0 Introduction

As well known, a fatigue life assessment of metals is usually divided into two types: low cycle fatigue (LCF)and medium/high cycle fatigue (HCF). Concerning HCF fatigue assessment, the Wöhler Curve Method is usually used. The Modified Wöhler Curve Method proposed by Susmel and Lazzarin ^[1], for example, is used to perform the HCF assessment of metals for conventional mechanical components subjected to multiaxial fatigue loading. While for LCF of metals, no doubt, the Coffin-Manson Curve Method is used to perform fatigue life assessment. The Modified Manson-Coffin Curve Method by Susmel et al.^[2], for example, is employed to evaluate LCF lifetime of metals under multiaxial fatigue loading.

A key to fatigue life evaluation of metals is in substance that, on the basis of fatigue test data of metallic specimens and mechanical analyses for those specimens, a proper stress intensity parameter, S, is taken to establish a relationship between the stress intensity parameter and the fatigue life, N. For example, the Wöhler Curve Method and the Coffin-Manson Curve Method are two typical examples of the relationship. When a different stress intensity parameter, S, is taken, naturally, a different relationship equation is obtained, which has usually different accuracy and efficiency when to be employed to perform fatigue life assessment. Both accurate and high efficient fatigue life evaluation equation is to be searched for by all field investigators.

Recently, a description on a practicability of the Wöhler Curve Method for low cycle fatigue of metals was given by Yan ^[3]. By the description and the low cycle fatigue test data of 16MnR steel ^[4], it is important to show that, for low cycle fatigue of the metallic material, such a way that a stress-based intensity parameter calculated by the linear-elastic analysis is taken to be a stress intensity parameter, S, to establish a relationship between the stress intensity parameter, S, and the fatigue life, N, is practicable. In this work, many metallic materials from the literature are given to show that the Wöhler Curve Method is well suitable for low cycle fatigue analysis of metals.

1 Experimental Verifications

In order to well show that the Wöhler Curve Method is well suitable for low cycle fatigue analysis of metals, "Uniaxial fatigue assessment according to Manson and Coffin's idea" appearing in Ref.[5] is rewritten as follows:

The strain-based approach postulates that fatigue lifetime can be estimated accurately by simultaneously considering the contributions to fatigue damage of both the elastic and plastic parts of the total strain amplitude (Coffin ^[6]; Manson ^[7]; Morrow ^[8]).

Consider the plain specimen sketched in Fig. 1 and assume that it is subjected to fully reversed uniaxial fatigue loading. According to Basquin's idea^[9], the relationship between the stress amplitude, σ_{x, a}, and the number of reversals to failure, 2N_f, can be expressed as follows:

$ \sigma_{x, a}=\sigma_{\mathrm{f}}^{\prime}\left(2 N_{\mathrm{f}}\right)^b $

(1)

where the fatigue strength coefficient σ^'_f, and the fatigue strength exponent b are material constants to be determined by running appropriate experiments.

By taking full advantage of Eq. (1), it is trivial to calculate the amplitude of the elastic part of the total strain ε_{x, a}^e, as the number of reversals to failure N_f increases, that is:

$ \varepsilon_{x, a}^e=\frac{\sigma_{x, a}}{E}=\frac{\sigma'_{\mathrm{f}}}{E}\left(2 N_{\mathrm{f}}\right)^b $

(2)

Similarly, the relationship between the plastic strain amplitude, ε_{x, a}^p, and N_f can be expressed as follows:

$ \varepsilon_{x, a}^p=\varepsilon^{'}_{\mathrm{f}}\left(2 N_{\mathrm{f}}\right)^c $

(3)

where the fatigue ductility coefficient ε^'_f, and the fatigue ductility exponent c, are also material constants to be determined experimentally.

Finally, if the elastic contribution to the overall fatigue damage is added to the corresponding plastic contribution, the relationship between the total strain amplitude ε_{x, a}, and the number of reversals to failure can be written directly in the following explicit form:

$ \varepsilon_{x, a}=\varepsilon_{x, a}^e+\varepsilon_{x, a}^p=\frac{\sigma_{\mathrm{f}}^{\prime}}{E}\left(2 N_{\mathrm{f}}\right)^b+\varepsilon_{\mathrm{f}}^{\prime}\left(2 N_{\mathrm{f}}\right)^c $

(4)

This relationship is usually called the Manson-Coffin equation.

In this paper, Eq. (1) or (2) is called the Basquin's equation.

Here, a comparison of the Wöhler equation with the Basquin's equation is given below:

Eq.(1) is rewritten as:

$ \lg \left( {{\sigma _{x, a}}} \right) = b\lg \left( {{N_{\rm{f}}}} \right) + \lg \left( {{{\sigma '}_{\rm{f}}}} \right) + b\lg \left( 2 \right) $

(5)

Here, the Wöhler equation is usually written as:

$ \lg \left( {{\sigma _{x, a}}} \right) = {A_1}\lg \left( {{N_{\rm{f}}}} \right) + {C_1} $

(6)

Obviously, Eq.(6) is the same as Eq.(1) by letting

$ A_1=b, \quad C_1=\lg \left(\sigma_{\mathrm{f}}^{'}\right)+b \lg (2) $

(7)

By comparing the Wöhler equation with Basquin's equation, thus, it is seen that the Wöhler equation is the same as Basquin's equation expressed in terms of Eq.(1) or (2). Based on the same, it is concluded that, in performing low cycle fatigue analysis of a metallic material by the Manson-Coffin Curve Method, if the relationship between σ_{x, a} and N_f is well revealed by the Basquin's Eq. (1), or if the relationship between ε_{x, a}^e and N_f is well revealed by using the Basquin's Eq. (2), low cycle fatigue analysis of the metallic material can be well calculated by using the Wöhler equation. Here, a metallic material with well linear fitting relationship between lg(σ_{x, a}) and lg(N_f) or lg(ε_{x, a}^e) and lg(N_f) is called the Basquin's material. Obviously, for the Basquin's material, its LCF analysis is well calculated by the Wöhler equation, which will be illustrated by means of the following examples from the literature.

In order to illustrate that the low cycle fatigue life of Basquin's material is to be well predicted by the Wöhler equation, we searched for relevant studies on low cycle fatigue of metals. Here, such studies are divided into three types: Type Ⅰ, Type Ⅱ, and Type Ⅲ.

1.1 Type Ⅰ

Both low cycle fatigue test data of a metallic material and Strain-life curves obtained by using the Manson-Coffin equation are included in the literature. Then the Basquin's curve in the Strain-life curves is used to show that the metallic material is the Basquin's material, its LCF life is to be well predicted by the Wöhler equation, and at the same time, low cycle fatigue test data of the metallic material are analyzed by using the Wöhler equation to further show that LCF life of the metallic material can be calculated by the Wöhler equation. Such type materials include EN AW-2007 aluminum alloy^[10] and EN AW-2024-T3 aluminum alloy^[11].

1.1.1 Example 1: Low cycle fatigue of EN AW-2007 aluminum alloy

Here, low cycle fatigue test data (see Table 1 and Table 2) of EN AW-2007 aluminum alloy reported by Szusta and Seweryn ^[10] is employed to show that, for the Basquin's material, its low cycle fatigue life is to be well predicted by the Wöhler Curve Method. Figs. 2, 3 and 4 illustrate the monotonic tension curve, the fatigue life strain curve for the symmetrical tension-compression, the fatigue life curve for symmetrical torsion, respectively. From Figs. 3 and 4, it can be seen that low cycle fatigue life of the metallic material is characterized by using the Manson-Coffin equation, and that low cycle fatigue life of the metallic material is characterized also by the Basquin's equation because the relationship between ε_a^e and N_f is well described by Basquin's equation. Thus the metallic material is the Basquin's material.

Here, an attempt is made that the Wöhler Curve Method is employed to analyze low cycle fatigue test data of EN AW-2007 aluminum alloy. The obtained results are given in Tables 1-3 and Fig. 5. From Fig. 5, it can be seen that the test fatigue lives are in good agreement with those predicted by the Wöhler curve Method, with error indexes : M.E.=1.0%, E.R.=[-21.4, 21.1]%, and M.E.=10.7%, E.R.=[-47.7, 87.8]% for tension and torsion fatigue, respectively. R.E., M.E., and E.R. are the abbreviation of relative error, mean error and range of error, respectively.Thus, it can be concluded that for the Basquin's material, EN AW-2007 aluminum alloy, its low cycle fatigue life is to be well calculated by the Wöhler Curve Method.

1.1.2 Example 2: Low cycle fatigue of aluminum alloy EN AW-2024-T3

Here, low cycle fatigue test data (as shown in Table 4 and Fig. 6) of aluminum alloy EN AW-2024-T3 at room temperature(RT) and elevated temperature reported by Szusta and Seweryn ^[11] is employed to further show the author's viewpoint that, for Basquin's material, its low cycle fatigue life is to be well predicted by the Wöhler Curve Method. Fig. 7 is a comparison of monotonic tension curves of EN AW-2024T3 aluminum alloy at 20°, 100°, 200°, 300°.

From Fig. 6, it can be seen that low cycle fatigue life of the metallic material is characterized by using the Manson-Coffin curve, and that low cycle fatigue life of the metallic material is characterized also by the Basquin's equation because the relationship between ε_a^e and N_f is well expressed by Basquin's equation. Thus the metallic material is Basquin's material.

Here, low cycle fatigue test data (see Table 4) of aluminum alloy EN AW-2024-T3 at room temperature (RT) and elevated temperature reported by Szusta and Seweryn ^[11] are analyzed by using the Wöhler Curve Method. The obtained results are given in Table 4, see also Fig. 8. From Fig. 8, it can be seen that the test fatigue lives are in good agreement with those calculated by the Wöhler Curve Method, with error indexes: M.E.=3.4%, E.R.=[-22.2, 46.2]%; M.E.=2.7%, E.R.=[-38.1, 23.5]%; M.E.=2.5%, E.R.=[-27.5, 38.7]% and M.E.=5.8%, E.R.=[-37.3, 58.2]% for 20°, 100°, 200°, and 300°respectively.

Table 5 gives the fitting constants in the Wöhler equation of aluminum alloy EN AW-2024-T3. It can be seen from the various values of R square shown in Table 5 that low cycle fatigue life of the metallic material is well calculated by using the Wöhler Curve Method.

1.2 Type Ⅱ and Type Ⅲ

Type Ⅱ: Low cycle fatigue test data of a metallic material are given but Strain-life curves are not reported in the literature. Then low cycle fatigue test data of the metallic material are analyzed by using the Wöhler equation to show that low cycle fatigue life of the metallic material can be computed by the Wöhler equation. Such type metals include A533B ^[12], Inconel 718^[13], SAE1045^[14], AISI304^[15], Ti-6Al-4V^[16], 6061-T6^[17], 1Cr-18Ni-9Ti^[18], AISI304^[19], AISIH11^[20], 34CrNiMo6^[21], RAFM steels ^[22]. The details are shown in Appendix A.

Type Ⅲ: Strain-life curves of a metallic material are reported but low cycle fatigue test data are not given in the literature. Then the Basquin's curve in the Strain-life curves is used to show that the metallic material is the Basquin's material, its low cycle fatigue life is to be well predicted by Wöhler equation. Such type metallic materials include: P92 ferritic-martensitic steel ^[23], Cr-Mo-V low alloy steel ^[24], S35C carbon steel and SCM 435 alloy steel ^[25-26], high strength spring steel with different heat-treatments ^[27], 316 L(N) stainless steel at room temperature^[28], CLAM steel at room temperature^[29]), Eurofer97^[30], F82H (a ferritic-martensitic steel)^[31], En3B (a commercial cold-rolled low-carbon steel)^[32], nodular cast iron^[33] and low carbon grey cast iron^[34]. The details are reported in Appendix B.

2 Conclusions and Further Work

From this work, the conclusions can be made: Wöhler Curve Method is well suitable for LCF life analysis of metallic materials, in which the stress-based intensity parameter calculation is on the basis the linear-elastic analysis without the need for carrying out complex and time-consuming the elastic-plastic analysis.

In view of the fact that medium/high cycle fatigue lives of metallic materials are well calculated by the Wöhler Curve Method, thus, from this work it appears to be possible that the Wöhler Curve Method is well suitable for fatigue life assessment for low/medium/high cycle fatigue of metallic materials. This work will be reported in another paper.

Appendix A

Experimental investigations: The Wöhler Curve Method is well suitable for low cycle fatigue life analysis of metals by using low cycle fatigue test data of the metals from the literature.

Here, low cycle fatigue test data of metals reported in the literature are analyzed by the Wöhler Curve Method to show that low cycle fatigue life of the metals can be calculated by the Wöhler Curve Method. A summary of the obtained results is given in Table A-1, the test data of different metals and the corresponding results are shown in Tables A-2 to A-14. Comparison of the test low cycle fatigue lives with those calculated by the Wöhler Curve Method are shown in Figs. A-1 and A-2. Such type materials are A533B ^[12](Table A-2), Inconel 718^[13](Table A-3), SAE 1045 ^[14](Table A-4), AISI 304 ^[15] (Table A-5), Ti-6Al-4V^[16](Table A-6), 6061-T6 ^[17] (Table A-7, Table A-8), 1Cr-18Ni-9Ti ^[18] (Table A-9, Table A-10), AISI 304 ^[19] (Table A-11), AISI H11 ^[20](Table A-12), 34CrNiMo6^[21](Table A-13), RAFM steels ^[22](Table A-14).

From Tables A-1 to A-14 and Figs. A-1 and A-2, it can be seen that the test low cycle fatigue lives of the metals are in good agreement with those calculated by the Wöhler Curve Method, which illustrates that low cycle fatigue life of the metals can be well calculated by the Wöhler Curve Method.

Appendix B

Experimental investigations: The Wöhler Curve Method is well suitable for low cycle fatigue life analysis of metals by using the Basquin's curve in the Strain-life curves of metals from the literature.

Here, the Basquin's curve in the Strain-life curves of a metallic material reported in the literature is employed to show that the metallic material is the Basquin's material, its low cycle fatigue life is to be well predicted by the Wöhler Curve Method. Such type materials are P92 ferritic-martensitic steel^[23](as shown in Fig.B-1), Cr-Mo-V low alloy steel^[24](as shown in Fig.B-2), S35C carbon steel and SCM 435 alloy steel ^[25-26](as shown in Fig.B-3), high strength spring steel with different heat-treatments^[27](as shown in Fig.B-4), 316 L(N) stainless steel at room temperature^[28](as shown in Fig.B-5), CLAM steel at room temperature^[29])(as shown in Fig.B-6), Eurofer97^[30](as shown in Fig.B-7), F82H (a ferritic-martensitic steel)^[31](as shown in Fig. B-8), En3B (a commercial cold-rolled low-carbon steel)^[32](as shown in Fig.B-9), nodular cast iron^[33](as shown in Fig. B-10) and low carbon grey cast iron^[34](as shown in Fig. B-11).

From Figs. B-1 to B-11, it can be seen that all Basquin's curves have well linear fitting relationship between lg(ε_{x, a}^e) and lg(N_f). Thus all the metals studied are the Basquin's materials, then their low cycle fatigue lives can be calculated by the Wöhler Curve Method.

From Fig.B-1, it can be seen that there is no well linear fitting relationship between lg(ε_{x, a}^p) and lg(N_f). So low cycle fatigue life of P92 ferritic-martensitic steel appears to be not well calculated by the Manson-Coffin equation.

From Fig.B-3, it can be seen that, from low-cycle to high-cycle fatigue of the two materials, the relationship between ε_a^e and N_f is expressed by the Basquin's equation. Thus it is very possible that the fatigue life of the two materials, from low cycle to high cycle fatigue, is well calculated by the Wöhler Curve Method.

In addition, it can be seen from Fig. B-10 (Strain-life curves of nodular cast iron) that, from low-cycle to high-cycle fatigue of the material at room temperature, the relationship between ε_a^e and N_f is expressed by Basquin's equation. Thus it is very possible that the fatigue life of the material at room temperature, from low cycle to high cycle fatigue, is well calculated by the Wöhler equation.