The material used in this study was 10CrNi3MoV high-strength steel, which was received in the quenched and tempered condition. Butt-welded joints with a V-groove were fabricated by Single Pulsed Gas Metal Arc Welding (SP-GMAW) processing for evenmatched configuration using corresponding wire (wire diameter Φ 1.2 mm). In contrast, Gas Metal Arc Welding (GMAW) processing was performed for undermatched welded joints by lower strength filler metal (wire diameter Φ 1.2 mm). The chemical compositions of the evenmatched and undermatched filler material are presented in Table 1.

The LCF tests were performed using the Instron-8802 hydraulic servo system machine, presented in Figure 1. The fatigue specimens are cut by wire cutting machine and manufactured by general mechanical equipment, and the detailed dimensions were given in Figure 1. It should be noticed that the strain gauge system kept running and collecting data throughout the cycling tests. To eliminate the potential residual stress in welded specimens, all of the samples, including the base metal specimens, were subjected to heat treatment at 550°C for 4 h and cooling in the furnace. Figure 2 presents the electron backscattered diffraction (EBSD) maps, including normal direction–inverse pole figure (IPF-Z) maps, grain orientation spread (GOS, 0–6°) maps, and kernel average misorientation (KAM, 0–5°) maps, of the as-received base metal and its undermatched weldments. The GOS and KAM values can be used to indicate the local residual plastic deformation (higher GOS and KAM values indicate higher strain level) as they count the local crystallographic misorientation in the grains. Therefore, these EBSD maps can visualize strain distribution in the scanned area (Liu et al., 2019). The base metal shows a dominating martensitic structure with a large number of laths and a relatively higher residual strain level, while the weldment has a more equiaxed grain shape and a relatively lower level of deformation. The small micron-sized blue grains in Figure 2E show a misorientation-free (deformation-free) structure, as can also be read from Figure 2F, which might be the embryos for recrystallization during the welding procedure.

The basic mechanical properties have been given by Song et al. (2018b). Figure 3 shows the monotonic (blue dash lines) and uniaxial cyclic (red solid lines) stress–strain curves under symmetrical strain-controlled cycling conditions. It can be seen that the ratio of 0.2% offset yield strength to ultimate tensile strength is higher than 0.9 for 10CrNi3MoV steel and the undermatched welds. From the stress–strain comparison of monotonic and cyclic loadings, both of them display cyclic softening behavior. For the base metal, it exhibits prominent softening characteristics under each strain ratio. While the undermatched welds reveal significant softening behavior below 0.8% strain amplitudes, in over 0.8% strain amplitudes, the monotonic and uniaxial cyclic stress–strain values seem to be consistent. Therefore, the monotonic and cyclic mechanical match ratios of base metal and weldments exist discrepancy. Table 2 summarized the cyclic strain components and stress results of fatigue tests carried out with smooth specimens for both materials under strain-controlled conditions (Song et al., 2018b).

The Lemaitre-Chaboche model (Chaboche, 1989) is often used to characterize the nonlinear mixed kinematic hardening behavior. As implemented in the ABAQUS FE code, the isotropic and nonlinear kinematic parts of a cyclic hardening model describe the yield surface’s translation in the stress space by the backstress tensor α. The function of the yield surface can be expressed as follows: f ( σ − α ) = σ 0 , (1) where σ is the stress tensor, σ ⁰ is the radius of the yield surface, and f ( σ − α ) can be shown as follows: f ( σ − α ) = 3 2 ( S − α d e v ) : ( S − α d e v ) , (2) here α^dev strands for the deviatoric part of the back stress tensor and S is the deviatoric stress tensor.

On the other hand, the kinematic hardening components can be defined by combining a purely kinematic term and a relaxation term, which is shown as the following equation: α = ∑ i [ C i 1 σ 0 ( σ − α ) ε p − γ i α ε p ] , (3) where C _i and γ_i are the material constants obtained from the cyclic experimental data, C _i presents the initial kinematic hardening modulus, and γ _i determines the variation rate of kinematic hardening modulus with the increasing plastic deformation. ε^p is the equivalent plastic strain rate. The deviatoric term α expresses the deviatoric part depending on the material hardening behavior. i is the number of decomposed A-F rules utilized, which is set to five in this study. The effective cumulative plastic strain ε^p can define the isotropic hardening components: σ 0 = σ | 0 + Q inf ( 1 − exp ( − b ⋅ ε p ) ) , (4) where σ | 0 is the yield stress at the zero plastic strain, Q inf denotes the maximum change in the size of the yield surface, and b refers to the rate at which the size of the yield surface changes as plastic straining evolution. These two material parameters well describe the nonuniformity of the isotropic hardening through loading cycles of the material. The Bauschinger effect, which is demonstrated with the yield surface transition, can be illustrated by the backstress tensor under cyclic loading.

According to the kinematic hardening theory, the corresponding parameters are fitted from the cyclic stress–strain curves of base metal and undermatched welds from the initial cycle to the 0.5N _f cycle, which is given in Table 2. It should be noted that these plastic parameters can be used to characterize the different strain amplitudes in the range of strain amplitudes from 0.2 to 1.2%. This model has robust advantages of calculating the cyclic material behavior by the fitting parameters, which can be derived from uniaxial cyclic test data. However, some limitations of the kinematic hardening model have been mentioned for the residual stress simulation (Smith et al., 2019).

Although the isotropic hardening parameters Q _inf and b are fitted from a single stress–strain curve, they are applied for all strain ranges. There are no significant cyclic hardening variations with the changing of strain ranges for structural steels; however, it can simulate the cyclic plastic behavior representing the real strain ranges. On the other hand, the kinematic parameters C _i and γ_i are obtained from a single cyclic response of the material presented in Table 3. It cannot characterize the monotonic and cyclic response simultaneously. Thus, these parameters can only be used to fit the cyclic mechanical behaviors of the materials.

For LCF, the damage development initiates once the incubation period has been reached. It means the materials will be subjected to cyclic plastic strain when the nominal loading levels exceed the yield stress. However, it is difficult to measure the material damage under cyclic tests by nondestructive direct measurements. From the perspective of phenomenological observation, the damage measurement technology proposed by Lemaitre and Desmorat (Lemaitre, 1985; Lemaitre and Desmorat, 2005) can be employed to quantify the isotropic damage based on the relationship between elasticity, plasticity, and damage, which is expressed in an inverse form as follows: D i = 1 − E i E ′ , (5) D i = 1 − σ i σ ′ . (6)

These equations can be demonstrated by Eq. 5, 6 in Figure 4. The damage evolution of material under cyclic loading is measured by Young’s modulus variation. Thus, the D _i represents the ith cycle, the E′ is the initial Young’s modulus, and E _i is the apparent Young’s modulus of the ith cycle. Considering the damage from the effective stress, the σ is the damaged stress tensor, and the σ′ strands for the initial stress tensor from the stabilized cycle. In our study, the damage expresses are based on Young’s modulus evolution.

Regarding the determination of fatigue damage by strain energy density parameter, the material energy at any moment during the strain-controlled cycling period can be expressed as follows, which is similar to Eq. 5, 6: D i = 1 − Δ w i Δ w 0 , (7) where Δw ₀ is the hysteresis energy density after cyclic softening and Δw _i is the apparent hysteresis energy density of the ith cycle.

To predict the material total fatigue life, we adopt the continuum mechanics theory proposed by Darveau (Darveau, 2002) and Lau et al. (Lau et al., 2002) for the LCF calculation in ABAQUS. The FE model of a cyclic tensile bar is established in this software. The direct cyclic analysis setting in the software is used to realize the cycle-by-cycle iteration. A 2D axisymmetric model with a 3.5 mm radius and 6.25 mm height was meshed using CAX4R elements to calculate damage results, which is shown in Figure 5. The displacement-controlled loading boundary is applied to the top part of the model corresponding to the mechanical strain range. The asymmetrical boundary of the one-quarter model is also shown in Figure 5.

The plasticity parameters for base metal and undermatched weldments are obtained by the least square method in Table 3. These parameters are validated in terms of hysteresis loops at the first cycle. Figure 6 shows the comparison between experimental data and simulation results of fully reversed strain cyclic tests for base metal and undermatched welds under uniaxial fatigue loading. The calculation results of the first hysteresis cyclic stress–strain curves and the peak stress evolution curves at the strain amplitudes of 0.6 and 0.8% agree with the experimental results perfectly. It indicates that the Chaboche plastic model has an excellent prediction capability for the cyclic hardening behavior for uniaxial fatigue loading. Furthermore, the experimental data and calculation results of maximum/minimum stress relaxation curves for the first 100 cycles are given in Figure 6. It can be seen that a good agreement between the experiments and the simulations has been achieved. The maximum stress drops slowly and keeps stable during stress relaxation, which demonstrates stable cyclic softening behavior.

Similarly, the comparison results between experimental data and simulation results regarding hysteresis cyclic stress–strain curves and stress relaxation curves for undermatched welds demonstrate good agreements under 0.6 and 0.8% strain ratios in Figure 7.

For a more explicit illustration of the accuracy of simulation results, the comparison is made between the calculation results and experimental data by stable hysteresis loop at half-life cycle from the strain range from 0.6 to 1.2% for base metal and weldments. The comparison of simulation and experimental results of hysteresis stress–strain curves of base metal and undermatched welds in the stabilization stage is given in Figure 8A,B, respectively. The calculation results exhibit good prediction at different strain amplitudes. Therefore, these plasticity parameters used in the research are calibrated reasonably.

The cyclic number of damage initiation and final failure should be defined to predict the LCF life based on the CDM approach. However, the number of cycles for the damage initiation and stabilization cannot be determined in a procedural standard. According to previous references, a quotient curve method was used to estimate the number of cycles for the stage of damage initiation and rupture (Rao et al., 1988). Hormozi et al. chose the strain energy density to obtain the damage parameters in cyclic stress–strain curves (Hormozi et al., 2014; Hormozi et al., 2015).

From the analysis of the material, the evolution of stress amplitudes for each cycle demonstrates different patterns. Therefore, various materials may provide different stress evolution response and further give dissimilar stabilized plateau. Before the stabilized cycle, the damage initiation could be estimated by the inelastic strain energy density per cycle (i.e., ∆w vs. N). The parameter ∆w is widely used to describe the damage initiation (Hormozi et al., 2014) and crack propagation (Korsunsky et al., 2007) in strain-controlled LCF investigation. The following equation can define the inelastic strain density (plastic strain energy density): Δ w = ∫ ε i e 1 ε i e 2 σ i j d ε i f i e , (13) where σ i j is the stress tensor, ε i f i e is the inelastic strain tensor, ε i e 1 is the inelastic strain tensor at the beginning of the cycle, and ε i e 2 is the inelastic strain tensor at the end of the cycle. It should be noted that Eq. 13 could be employed to derive the inelastic strain by subtracting the strain results from the LCF experimental data. The inelastic strain is equivalent to the plastic strain for the LCF tests if other strains (i.e., creep strain) are not considered. Thus, the relationship of inelastic strain and plastic strain can be expressed as follows: ε i e = ε p . (14)

The inelastic strain energy density is equivalent to the plastic strain energy density in this study. The relationships among cyclic stress, total strain, and plastic strain in the stabilized cyclic state can be demonstrated typically under 0.8% total strain amplitudes in Figure 9. Finally, the value of ∆w is equivalent to the critical area of the hysteresis loop by the red circle.

The evolution of plastic strain energy density with per cycle (denoted as ∆w ₀) of LCF tests under 0.6% strain amplitudes is illustrated in Figure 10A. It can be seen that the plastic strain energy density per cycle seems to be the same pattern as the stress response evolution plots. In Figure 10, the ∆w ₀ line can be divided into three stages according to the material behavior. The first is the initial cycle stage that the ∆w ₀ decreases with the increase of cycle due to the cyclic softening behavior. After the initial stage, the plastic strain energy density values reach a stable response since the energy dissipated per cycle keeps constant. In the last stage, the ∆w ₀ starts falling once the damage is initiated. No detailed rules or standards define the exact number of the cycle when the damage initiates. In a previous study (Hormozi et al., 2015), the start number of cycles after the noticeable cyclic softening performance is defined as stabilized points (N _s). Simultaneously, a 0.5% drop of the ∆w from the maximum value of plastic strain energy density is considered a reliable value at which the damage was initiated (N _i). Similarly, a 10% drop of stabilized ∆w value from the initial damage point is defined as the final fatigue fracture.

Figure 10B is the evolution of ∆w ₀ with per cycle under 0.8% strain amplitudes for the base metal. It can be seen that the values of ∆w ₀ in different stages are higher than the results of 0.6% strain amplitudes. In contrast to the undermatched welds results, the ∆w ₀ degradation under 0.6 and 0.8% strain amplitudes are depicted in Figure 10C,D. Comparing the ∆w ₀ of base metal and undermatched welds, the percentage of initial stage in the whole stage of weldments seems to be shorter than that of base metal. It implied that the weldments demonstrate a higher possibility for fatigue damage initiation. Afterward, the damage accumulation stage of undermatched welds shows more cycles than the base metal. It can be proved that the material degradation grows slower than the base metal. If we remove the fatigue damage standard of an initial 0.5% drop and the final 10% drop, the damage deterioration of base metal exhibits higher stability than the weldment. According to the SEM micrographs of the base metal and undermatched welds shown in Figure 11A, the inner base metal does not have significant defects and shows a generally homogeneous microstructure before welding. In contrast, the formation of some micron-sized precipitated particles in the undermatched welds is visible in Figure 11B. These small defects lead to the formation and propagation of microcrack and resulting in damage failure. Furthermore, according to the EBSD characterizations in Figure 2, the undermatched weldment has some micron-sized recrystallization nuclei. The local residual deformation in these nuclei (almost zero) is significantly different from the grain matrix (higher deformation level and thus higher strength due to work-hardening), leading to an intense strength difference between them. The stress and strain incompatibility between the recrystallized cells and the matrix grains with higher deformation levels can contribute to the local stress concentration and initiating microcracks. As the loading continues, the recrystallized small deformation-free grains in the weldment have a higher chance of being work hardened, consuming more plastic strain energy and slowing down the damage accumulation procedure. It can be the reason that the weldment shows more cycles than the base metal.

The values obtained for ∆w, ∑ΔD, N _f, N _0, and ∑ΔD/∑ΔN from the LCF experimental tests are summarized in Table 4. It should be noted that the damage accumulations (∑ΔD) are acquired by stabilization and the evolution of plastic strain energy density magnitudes.

Linear log–log relationships between the plastic strain energy density and fatigue life in the initial stage for base metal and undermatched welds are shown in Figure 12. The related equations are given as follows: { N 0 = 3162.3 ⋅ ( Δ w ) − 1.126 ,   for base metal, N 0 = 135.8 ⋅ ( Δ w ) − 0.53 ,   for undermatched welds . (15)

Furthermore, the linear relationships of ∆w and damage value per cycle for two materials are shown in Figure 13, and the fitting equations are depicted as follows: { Δ D / Δ N = 1.39 E − 3 ⋅ ( Δ w ) − 0.095 ,   for base metal, Δ D / Δ N = 4.35 E − 5 ⋅ ( Δ w ) 1.01 ,   for undermatched welds . (16)

It should be noted that the c ₃ parameter should be calculated by the characteristic length L, which is associated with the adjacent distance between integration points. The L is set as 1.75 mm in fine mesh portion by ABAQUS. Finally, the damage parameters for LCF are summarized in Table 5.

The direct cyclic FE analysis is conducted in ABAQUS under different loading conditions by combining the Chaboche cyclic plastic parameters with CDM parameters of base metal and undermatched weldments. Figure 14 illustrates the LCF damage evolution at the critical material point in the specimen for base metal and undermatched welds under 0.6 and 0.8% symmetrical strain amplitudes. FE hysteresis damage growth under 0.6% strain amplitude, as shown in Figure 14A, is overestimated compared with the experimental data of base metal. Due to the intermediate processing of damage parameters, the deviation of total experimental damages leads to the overestimated FE damage. Figure 14C,D compare the fatigue damage of experimental data and computation results by the LCF damage model for undermatched welds in ABAQUS. The excellent agreement of experimental and FE results in damage initiation and degradation stages validates the damage model’s accuracy in ABAQUS. Despite the minor deviations in the FE results, it can be concluded that the FE damage model is capable of accurately predicting the damage magnitude based on energy damage evolution. It implies that the damage laws are feasible, and the effect of damage on the fatigue initiation and degradation of FE material characteristics can demonstrate the experimental observations accurately.

Based on validated fatigue damage parameters proposed from the experimental data, the LCF damage predictions of different strain amplitudes from 0.6 to 1.2% are given in Figure 15. As can be observed from Figure 15A, the damage initiation life for base metal decreases with the increase of strain amplitude. Regarding damage degradation after the damage initiation, it overgrows with the decrease of strain amplitude. However, Figure 15B shows the same damage initiation state for undermatched welds; the speed of damage degradation seems to be faster with the increase of strain amplitudes. As defined by the actual damage boundary of material rupture, the calculation damage limitation is also set to 0.1. Thus, we can obtain the predicted fatigue life based on the damage simulation results.

Figure 16 plots the predicted fatigue life based on the CDM approach against the experimental values for base metal and undermatched welds. It can be seen that all data points of the predicted results fall within the factor of two lines. For the undermatched welds, most of the predicted fatigue lives from the damage model are nonconservative. Therefore, the calibrated damage material parameters can be further applied to assess the fatigue life for more complicated structures.