Low Cycle Fatigue Life Assessment Based on the Accumulated Plastic Strain Energy Density

The accumulated plastic strain energy density at a dangerous point is studied to estimate the low cycle fatigue life that is composed of fatigue initiation life and fatigue crack propagation life. The modified Ramberg–Osgood constitutive relation is applied to characterize the stress–strain relationship of the strain-hardening material. The plastic strain energy density under uni-axial tension and cyclic load are derived, which are used as threshold and reference values, respectively. Then, a framework to assess the lives of fatigue initiation and fatigue crack propagation by accumulated plastic strain energy density is proposed. Finally, this method is applied to two types of aluminum alloy, LC9 and LY12 for low-cycle fatigue, and agreed well with the experiments.


Introduction
The failure of machinery under cyclic loading is a major engineering concern in practice, and the fatigue properties of metal has long been a focus [1,2]. Many models were established by using equivalent stress [3][4][5], J-integral [5] and strain energy density [6,7] as a failure criterion. Among them, the strain energy density criterion has attracted much attention because of its wide applicability.
On one hand, the use of the energy theory to study the fatigue goes back a long way. Most of the works have been conducted based on the total strain energy density (TSED), which has been widely used to evaluate material failure [8]. Oliferuk and Maj [9] described the energy storage during the plastic deformation based on the experimental results, and gave the corresponding mathematical expression of the plastic strain energy. Shahrooi et al. [10] studied the damage criterion based on plastic strain energy for ratcheting under multiaxial loading and predicted the fatigue lives with compression to experimental data. Wang et al. [11] introduced the main fatigue life prediction models, including the Manson-Coffin formula, Basquin formula and the strain damage model. It was pointed out that the coefficients of the fatigue life prediction formulae were dependent on the material properties. Therefore, the constant coefficients used in the Manson-Coffin formula and the strain damage model was not reliable. Sun et al. [12] carried out the fatigue test of the nickel alloy GH4169 at 650 • C under the combination of proportional and non-proportional tension and torsion loadings, thereby establishing the damage model which could be simplified to the uniaxial Manson-Coffin equation. Xu et al. [13] proposed a damage evolution model for low-cycle fatigue considering that the fatigue damage accumulation is mainly caused by cyclic plastic strain. Martins et al. [14] studied the low-cycle fatigue life of bainitic steels based on the cumulative strain energy density and developed a new predictive model to estimate the fatigue life. Roy and Song et al. [15,16] gave the expression of plastic strain energy for Masing material under cyclic loading. Then, the plastic strain energy density was derived both analytically and experimentally for different materials. From these works, the fatigue damage in low cycle fatigue is mainly caused by temperature [33] and strain rate [34] have confirmative influence on the relation between stress and strain, the macroscopic phenomenological models are still popular. Among these constitutive relations, the Ramberg-Osgood model [35] is widely used. Hertelé et al. [36] modified the Ramberg-Osgood constitutive model and proposed a constitutive model which was applicable to both aluminum alloy and stainless steel. Their study revealed that the modified model could be applied to other non-linear metallic materials with different parameters.

Modified Ramberg-Osgood Constitutive Equation
The modified Ramberg-Osgood constitutive relation could more precisely describe the nonlinear stress-strain relationship of strain-hardened ductile metals. The stress-strain curve is shown in Figure 1.

Plastic Strain Energy Density under Uni-Axial Tension
An accurate description of the constitutive relation is critical for the calculation accuracy of the plastic strain energy density. Although, factors like microstructure [32], temperature [33] and strain rate [34] have confirmative influence on the relation between stress and strain, the macroscopic phenomenological models are still popular. Among these constitutive relations, the Ramberg-Osgood model [35] is widely used. Hertelé et al. [36] modified the Ramberg-Osgood constitutive model and proposed a constitutive model which was applicable to both aluminum alloy and stainless steel. Their study revealed that the modified model could be applied to other non-linear metallic materials with different parameters.

Modified Ramberg-Osgood Constitutive Equation
The modified Ramberg-Osgood constitutive relation could more precisely describe the nonlinear stress-strain relationship of strain-hardened ductile metals. The stress-strain curve is shown in Figure 1. The modified Ramberg-Osgood stress-strain relation is expressed as follows: where E0 is the elastic modulus, E0.2 is tangent modulus at σ0.2 which is the nominal yield stress, n is the strain hardening index, ε0.2 is the total strain at yielding, σf is the true stress upon the damage of material under monotonic loading, εf is the true fracture strain under monotonic loading, m is the shape correction parameter of the stress-strain curve. m is related to the yield strength and fracture strength. According to the literature [35], the value of m may be taken as: The stress-strain relation expressed in Equation (1) could be presented by the curve shown in Figure 1 where εe is elastic strain, εp is the plastic strain, εft is the total fracture strain. The modified Ramberg-Osgood stress-strain relation is expressed as follows: where E 0 is the elastic modulus, E 0.2 is tangent modulus at σ 0.2 which is the nominal yield stress, n is the strain hardening index, ε 0.2 is the total strain at yielding, σ f is the true stress upon the damage of material under monotonic loading, ε f is the true fracture strain under monotonic loading, m is the shape correction parameter of the stress-strain curve. m is related to the yield strength and fracture strength. According to the literature [35], the value of m may be taken as: The stress-strain relation expressed in Equation (1) could be presented by the curve shown in Figure 1 where ε e is elastic strain, ε p is the plastic strain, ε ft is the total fracture strain.

Plastic Strain Energy Density
The plastic strain energy density (shaded area) should be the total strain energy minus the elastic strain energy density (the triangle on the right), as shown in Figure 1. Based on the constitutive equations Equations (1) and (2), the strain is used as the variable of integration. The accumulated plastic strain energy density at fracture under uni-axial tensile loading could be obtained. The loading curve is expressed by Equation (1). The unloading curve is ε = σ/E 0 + ε f . Since the strain is taken as the variable of integration, Equation (1) needs to be rewritten as the formula of stress in terms of strain. To do that, the strain is divided into elastic and plastic parts. Then the stress could be written in terms of plastic strain: The plastic strain and stress relationship is shown in Figure 2, where the shaded area is the plastic strain energy density that could be calculated by integrating Equation (3) with respect to strain, as shown in Equation (4): The loading curve is expressed by Equation (1). The unloading curve is ε = σ/E0 + εf. Since the strain is taken as the variable of integration, Equation (1) needs to be rewritten as the formula of stress in terms of strain. To do that, the strain is divided into elastic and plastic parts. Then the stress could be written in terms of plastic strain: The plastic strain and stress relationship is shown in Figure 2, where the shaded area is the plastic strain energy density that could be calculated by integrating Equation (3) with respect to strain, as shown in Equation (4): Two metals, LY12 and LC9, are used for subsequent theoretical analyses. The materials' parameters are listed in Table 1. By applying the method mentioned above, the parameters of the constitutive relation and plastic strain energy density under uniaxial tension could be obtained. Two metals, LY12 and LC9, are used for subsequent theoretical analyses. The materials' parameters are listed in Table 1. By applying the method mentioned above, the parameters of the constitutive relation and plastic strain energy density under uniaxial tension could be obtained. Table 1. The mechanical parameters for two materials [37]. In order to calculate the plastic strain energy density of the dangerous point, ε 0.2 , E 0.2 , m and other parameters should be determined first according to Equation (1). The first two could be get directly by E 0.2 = dσ/dε while σ = σ 0.2 and ε 0.2 = σ 0.2 /E 0.2 + 0.002. m is obtained from Equation (2). These parameters and the plastic strain energy density of the two materials are listed in Table 2. where ε 0.2 is the yielding strain, E 0.2 is the tangent modulus at σ 0.2 , m is the shape correction parameter.

Accumulated Plastic Strain Energy Density under Cyclic Loading
The plastic strain energy density accumulated at the dangerous point within a single cycle under uniaxial cyclic loading is considered. When applying the modified Ramberg-Osgood model, the unloading curve can be formulated by ε = (σ − σ a )/E 0 + ε a , as shown in Figure 3. The shaded area is the plastic strain energy density accumulated within one cycle.
In order to calculate the plastic strain energy density of the dangerous point, ε0.2, E0.2, m and other parameters should be determined first according to Equation (1). The first two could be get directly by E0.2 = dσ/dε while σ = σ0.2 and ε0.2 = σ0.2/E0.2 + 0.002. m is obtained from Equation (2). These parameters and the plastic strain energy density of the two materials are listed in Table 2.

Accumulated Plastic Strain Energy Density under Cyclic Loading
The plastic strain energy density accumulated at the dangerous point within a single cycle under uniaxial cyclic loading is considered. When applying the modified Ramberg-Osgood model, the unloading curve can be formulated by ε = (σ − σa)/E0 + εa, as shown in Figure 3. The shaded area is the plastic strain energy density accumulated within one cycle. Then, the formula of single-cycle plastic strain energy density is expressed as follows: when, Then, the formula of single-cycle plastic strain energy density is expressed as follows: when, 0.002 < ε pa ≤ ε f , where ε pa is the plastic strain amplitude of the unloading/reloading. σ' f is the fatigue strength coefficient. ε' f is the fatigue ductility coefficient. m' is the shape parameter under cyclic loading. σ' 0.2 is the yield stress under cyclic loading. ε' 0.2 is the total strain under cyclic loading when material yield. E' 0.2 is tangent modulus at σ' 0.2 .
To calculate the plastic strain energy density accumulated in one cycle for the two materials LY12 and LC9, the fatigue parameters required are listed in Table 3. Table 3. Fatigue parameters of LY12 and LC9 [37]. In order to apply the modified Ramberg-Osgood stress-strain relation to calculate the plastic strain energy density, the same method should be adopted for the parameters of Equations (5) and (6) as in the previous section: The results are listed in Table 4. According to the fatigue parameters of the two materials and calculation parameters given by Table 4, Equation (6) could be used to calculate the accumulated plastic strain energy densities of a single cycle for the two materials above under the cyclic loading. The results are listed in Table 5.

Fatigue Life Estimation
In this paper, we try to establish a method for low cycle fatigue life assessment from the energy dissipation point of view; that is, along with the energy dissipation caused by plasticity, the material damages and fractures. For this purpose, two hypotheses should be addressed: (1) Accumulated plastic strain energy density causes the damage and fracture of material; (2) Material damages or fractures when the accumulated plastic strain energy density reaches a critical value.
From here on, these hypotheses are applied to assess the fatigue life of metals. The critical value of the accumulated plastic strain energy density could be obtained by uni-axial tension. Despite the plastic strain energy densities under the uni-axial tension and cyclic loading are different, they are related to each other closely since both of them represent the energy dissipation and material degradation. It should be pointed out that the fatigue failure study does not mean the fracture of the whole material, but the failure at the dangerous point, that leads to the material separation (or forming micro cracks). Therefore, the linear cumulative damage theory based on plastic strain energy density is used to predict the fatigue life of the material under cyclic loading.
The process of fatigue is divided into two stages, fatigue crack initiation and fatigue crack propagation. The first stage is the process from damage (micro crack nucleation) to a macro crack formed, and the second stage is the crack stable propagation to final fracture.

Estimation of Crack Initiation Life
With applying the modified Ramberg-Osgood relationship, the plastic strain energy density in a single cycle could be formulated from Equation (5) as: From Equation (8), the relation between fatigue life and plastic strain range could be established as follows: where N is the number of cycle, ∆ε p is the change of plastic strain. Equation (9) is derived by applying the linear damage accumulation principle which considers the plastic strain energy density accumulated in every cycle is a constant. Furthermore, when subjected to cyclic loading with non-zero mean stress, the stress-strain curve of material in one cycle is almost antisymmetric, indicating the area enclosed in a loop's curve is almost twice the area above the mean stress. This is the reason why the ε pa is replaced by 2 × ∆ε p in Equation (9). Then the relation between the plastic strain energy density and the number of cycles is expressed as follows: It can be seen from the above equation that the cyclic plastic strain energy density range has an exponential relationship with the fatigue life, which is denoted as: where α and b are constants. Equation (11) shows the relationship between the single cycle plastic strain energy density and fatigue life.
According to the linear damage accumulation principle that assumes the plastic strain energy density accumulated in every cycle is not changed with the increase of cycle times: Substituting Equation (11) to Equation (12): Meanwhile, the total plastic strain energy density under the uni-axial tension could be regarded as a special case of fatigue loading, that is, the corresponding total plastic strain energy density when the fatigue life is 1. Thus, constant α should be equal to the plastic strain energy density calculated in the uni-axial tension conditions in this article. Then the constants b could be fitted from Equation (11).

Estimation of Crack Propagation Life
For brittle materials, the fracture criterion is given by the Griffith criterion.
πa (14) where, σ f is the nominal fracture stress, γ is the surface free energy and a is the crack length. The elastic strain energy should decrease accompanying the crack propagating and the reduction of the elastic strain energy should reach the energy needed to form a new crack surface at least. If the specimen is subjected to two different stress aptitudes σ 1 and σ 2 , respectively, the relation of critical crack length a 1 and a 2 can be deduced from Equation (14) as: When the stress is small, the final fracture of fatigue is caused by a longer crack in advance, and vice-versa.
Equation (14) describes a fracture criterion not only for fatigue loading but also for uni-axial tensile loading. However, the mechanism and the process of reaching breaking point is essentially different. Under uni-axial tensile loading, the stress increases gradually before the fracture occurs. While under fatigue loading, the stress increases due to the gradual expansion of the crack, and the bearing area is decreased. This indicates that crack propagation can produce the critical stress causing the final fracture. In the following discussion, the cyclic plastic strain energy will be used as a criterion for crack propagation.
There exists a plastic zone at the crack tip if the material with a crack is loaded. The energy required for the plastic deformation near the crack depends on the volume of the plastic zone and the distribution of the strain in the plastic zone. It is assumed that the volume of the plastic deformation zone is proportional to γ sub square of the crack length. Then, when two cracks with different length expand, the relationship between the plastic strain energy density required by the expansion is [38]: To simplify the fatigue problems, the following approximate simplification is proposed in this paper. Let σ f and W f represent the true fracture stress and critical plastic strain energy density corresponding to the uniaxial tensile fracture, respectively. σ a and W p denote the cyclic stress amplitude and the total cyclic plastic strain energy density corresponding to the fatigue fracture, respectively. From Equations (15) and (16), we have: For the calculation of total plastic strain energy density under fatigue fracture, the linear damage accumulation principle is adopted. It is assumed that the plastic strain energy density accumulated in every cycle is not changed with the increase of cycle times, that is, the hysteresis loops of the material at different cycles are the same. The cumulative plastic strain energy density of each cycle is ∆W p . The total cyclic plastic strain energy density is: By substituting Equation (17) into Equation (18), the formula of the fatigue crack growth life can be obtained: According to the modified Ramberg-Osgood constitutive relation, the plastic strain energy density of a single cycle is ∆W p , as shown in Equations (5) and (6). The total cyclic plastic strain energy density W f can be obtained based on Equation (4). Then the life of fatigue crack growth can be obtained by Equation (19).
For a given material, the γ value could be determined. Then the fatigue life of materials under the given stress is estimated. According to Equation (17), σ a -W p line could be drawn on the double logarithm coordinate to check and calculate the value of γ. The line passes through W f and σ f point, its slope equal to the exponent of the equation, −1/2γ. For most metals, the slope is between −1/8 and −1/3 [38].

Examples
The modified Ramberg-Osgood stress-strain relationship is used in the study. Given the stress ratio of 0, the fitting is conducted from the relationship between fatigue life and accumulative plastic strain energy density for the two materials LY12 and LC9, as shown in Figures 4 and 5. The constants α and b could be numerically fitted. By fitting, α = 106.94 and b + 1 = 0.2927 could be obtained for LY12 material, and α = 200.42, b + 1 = 0.2147 for LC9 material.
fatigue crack growth can be obtained by Equation (19).
For a given material, the γ value could be determined. Then the fatigue life of materials under the given stress is estimated. According to Equation (17), σa-Wp line could be drawn on the double logarithm coordinate to check and calculate the value of γ. The line passes through Wf and σf point, its slope equal to the exponent of the equation, −1/2γ. For most metals, the slope is between −1/8 and −1/3 [38].

Examples
The modified Ramberg-Osgood stress-strain relationship is used in the study. Given the stress ratio of 0, the fitting is conducted from the relationship between fatigue life and accumulative plastic strain energy density for the two materials LY12 and LC9, as shown in Figures 4 and 5. The constants α and b could be numerically fitted. By fitting, α = 106.94 and b + 1 = 0.2927 could be obtained for LY12 material, and α = 200.42, b + 1 = 0.2147 for LC9 material.   According to the known parameters of LY12 and LC9, the relationship between stress and the total accumulated plastic strain energy density could be fitted. These parameters conform to the linear relation in the double logarithmic coordinates. The fitted constant γ is expressed as: −1/2γ = −0.2105 for LY12, and −1/2γ = −0.3348 for LC9. By substituting these parameters into Equation (19), the crack propagation life of the corresponding material could be obtained.
According to the α, b and γ determined above, Equations (11) and (19) are used to predict the fatigue crack initiation life and propagation life under different loading conditions, with the results shown in Tables 6 and 7.
To verify the validation of current approach, experimental results are necessary. Table 6. Life estimation of LY12. According to the known parameters of LY12 and LC9, the relationship between stress and the total accumulated plastic strain energy density could be fitted. These parameters conform to the linear relation in the double logarithmic coordinates. The fitted constant γ is expressed as: −1/2γ = −0.2105 for LY12, and −1/2γ = −0.3348 for LC9. By substituting these parameters into Equation (19), the crack propagation life of the corresponding material could be obtained.
According to the α, b and γ determined above, Equations (11) and (19) are used to predict the fatigue crack initiation life and propagation life under different loading conditions, with the results shown in Tables 6 and 7.
To verify the validation of current approach, experimental results are necessary.

Experimental Verification
The first fatigue test was carried out on the LC9 aluminum alloy using the PA-100 electro-hydraulic servo fatigue testing machine. The specimens were machined to the funnel-shaped round bar. The loading conditions were 8 ± 4 kN, 8 ± 5 kN and 8 ± 6 kN, respectively. The maximum values of σ max loaded were 71.12%, 77.04% and 82.97% of the yield stress σ 0.2 , respectively. The loading frequency was 10 Hz. According to the loading and unloading curves in the experiment, the stress-strain curve within one cycle is used to calculate the enclosed area, and thus the accumulated plastic strain energy in one cycle is obtained. The formula established in this paper was used to predict the fatigue life and compared with the experimental results. The results are shown in Table 8. The second fatigue test was carried out on the LY12 aluminum alloy. The loading conditions were 6 ± 4 kN, 6 ± 5 kN and 6 ± 6 kN, respectively. The maximum values of σ max loaded were 73.75%, 81.04% and 88.54% of the yield stress σ 0.2 , respectively. The comparison between the proposed method and the experimental results is shown in Table 9.
The data in Tables 8 and 9 indicate that the fatigue life estimated by the formula of this article is close to the experimental value. The calculation and experimental error is caused by the factors such as internal defects of materials. The fatigue life estimated by the formula of the current model is closer to the experimental value compared with the result by the Manson formula.  In general, the predicted results agree well with the experiments. The average errors between the proposed method and experiments are 14.7% and 20.6% in Tables 10 and 11, respectively. In addition, the experimental results are always smaller than calculated values. It can be understood that there must exist some defects or inhomogeneities in the test specimen, and surface defects from manufacturing are hard to avoid.

Conclusions
In this paper, a model to assess the low cycle fatigue life based on the accumulated plastic strain energy density is proposed. In this model fatigue is divided into two stages, fatigue initiation and fatigue propagation. Both stages are evaluated by the accumulated plastic strain energy density. The experimental results agree well with the current model on fatigue life, implying the validity of this model.
In particular, the threshold value of accumulated plastic strain energy density is defined by the static tensile experiment, bridging the fatigue life to a simple standard test and clarifying the threshold value of the fatigue parameter which has long been a thorny issue. The results confirm the feasibility of this hypothesis.