loading, the strain saturation value increases gradually, and the generation rate of irreversible damage accelerates, which is caused by increased plastic deformation with large cyclic loading. The comparison of infrared temperature evolution shows that the increases in magnesium alloy surface temperature with cyclic loading can correspond to rapid accumulation stage strains in the plastic strain process.

Equation (10) shows that entropy generation is closely related to plastic strain energy density. This paper explores the evolution law of plastic strain energy first in the fatigue process of magnesium alloy to explore entropy generation preferably. Figure 9 shows the change in plastic strain energy under different cyclic loadings. The plastic strain energy of magnesium alloy has the same evolution law under different cyclic loadings in the fatigue process, which is predominantly divided into the following three stages. The Stage I initial rapid increase stage which requires only thousand cycles(<1000). A large amount of plastic deformation occurs when cyclic stress is loaded into the static specimen suddenly, resulting in the rapid accumulation of plastic strain energy. Stage II is the decline stage. During the plastic deformation of metal, grain slip and dislocation entanglement occur, resulting in grain elongation, break, and breakage. Additionally, fibrosis and residual stress occur in metal–work hardening, thus slowly increasing the plastic strain. According to Equation (12), stress remains unchanged, strain increases slightly, and the work hardening index increases. Thus, the overall strain energy density curve shows a decreasing trend. Stage III is the stable stage. Dislocation stacking occurs when dislocations accumulate to a certain extent, and the degree of work hardening is stable. Thus, the plastic strain energy density remains almost constant. As shown in Figure 9, Stage III occupies most of the fatigue life of the material.

Thermodynamic entropy evolution of the magnesium alloy during the fatigue process

Thermodynamic entropy can represent the irreversible process and is related to the irreversible variables of fatigue. The total entropy of the system is composed of entropy generation and entropy flow. According to the calculation relationship among entropy generation, plastic strain density, and temperature, the evolution law of entropy generation is obtained in the fatigue process of magnesium alloy. Entropy generation represents the change rate of system disorder, and cumulative entropy generation represents the degree of disorder. Figure 10(a) shows the change in entropy generation under different cyclic loadings, and the same evolution law is present under different cyclic loadings. For the specific change trend, the initial entropy generation has an upward trend, a small downward trend, and a stable stage. This stage also accounts for most of the fatigue life. For the fatigue process with different cyclic loadings, a high cyclic loading results in high entropy generation. The entropy generation of stress below the fatigue limit is almost stable because internal friction plays a dominant role.

The cumulative entropy generation is obtained by accumulating the previous entropy generation, showing the irreversible characteristics of fatigue with increased cumulative entropy generation. The accumulation process of thermodynamic entropy in the fatigue process can be seen as a quasi linear accumulation process, as shown in Figure 10(b). When the specimen breaks, the cumulative entropy generation reaches the final value in the fatigue process, which is the FFE of the specimen. For the fatigue process with different cyclic loadings, a high cyclic loading, results in fast entropy accumulation rate, high slope of the cumulative entropy generation curve, and minimal time to reach the peak of cumulative entropy generation. At the same time, a large cyclic loading results in fast sample fractures, short entropy generation accumulation time, and small final FFE.

At present, the change law of internal entropy generation of AZ31 magnesium alloy has been explored. Entropy flow represents the rate of heat exchange between the system and the environment, that is, the impact of the environment on the system, and the influence of the external environment on the system will be discussed at room temperature. Figure 11(a) shows the entropy flow changes under different cyclic loadings. The entropy flow is determined by the temperature change across the boundary of the system [3]. Similar to the temperature evolution, entropy flow increases first, then falls, and finally stabilises. With increasing cyclic load, the overall entropy flow increases gradually. Compared with entropy generation, the entropy flow value is extremely small.

The cumulative entropy flow is accumulated to represent the magnitude of environmental impact qualitatively. The evolution trend of cumulative entropy flow shows two stages in Figure 11(b), which consists of two straight lines with different slopes. In the first stage, the entropy flow value is large, resulting in a large cumulative entropy flow slope. In the second stage, the entropy flow is stable, resulting in linear cumulative entropy flow. Compared with accumulated entropy generation, the accumulated entropy flow value is extremely small, which indicates that the environment has little effect on the sample at room temperature.

Fatigue damage of the AZ31B magnesium alloy based on thermodynamic entropy

Figure 12 shows the normalisation results of cumulative entropy generation, and the effects of loading amplitude and sample size are excluded. Normalisation results show a linear relationship between normalised entropy generation and normalised cycle times.

The normalised entropy generation and normalised cycle times are approximately linear [16]. (19) (20)

This shows that the process of metal fatigue damage can be described by such a curve regardless of the cyclic loading. Different from the Miner's linear damage rule [31], the life ratio is added simply. Although the cyclic loading is not directly reflected in the equation, the cumulative entropy generation can measure the irreversibility of the system process and the damage of the system in thermodynamics. The damage evolution of magnesium alloy under different cyclic loads is determined in accordance with Equation (20) [16], as shown in Figure 13. D is the damage, and Dc is the critical damage. Figure 13 shows that different load levels cause different entropy generation and entropy generation accumulation speeds and make the damage variable approach failure speed differently. A large load results in a short time for D to approach 1 and short fatigue life. Calculation results in Figure 13 show that when the fatigue life of the specimen is 90%, that is, when D is almost equal to 0.3, the material enters the stage of rapid crack growth before failure. In the last 10% of life, D increases almost steeply, and this phenomenon can play an early warning role for fatigue damage protection.

Prediction of fatigue life and fatigue limit based on the FFE model

As shown in the previous test, FFE can be obtained by the test. At the same time, the FFE of the model can be calculated in accordance with the method in Section 4.1, and material parameters and model details are shown in Table 2.

Table 2

Material parameters and model details.

_max (MPa)	p (g m⁻³)	c (J kg⁻¹K⁻¹)	a	b	d	f (Hz)	n ’	L (mm)	K (W/m K)	T (K)
150	1770	1000	0.47	−0.31	0.00041	10	0.24	40	100	299
140			0.46	−0.29	0.00048
130			0.27	−0.15	0.00049
120			0.16	−0.09	0.00046
110			0.15	−0.09	0.00012
100			0.09	−0.05	0.00002

Figure 14(a) shows the compared histogram of the results of the experimental FFE and the FFE model. The error between FFEs calculated by the test and the model is within 10%. Therefore, the FFE calculated by the model can predict the fatigue life effectively, thus connecting the cyclic load with fracture times. FFE is composed of plastic deformation (unrecoverable microstructure movement) and internal friction (caused by recoverable microstructure movement, such as the oscillation of dislocation rings) [27]. Plastic deformation contributes to fatigue damage. Therefore, many studies considered the influence of internal friction and subtracted the FFE value caused by internal friction during calculations. FFE increases slightly and can be roughly considered as constant [18,27,32]. Plastic deformation is dominant in the process of low-cycle fatigue, and the proportion of internal friction in high-cycle fatigue cannot be ignored, thereby providing a basis for exploring fatigue limit. Therefore, the FFE value caused by internal friction is not subtracted in the calculation of FFE in this paper. The FFE calculated by the model and the loading amplitude can be fitted into a linear relationship, as shown in the following equation and Figure 14(b). (21) The number of cycles and the FFE calculated by the model can be fitted into a linear relationship, as shown in the following equation and Figure 14(c). (22) By establishing the relationship between entropy generation and cyclic loading, an evident threshold, which can be defined as the fatigue limit, is observed to divide the curve into two segments, as shown Figure 15(a).

Under high cyclic stress, a large amount of rapid plastic deformation occurs in the material. Thus, the atomic disorder rate is inferred to be extremely large in the system, that is, the entropy generation value is extremely large. In the case of low stress, the entropy generation is almost completely caused by internal friction, and the entropy generation value is small and stable. An inflection point that can predict the fatigue limit is observed. As shown in Figures 15(a,b), the fatigue limits predicted by entropy generation and the curve of S–N are 102.25 and 96.65 MPa, respectively, and the error is 6.8%.

From the perspective of entropy generation composition, plastic deformation can be ignored for the alternating stress below the fatigue limit, because the mechanical behaviour of the material is predominantly related to the recoverable microstructure movement. The entropy generation under this stress condition is almost completely caused by internal friction, and the fatigue life is considered infinite. For alternating stresses above the fatigue limit, plastic deformation and fatigue damage occur. In this case, the entropy generation includes two parts: one is caused by internal friction, and the other is caused by plastic deformation [33]. The latter directly leads to increased disorder degree. At this critical value, the generation mechanism of entropy generation changes from internal friction to the comprehensive action of internal friction and plastic deformation.

Conclusion

In the process of fatigue, the evolution of entropy generation is divided into three stages: rising, falling and stable stages. With increased cyclic loading, the overall entropy generation increases. The cumulative entropy generation shows a quasi-linear relationship. The evolution trend of entropy flow is the same as that of temperature, and the cumulative entropy flow is small, suggesting that the external environment has little influence on the system at room temperature.

Damage presents an exponential evolution law in the fatigue process. In the first stage, damage accumulates slowly. When D reaches 0.3, the damage increases rapidly, indicating that the stage of rapid crack growth is reached.

The FFE model is established under fatigue loading, and the fatigue life is predicted by this model. The model error is within 10%. The fatigue limit measured by the entropy generation double line method is 102.25 MPa, and the error is 6.8%.

Footnotes

Disclosure statement

No potential conflict of interest was reported by the author(s).

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