Multiaxial Cycle Deformation and Low-Cycle Fatigue Behavior of Mild Carbon Steel and Related Welded-Metal Specimen

The low-cycle fatigue experiments ofmild carbonQ235B steel and its relatedwelded-metal specimens are performedunder uniaxial, in-phase, and 90 out-of-phase loading conditions. Significant additional cyclic hardening for 90 out-of-phase loading conditions is observed for both base metal and its related weldment. Besides, welding process produces extra additional hardening under the same loading conditions compared with the base metal. Multiaxial low-cycle fatigue strength under 90 out-of-phase loading conditions is significantly reduced for both base-metal and welded-metal specimens. The weldment has lower fatigue life than the base metal under the given loading conditions, and the fatigue life reduction of weldment increases with the increasing strain amplitude. The KBM, FS, and MKBM critical plane parameters are evaluated for the fatigue data obtained. The FS and MKBM parameters are found to show better correlation with fatigue lives for both base-metal and welded-metal specimens.


Introduction
Engineering components are always subjected to complex cycle loading during the service period, and the failure eventually occurs due to the accumulated fatigue damage [1].In engineering applications, fatigue failures occur in local regions, where stress concentrations generate multiaxial stress/strain states [2].The multiaxial stress/strain states commonly arise from multidirectional external loads, notch effects, and complex geometric features, which indeed influence the fatigue strength of engineering components [3].
Understanding of multiaxial fatigue strength of metallic materials is always based upon the experimental observations from thin-walled tubular specimens under axial-torsional loading.For a long period, the multiaxial fatigue tests are mainly focused on the base material of metals.A review of the multiaxial fatigue experiments for metallic materials can be found in [4,5].In the last decade, Chen et al. [6] studied the multiaxial strength of type 304 stainless steel under sequential biaxial loading.Gao et al. [7,8] tested the multiaxial fatigue strength of 16MnR steel and 7075-T651 aluminum alloy under various multiaxial loading paths.Shang and Wang [9] conducted the fatigue tests on hot-rolled medium-carbon 45 steel under the axial-torsional loading using sinusoidal wave forms.The multiaxial cycle deformation and fatigue behaviors of type 304 stainless steel and medium-carbon 1050 steel are studied by Shamsaei [10].Gladskyi and Shukaev [11] conducted the contrastive analysis of the uniaxial and multiaxial low-cycle fatigue strength of type BT1-0 titanium alloy.
For welded steel structures, residual stresses, welding defects, material inhomogeneity in the weld zone, and so forth caused by the welding process can significantly reduce the fatigue strength of the welded joints [12][13][14].At present, the fatigue design of welded joints is mainly based on the fatigue resistance S-N curves, which are obtained from the statistical analysis of the uniaxial fatigue test results of classified welding structural details [15][16][17].However, multiaxial fatigue of welded joints is rarely investigated.Chen et al. [18] conducted the low-cycle fatigue experiments on 1Cr-18Ni-9Ti stainless steel and related weld metal under axial, torsional, and 90 ∘ out-of-phase loading.Fatigue of welded components under bending-torsion proportion and nonproportion loading was studied in [19].
The multiaxial fatigue of metallic materials has attracted a widespread attention in the past decades.Investigations on multiaxial fatigue are often conducted by using an equivalent parameter which makes it possible to compare the multiaxial   loading with uniaxial one [20].Then the fatigue analysis methods developed for the uniaxial case can be employed to solve the multiaxial fatigue problems.The well-known Manson-Coffin criterion, which is widely used in the uniaxial low-cycle fatigue analysis, is modified for the multiaxial loading condition.In the recent decades, the Manson-Coffin criterion in terms of critical plane parameters, such as KBM and FS parameters, plays an important role in the multiaxial fatigue damage evaluation (more details are given in [21,22]).
To study the multiaxial cycle deformation and low-cycle fatigue behaviors of mild carbon Q235B welded joints, which are more and more widely used in the steel constructions in China, the fatigue experiments are conducted on Q235B steel and its weldment by using thin-walled tubular specimens under fully reversed strain-controlled loading conditions with uniaxial, in-phase, and 90 ∘ out-of-phase loading.The KBM, FS, and MKBM critical plane parameters are evaluated for the experimental data gathered in this study.1.

Experimental Procedure
The base-metal specimen has a tubular geometry with the outside and inside diameters of 18 mm and 14 mm, respectively.The wall thickness in the gage section is 2 mm.The geometry of the base-metal specimen is displayed in Figure 1.
The welded specimen is made by the manual CO 2 gasshielded welding process.The welding wire of MG70S-6 with a diameter of 2 mm is used.The chemical composition of welding wire is presented in Table 2.
The manufacture of the welded metal specimen is followed by [18].A well-designed notch is first machined at the center of the base metal bar, and the notch is then filled with weld metal.Finally, the welded-metal specimen is machined to the shape in accordance with the base-metal specimen.The manufactured welded-metal specimen has an 18 mm long welded zone at the center of the gauge length.The geometry of the welded thin-walled tubular specimen tested is identical to the base one (see Figure 2).
The monotonic mechanical properties for the base-metal and welded-metal specimens are listed in Table 3.

Fatigue Tests.
Fatigue tests under uniaxial, in-phase, and 90 ∘ out-of-phase loading conditions are conducted under the fully reversed strain-controlled loading at constant amplitudes.The applied waveforms for both base-metal and welded-metal specimens are sinusoidal.The three test strain paths are displayed in Figure 3.The horizontal axis is the term of axial strain, , and the vertical axis is the term of shear strain, / √ 3. The correlation of the horizontal axis and the vertical axis is derived from the von Mises criterion of  = √ 2 + (1/3) 2 in which  is the equivalent von Mises strain.Fatigue tests were conducted on an MTS tension-torsion machine under strain-controlled loading using a tensiontorsion strain extensometer with the gauge length of 25 mm, which is mounted at the center of the outside of the specimen gauge section to measure the strain responses.The loading frequency for constant-amplitude tests is 1.0 HZ.Fatigue life is assumed as the number of cycles for which there is 30% reduction with respect to the maximum tensile or shear stress of the uniaxial test.

Results and Discussion
The stable hysteresis loops of base-metal and welded-metal specimens under uniaxial, in-phase, and 90 ∘ out-of-phase loading at different strain amplitude are presented in Figures 4-6, respectively.It can be observed that the multiaxial cycle deformation behavior for in-phase loading condition is basically the same as the uniaxial one, while the multiaxial cycle deformation behaviors for out-of-phase loading condition significantly changed.The maximum shear and axial stress responses as well as shear and axial strains are simultaneous under in-phase loading conditions for both base-metal and welded-metal specimen.However, the maximum values of cyclic stress and strain responses do not always occur at the same time under the 90 ∘ out-of-phase loading, which indicates that the metal's plastic yield flow for 90 ∘ out-ofphase loading condition is different from that under the uniaxial one.Crack initiation under uniaxial fatigue test is in the circumferential direction as usual.In the in-phase fatigue test, the patterns of the macroscopic crack are observed similar to that of uniaxial fatigue test.For the 90 ∘ out-ofphase fatigue test, the crack direction is irregular because the maximum principal stress direction changed with respect to the nonproportionality loading condition, and the fracture shape of the macrocrack for 90 ∘ out-of-phase fatigue test was jagged.The crack patterns for base metal and welded metal under in-phase and out-of-phase loading conditions are shown in Figures 7 and 8, respectively.

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The fatigue experimental and analytical results for the base-metal and welded-metal specimens under uniaxial, in-phase, and 90 ∘ out-of-phase loading are presented in Table 4.The table includes axial and shear stress and strain amplitudes, maximum shear strain amplitude acting on the maximum shear plane (critical plane), Δ max /2, strain ration, the normal strain range acting on the critical plane, Δ  , maximum normal stress acting on the critical plane,  ,max , and strain ration parameter, , for each fatigue test.
The stabilized cycle stress-strain relationship can be represented by the Ramberg-Osgood equation [23,24].
where   is the cyclic hardening coefficient and   is the cyclic hardening exponent,  is Young's modulus of the investigated material, and Δ and Δ are the equivalent strain range and the equivalent stress range, respectively.Δ  and Δ  are the equivalent elastic strain range and the equivalent plastic strain range, respectively.The equivalent elastic strain can be calculated based on Hooke's law.The equivalent plastic strain range can be then calculated by Comparison of cyclic stress-strain curves between base metal and welded metal for uniaxial, in-phase, and out-ofphase loading conditions is presented in Figure 9, in which the solid lines and the dotted lines are the fitted cyclic stressstrain curves for base-metal specimens and welded-metal   specimens, respectively.It can be seen that the fitted cyclic stress-strain curves for both base metal and welded metal under in-phase loading are similar to those obtained for uniaxial loading, while the fitted cyclic stress-strain curves for the 90 ∘ out-of-phase loading conditions are above those relative to the uniaxial loading case, which indicates that a significant additional cyclic hardening effect occurs for both base metal and welded metal under the out-of-phase loading conditions.It can be also seen that the stabilized cyclic stressstrain curves of the welded metal are over those of the base metal under the same loading condition.It can be concluded that the welding process produces extra additional hardening for the weldment compared with the base metal.

Fatigue Life Analysis
The Manson-Coffin equation in terms of the equivalent strain parameter for fatigue evaluation under uniaxial loading can be written as [20][21][22] where Δ is the equivalent von Mises strain range,   is the cycle numbers to fatigue failure, and    ,    , b, and  are the fatigue strength coefficient, the fatigue ductility coefficient, the fatigue strength exponent, and the fatigue ductility exponent, respectively.The fatigue properties fitted by (3) for the base metal and welded metal under the uniaxial, in-phase, and 90 ∘ out-of-phase loading are listed in Table 5.
The relationships of equivalent strain parameters versus fatigue life under the investigated loading conditions are shown in Figure 10(a) for the base-metal specimens and in Figure 10(b) for the welded-metal specimens, respectively.It can be seen that the fatigue life under in-phase loading conditions is slightly longer than that under uniaxial loading conditions for both base metal and welded metal, while the fatigue life for 90 ∘ out-of-phase loading conditions is significantly reduced for both base metal and welded metal compared with the uniaxial case.Therefore, the following conclusion can be drawn: the equivalent strain parameter is not well correlated with fatigue life in the case of out-of-phase loading.
Fatigue lives of base-metal and welded-metal specimen for the three investigated loading conditions are compared in Figure 11.It can be observed that the fatigue life of weldedmetal specimen is greatly reduced compared with that of base metal under the same loading conditions, and the fatigue life reduction of weldment increases with increasing strain amplitude.

Critical Plane Parameters
Three strain-based critical plane parameters, which are adopted to correlate the fatigue life for the different loading conditions, are investigated in this section.
Kandil-Brown-Miller [22] takes the linear combination of the shear and the normal strain ranges acting on the critical plane as the fatigue parameter.The correlation of the KBM parameters and fatigue lives can be obtained by employing where Δ max and Δ  are the maximum shear strain range and the normal strain range acting on the critical plane, respectively.]  and ]  are the elastic and plastic Poisson's ratios, respectively.Consistency of volume requires the elastic Poisson's ratio to typically equal 0.3 and the plastic Poisson's ratio to be 0.5.S is an experimental coefficient of the investigated materials.Fatemi and Socie [25] proposed a widely accepted critical plane concept using the maximum normal stress, which can reflect the effect of nonproportional cyclic additional hardening on multiaxial fatigue damage to replace the normal strain in KBM parameter.The Fatemi-Socie (FS) model can be written as where  ,max is the maximum normal stress acting on the critical plane.  is the yield strength for the investigated materials. is an experimental coefficient.As an approximation, one may simply assume the experimental coefficient in FS model to be 1.0 [10].
Based on the FS critical plane concept, Li and the coauthors [4] developed a stress-correlated factor to consider the effect of the nonproportional additional hardening on multiaxial fatigue damage, which can be used to modify the KBM parameter.The Modified KBM (MKBM) model can be rewritten as Figures 12-14 present the correlation of the KBM, FS, and MKBM parameters with the observed fatigue life.It can be seen that KBM parameters fit well fatigue life data of both base-metal and welded-metal specimens for the uniaxial and  in-phase tests.No correlation could be found for the 90 ∘ out-of-phase loading condition.It should be noted that the experimental coefficient  in KBM model is taken as 0.8 for the base metal and 1.0 for the welded metal, respectively, which are the averaged values over the fatigue test data.
FS parameters and fatigue life are well correlated for base metal and welded metal for all loading conditions.The correlation of the MKBM parameters with the fatigue life for the out-of-phase loading is greatly improved with respect to KBM, especially for the welded-metal specimens under 90 ∘ out-of-phase loading.The improving predictions of the FS and MKBM parameters for out-of-phase loading conditions can be attributed to the introduced stress term  ,max , which reflects the effect of the material additional cyclic hardening due to the nonproportionality of the cyclic loading on multiaxial fatigue damage.
A critical plane approach should be able to predict both the fatigue life and the critical planes, where cracks are predicted to initiate [26,27].Comparison of theoretical predictions with experimental data allows evaluating the validity of a proposed critical plane approach.However, due to the difficulties in defining cracking direction observed experimentally due to the roughness of the crack surface, limited work has been done in the evaluation of a critical plane approach for predicting the cracking directions.Jiang [26] investigated the cracking behavior predictions of FS model by using 35 thin-walled tubular specimens of S460N steel and found that only about 20% specimens are predicted correctly by the FS model for the cracking orientations, which reveals that the predictions for the cracking directions are far less desirable than the fatigue life predictions.The time-consuming cracking behavior observation may have

Figure 5 :
Figure 5: Axial (left) and shear (right) hysteresis loops for in-phase loading: (a) base metal and (b) welded metal.

Figure 6 :
Figure 6: Axial (left) and shear (right) hysteresis loops for out-of-phase loading: (a) base metal and (b) welded metal.

Figure 7 :
Figure 7: Patterns of crack growth for in-phase test: (a) base metal and (b) welded metal.

Figure 8 :Figure 9 :
Figure 8: Patterns of crack growth for out-of-phase test: (a) base metal and (b) welded metal.

Figure 10 :
Figure 10: Relationship of equivalent strain versus fatigue life: (a) base metal and (b) welded metal.

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Table 3 :
Mechanical properties of Q235B base and welded metal.

Table 4 :
Fatigue test and analysis results for base-metal and welded-metal specimens.

Table 5 :
Fatigue properties of base metal and welded metal.