multiaxial cycle deformation and low-cycle fatigue behavior of...

Research ArticleMultiaxial Cycle Deformation and Low-Cycle Fatigue Behaviorof Mild Carbon Steel and Related Welded-Metal Specimen

Weilian Qu Ernian Zhao and Qiang Zhou

Hubei Key Laboratory of Roadway Bridge amp Structure Engineering Wuhan University of Technology Wuhan Hubei 430070 China

Correspondence should be addressed to Ernian Zhao zhaoern126com

Received 4 July 2016 Revised 10 December 2016 Accepted 19 January 2017 Published 13 February 2017

Academic Editor Luciano Lamberti

Copyright copy 2017 Weilian Qu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Engineering components are always subjected to complexcycle loading during the service period and the failure even-tually occurs due to the accumulated fatigue damage [1] Inengineering applications fatigue failures occur in localregions where stress concentrations generate multiaxialstressstrain states [2]Themultiaxial stressstrain states com-monly arise from multidirectional external loads notcheffects and complex geometric features which indeed influ-ence the fatigue strength of engineering components [3]

Understanding of multiaxial fatigue strength of metallicmaterials is always based upon the experimental observationsfrom thin-walled tubular specimens under axial-torsionalloading For a long period the multiaxial fatigue tests aremainly focused on the base material of metals A review ofthe multiaxial fatigue experiments for metallic materials canbe found in [4 5] In the last decade Chen et al [6] studied themultiaxial strength of type 304 stainless steel under sequentialbiaxial loading Gao et al [7 8] tested the multiaxial fatiguestrength of 16MnR steel and 7075-T651 aluminum alloyunder various multiaxial loading paths Shang and Wang [9]conducted the fatigue tests on hot-rolled medium-carbon45 steel under the axial-torsional loading using sinusoidal

wave forms The multiaxial cycle deformation and fatiguebehaviors of type 304 stainless steel andmedium-carbon 1050steel are studied by Shamsaei [10] Gladskyi and Shukaev[11] conducted the contrastive analysis of the uniaxial andmultiaxial low-cycle fatigue strength of type BT1-0 titaniumalloy

For welded steel structures residual stresses weldingdefects material inhomogeneity in the weld zone and soforth caused by the welding process can significantly reducethe fatigue strength of the welded joints [12ndash14] At presentthe fatigue design of welded joints is mainly based onthe fatigue resistance S-N curves which are obtained fromthe statistical analysis of the uniaxial fatigue test results ofclassified welding structural details [15ndash17] However mul-tiaxial fatigue of welded joints is rarely investigated Chenet al [18] conducted the low-cycle fatigue experiments on1Cr-18Ni-9Ti stainless steel and related weld metal underaxial torsional and 90∘ out-of-phase loading Fatigue ofwelded components under bending-torsion proportion andnonproportion loading was studied in [19]

The multiaxial fatigue of metallic materials has attracteda widespread attention in the past decades Investigations onmultiaxial fatigue are often conducted by using an equivalentparameter which makes it possible to compare the multiaxial

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 8987376 12 pageshttpsdoiorg10115520178987376

2 Advances in Materials Science and Engineering

175 17540

55 55

185

18

R60

14120601120601 24120601

Figure 1 Geometry of the base-metal specimen

55 55

175 17540185

18

18 14120601120601

24120601

R60

Figure 2 Geometry of the welded-metal specimen

Table 1 Chemical composition of Q235B steel (wt)

C Mn Si P S V Alt Nb016 045 026 0021 0025 0006 0004 0003

loading with uniaxial one [20] Then the fatigue analysismethods developed for the uniaxial case can be employedto solve the multiaxial fatigue problems The well-knownManson-Coffin criterion which is widely used in the uniaxiallow-cycle fatigue analysis is modified for the multiaxialloading condition In the recent decades the Manson-Coffincriterion in terms of critical plane parameters such as KBMand FS parameters plays an important role in the multiaxialfatigue damage evaluation (more details are given in [21 22])

To study the multiaxial cycle deformation and low-cyclefatigue behaviors of mild carbon Q235B welded joints whichare more and more widely used in the steel constructions inChina the fatigue experiments are conducted on Q235B steeland its weldment by using thin-walled tubular specimensunder fully reversed strain-controlled loading conditionswith uniaxial in-phase and 90∘ out-of-phase loading TheKBM FS andMKBM critical plane parameters are evaluatedfor the experimental data gathered in this study

2 Experimental Procedure

21Materials and Specimens The investigatedmaterial in thepresent study is mild carbon Q235B structural steel which iswidely used in Chinarsquos steel constructions The investigatedQ235B steel has Youngrsquos modulus of 204GPa yield strengthof 270MPa ultimate strength of 390MPa Poissonrsquos ratio of03 and elongation of 369 The chemical composition ofQ235B steel is presented in Table 1

The base-metal specimen has a tubular geometry withthe outside and inside diameters of 18mm and 14mmrespectively The wall thickness in the gage section is 2mm

Table 2 Chemical composition of welding wire (wt)

C Mn Si P S V Ni Cr Mo Cu0077 154 092 0011 0012 0002 0006 0023 0004 0126

The geometry of the base-metal specimen is displayed inFigure 1

The welded specimen is made by the manual CO2 gas-shielded welding processThe welding wire of MG70S-6 witha diameter of 2mm is used The chemical composition ofwelding wire is presented in Table 2

The manufacture of the welded metal specimen is fol-lowed by [18] A well-designed notch is first machined at thecenter of the base metal bar and the notch is then filled withweld metal Finally the welded-metal specimen is machinedto the shape in accordance with the base-metal specimenThemanufacturedwelded-metal specimenhas an 18mm longwelded zone at the center of the gauge length The geometryof the welded thin-walled tubular specimen tested is identicalto the base one (see Figure 2)

The monotonic mechanical properties for the base-metaland welded-metal specimens are listed in Table 3

22 Fatigue Tests Fatigue tests under uniaxial in-phase and90∘ out-of-phase loading conditions are conducted under thefully reversed strain-controlled loading at constant ampli-tudes The applied waveforms for both base-metal andwelded-metal specimens are sinusoidal The three test strainpaths are displayed in Figure 3 The horizontal axis is theterm of axial strain 120576 and the vertical axis is the term ofshear strain 120574radic3 The correlation of the horizontal axis andthe vertical axis is derived from the von Mises criterion of120576 = radic1205762 + (13)1205742 in which 120576 is the equivalent von Misesstrain

Advances in Materials Science and Engineering 3

Table 3 Mechanical properties of Q235B base and welded metal

Youngrsquos Shear Yield Tensile Ultimate Elongationmodulus modulus strength strength strain 120575 ()E (GPa) G (GPa) 120590119910 (MPa) 120590119906 (MPa) 120576119906 ()

Base metal 204 814 269 3909 152 369Welded metal 198 763 265 3713 439 143

120574radic3

120576

(a)

120574radic3

120576

(b)

120574radic3

120576

(c)

Figure 3 Fatigue test loading paths (a) uniaxial (UA) (b) in-phase (IP) and (c) 90∘ out-of-phase (OP)

minus400minus300minus200minus100

100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

120576 = 0006120576 = 0008

120576 = 0010

minus0012 0012minus0006 00060

(a)

120576 = 0006120576 = 0008

120576 = 0010


100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

minus0012 0012minus0006 00060

(b)

Figure 4 Axial hysteresis loops for uniaxial loading (a) base metal and (b) welded metal

Fatigue tests were conducted on an MTS tension-torsionmachine under strain-controlled loading using a tension-torsion strain extensometer with the gauge length of 25mmwhich is mounted at the center of the outside of the specimengauge section to measure the strain responses The loadingfrequency for constant-amplitude tests is 10HZ Fatigue lifeis assumed as the number of cycles for which there is 30reduction with respect to themaximum tensile or shear stressof the uniaxial test

3 Results and Discussion

The stable hysteresis loops of base-metal and welded-metalspecimens under uniaxial in-phase and 90∘ out-of-phase

loading at different strain amplitude are presented in Figures4ndash6 respectively It can be observed that the multiaxial cycledeformation behavior for in-phase loading condition is basi-cally the same as the uniaxial one while the multiaxial cycledeformation behaviors for out-of-phase loading conditionsignificantly changed The maximum shear and axial stressresponses as well as shear and axial strains are simultaneousunder in-phase loading conditions for both base-metal andwelded-metal specimen However the maximum values ofcyclic stress and strain responses do not always occur atthe same time under the 90∘ out-of-phase loading whichindicates that the metalrsquos plastic yield flow for 90∘ out-of-phase loading condition is different from that under theuniaxial one


minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300

minus0008 0008minus0004 00040

radic3120591

(MPa

)

120574radic3

(a)

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300

minus0008 0008minus0004 00040radic3120591

(MPa

)

120574radic3

(b)

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

Crack initiation under uniaxial fatigue test is in thecircumferential direction as usual In the in-phase fatiguetest the patterns of the macroscopic crack are observedsimilar to that of uniaxial fatigue test For the 90∘ out-of-phase fatigue test the crack direction is irregular becausethemaximum principal stress direction changed with respectto the nonproportionality loading condition and the frac-ture shape of the macrocrack for 90∘ out-of-phase fatiguetest was jagged The crack patterns for base metal andwelded metal under in-phase and out-of-phase loadingconditions are shown in Figures 7 and 8 respective-ly

The fatigue experimental and analytical results for thebase-metal and welded-metal specimens under uniaxialin-phase and 90∘ out-of-phase loading are presented inTable 4 The table includes axial and shear stress and strainamplitudes maximum shear strain amplitude acting onthe maximum shear plane (critical plane) Δ120574max2 strainration the normal strain range acting on the critical planeΔ120576119899 maximum normal stress acting on the critical plane120590119899max and strain ration parameter 120582 for each fatiguetest

The stabilized cycle stress-strain relationship can be rep-resented by the Ramberg-Osgood equation [23 24]

Δ1205762 = Δ1205761198902 + Δ1205761199012 = Δ1205902119864 + ( Δ12059021198701015840)11198991015840 (1)

where1198701015840 is the cyclic hardening coefficient and 1198991015840 is the cyclichardening exponent119864 is Youngrsquos modulus of the investigatedmaterial and Δ120576 and Δ120590 are the equivalent strain range andthe equivalent stress range respectively Δ120576119890 and Δ120576119901 arethe equivalent elastic strain range and the equivalent plasticstrain range respectively The equivalent elastic strain can becalculated based on Hookersquos lawThe equivalent plastic strainrange can be then calculated by

Δ1205761199012 = Δ1205762 minus Δ1205761198902 = Δ1205762 minus Δ1205902119864 (2)

Comparison of cyclic stress-strain curves between basemetal and welded metal for uniaxial in-phase and out-of-phase loading conditions is presented in Figure 9 in whichthe solid lines and the dotted lines are the fitted cyclic stress-strain curves for base-metal specimens and welded-metal


minus300minus200minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400


100200300400

minus400

minus0008 0008minus0004 00040 minus0008 0008minus0004 00040

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3120591

(MPa

)

Shear strain 120574radic3

(a)

minus100


Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400

minus400

minus0008 0008minus0004 00040

minus300minus200

100200300400

minus400Shear strain

minus0008 0008minus0004 00040

120574

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3

radic3

120591(M

Pa)

(b)

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

(a) (b)

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

specimens respectively It can be seen that the fitted cyclicstress-strain curves for both base metal and welded metalunder in-phase loading are similar to those obtained foruniaxial loading while the fitted cyclic stress-strain curves

for the 90∘ out-of-phase loading conditions are above thoserelative to the uniaxial loading case which indicates that asignificant additional cyclic hardening effect occurs for bothbase metal and welded metal under the out-of-phase loading


(a) (b)

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

0 0002 0004 0006 0008 001 00120

100

200

300

400

500

UniaxialIn-phaseOut-of-phase


Welded metalBase metal

Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)

Equivalent strain amplitude Δ1205762

Figure 9 Comparison of cyclic stress-strain curves between basemetal and welded metal for uniaxial in-phase and out-of-phaseloading conditions

conditions It can be also seen that the stabilized cyclic stress-strain curves of the welded metal are over those of the basemetal under the same loading condition It can be concludedthat the welding process produces extra additional hardeningfor the weldment compared with the base metal

4 Fatigue Life Analysis

TheManson-Coffin equation in terms of the equivalent strainparameter for fatigue evaluation under uniaxial loading canbe written as [20ndash22]

Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)

where Δ120576 is the equivalent von Mises strain range 119873119891 isthe cycle numbers to fatigue failure and 1205901015840119891 1205761015840119891 b and119888 are the fatigue strength coefficient the fatigue ductilitycoefficient the fatigue strength exponent and the fatigueductility exponent respectively The fatigue properties fittedby (3) for the base metal and welded metal under theuniaxial in-phase and 90∘ out-of-phase loading are listed inTable 5

The relationships of equivalent strain parameters versusfatigue life under the investigated loading conditions areshown in Figure 10(a) for the base-metal specimens and inFigure 10(b) for the welded-metal specimens respectivelyIt can be seen that the fatigue life under in-phase loadingconditions is slightly longer than that under uniaxial loadingconditions for both base metal and welded metal whilethe fatigue life for 90∘ out-of-phase loading conditions issignificantly reduced for both base metal and welded metalcompared with the uniaxial case Therefore the followingconclusion can be drawn the equivalent strain parameter isnot well correlated with fatigue life in the case of out-of-phaseloading

Fatigue lives of base-metal and welded-metal specimenfor the three investigated loading conditions are compared inFigure 11 It can be observed that the fatigue life of welded-metal specimen is greatly reduced compared with that of basemetal under the same loading conditions and the fatiguelife reduction of weldment increases with increasing strainamplitude

5 Critical Plane Parameters

Three strain-based critical plane parameters which areadopted to correlate the fatigue life for the different loadingconditions are investigated in this section

Kandil-Brown-Miller [22] takes the linear combination ofthe shear and the normal strain ranges acting on the criticalplane as the fatigue parameter The correlation of the KBMparameters and fatigue lives can be obtained by employing


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

Strain pathsΔ1205762 Δ1205742 120582 Δ1205902 Δ1205912 Δ120574max2 Δ120576119899 120590119899max 119873119891() () (MPa) (MPa) () () (MPa)

Base-metal fatigue testUA 060 0 0 270 0 087 016 135 3443UA 060 0 0 273 0 087 016 137 4012UA 080 0 0 289 0 117 021 145 1643UA 080 0 0 288 0 117 021 144 1817UA 103 0 0 302 0 151 027 151 1257UA 101 0 0 299 0 149 027 149 1254IP 027 050 1852 194 88 063 008 119 16269IP 042 074 1762 196 105 096 011 107 4443IP 042 074 1762 195 107 096 011 102 4535IP 056 096 1714 214 108 126 015 119 2929IP 056 097 1732 213 108 127 015 118 2971IP 071 125 1761 219 111 163 019 123 1475IP 071 118 1662 218 110 158 019 117 2053OP 040 069 1725 305 176 069 017 313 1494OP 040 070 1750 306 170 070 017 312 1519OP 051 085 1667 326 182 081 022 322 1002OP 050 081 1620 321 181 085 022 326 992OP 061 101 1656 336 189 101 027 336 703OP 061 099 1623 329 186 099 027 329 642

Welded-metal fatigue testUA 061 0 0 295 0 089 017 147 289UA 060 0 0 293 0 087 016 146 432UA 081 0 0 313 0 118 022 156 118UA 080 0 0 315 0 117 022 157 177UA 102 0 0 329 0 150 027 164 70UA 100 0 0 334 0 147 027 167 60IP 028 049 1750 189 104 063 008 102 2697IP 042 075 1786 221 121 096 012 118 1262IP 042 074 1762 220 122 097 012 120 701IP 057 101 1772 230 131 131 015 127 384IP 056 100 1786 241 134 129 015 129 393IP 071 125 1761 253 128 163 019 142 160IP 070 127 1814 243 129 163 019 135 145OP 040 069 1725 345 195 069 018 345 359OP 041 068 1659 335 190 068 019 335 388OP 049 088 1796 350 198 091 023 362 220OP 050 091 1820 362 207 088 024 356 172OP 063 103 1635 405 234 088 023 390 98OP 056 107 1911 410 237 103 030 383 76

theManson-Coffin equationThus the Kandil-Brown-Miller(KBM) model can be given by

Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)

where Δ120574max and Δ120576119899 are the maximum shear strain rangeand the normal strain range acting on the critical planerespectively ]119890 and ]119901 are the elastic and plastic Poissonrsquosratios respectively Consistency of volume requires the elasticPoissonrsquos ratio to typically equal 03 and the plastic Poissonrsquosratio to be 05 S is an experimental coefficient of theinvestigated materials


Table 5 Fatigue properties of base metal and welded metal

Fatigue properties Uniaxial tests In-phase tests 90∘ out-of-phase testsBase-metal specimen1205901015840119891 (MPa) 40759 43840 53509

1205761015840119891 08091 06196 05630119887 minus00424 minus00520 minus00702119888 minus05827 minus05277 minus06723

Welded-metal specimen1205901015840119891 (MPa) 48114 50827 519351205761015840119891 00375 00654 00202119887 minus00755 minus00708 minus03555119888 minus03154 minus00624 minus03181

0001

0002

0004

000600080010

0020Base-metal specimen

Axial test dataIn-phase test data

Scatter band of 3out-of-phase test data90∘

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02

Cycles to failure Nf

(a)

0001

0002

0004

0006

00080010

0020Welded-metal specimen



Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)

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

Fatemi and Socie [25] proposed a widely accepted criticalplane concept using the maximum normal stress which canreflect the effect of nonproportional cyclic additional harden-ing onmultiaxial fatigue damage to replace the normal strainin KBM parameter The Fatemi-Socie (FS) model can bewritten as

Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)

where 120590119899max is the maximum normal stress acting on thecritical plane 120590119910 is the yield strength for the investigated

materials 119896 is an experimental coefficient As an approxima-tion one may simply assume the experimental coefficient inFS model to be 10 [10]

Based on the FS critical plane concept Li and the coau-thors [4] developed a stress-correlated factor to consider theeffect of the nonproportional additional hardening on multi-axial fatigue damage which can be used to modify the KBMparameter The Modified KBM (MKBM) model can berewritten as

Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)

Figures 12ndash14 present the correlation of the KBM FS andMKBM parameters with the observed fatigue life It can beseen that KBM parameters fit well fatigue life data of bothbase-metal and welded-metal specimens for the uniaxial and


0001

0002

0004

000600080010

0020

Base-metal test dataWelded-metal test data

Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)

Figure 11 Comparison of fatigue life between base-metal and welded-metal specimen (a) uniaxial loading (b) in-phase loading and (c) 90∘out-of-phase loading

in-phase tests No correlation could be found for the 90∘out-of-phase loading condition It should be noted that theexperimental coefficient 119878 in KBM model is taken as 08 forthe base metal and 10 for the welded metal respectivelywhich are the averaged values over the fatigue test data

FS parameters and fatigue life are well correlated forbase metal and welded metal for all loading conditions Thecorrelation of the MKBM parameters with the fatigue life forthe out-of-phase loading is greatly improved with respect toKBM especially for the welded-metal specimens under 90∘out-of-phase loading The improving predictions of the FSand MKBM parameters for out-of-phase loading conditionscan be attributed to the introduced stress term 120590119899max whichreflects the effect of the material additional cyclic hardeningdue to the nonproportionality of the cyclic loading onmultiaxial fatigue damage

A critical plane approach should be able to predict boththe fatigue life and the critical planes where cracks arepredicted to initiate [26 27] Comparison of theoreticalpredictions with experimental data allows evaluating thevalidity of a proposed critical plane approach However dueto the difficulties in defining cracking direction observedexperimentally due to the roughness of the crack surfacelimited work has been done in the evaluation of a criticalplane approach for predicting the cracking directions Jiang[26] investigated the cracking behavior predictions of FSmodel by using 35 thin-walled tubular specimens of S460Nsteel and found that only about 20 specimens are predictedcorrectly by the FS model for the cracking orientationswhich reveals that the predictions for the cracking directionsare far less desirable than the fatigue life predictions Thetime-consuming cracking behavior observation may have

10 Advances in Materials Science and EngineeringKB

M p

aram

eter

Base-metal specimen

UniaxialIn-phaseOut-of-phaseScatter band of 3


10minus3

10minus2

10minus1

105104103102

(a)

Welded-metal specimen


KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)

Figure 12 Correlation of KBM parameters with the fatigue life (a) base metal and (b) welded metal

FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)

Figure 13 Correlation of FS parameters with the fatigue life (a) base metal and (b) welded metal

prevented more detailed work on the cracking directionexaminations [26] Accordingly it is worth pointing out thatthe predictions for cracking directions using different criticalplane models certainly warrant investigations in the future

6 Conclusions

The multiaxial cycle deformation and fatigue behavior ofQ235B mild carbon steel and its related welded metal are

experimentally investigated in the present paper The follow-ing conclusions can be drawn

(1) Significant additional cyclic hardening effect is ob-served for both base steel and welded metal underout-of-phase loading conditions Besides weldingproc-ess produces extra additional hardening com-pared with the base metal

(2) Fatigue strength under in-phase loading is slightlyhigher than that under uniaxial loading for both base

Advances in Materials Science and Engineering 11M

KBM

par

amet

er

Base-metal specimen

AxialCycles to failure Nf

10minus3

10minus2

10minus1

105104103102

In-phaseOut-of-phaseScatter band of 3

(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)

Figure 14 Correlation of MKBM parameters with the fatigue life (a) base metal and (b) welded metal

metal and welded metal while the fatigue strengthunder 90∘ out-of-phase loading is significantly re-duced

(3) Fatigue strength for welded-metal specimens isgreatly reduced compared with the base metal underthe same loading conditions and the fatigue lifereduction of the weldment increases with increasingapplied strain amplitudes

(4) The FS and the MKBM parameters show bettercorrelation with fatigue life for both base-metal andwelded-metal specimensThe good correlation can beattributed to the introduction of maximum normalstress acting on the critical plane which can reflectthe influence of the nonproportional hardening onmultiaxial fatigue damage

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Keshi Zhang inGuangxi University for providing the fatigue test equipmentand invaluable comments on the experimental study Theauthors also gratefully acknowledge the financial support forthiswork fromNationalNatural Science Foundation ofChina(nos 51378409 and 51438002)

References

[1] A Karolczuk and E Macha ldquoA review of critical plane orienta-tions in multiaxial fatigue failure criteria of metallic materialsrdquo

International Journal of Fracture vol 134 no 3-4 pp 267ndash3042005

[2] J Das and S M Sivakumar ldquoAn evaluation of multiaxialfatigue life assessment methods for engineering componentsrdquoInternational Journal of Pressure Vessels and Piping vol 76 no10 pp 741ndash746 1999

[3] C M Sonsino ldquoEffect of residual stresses on the fatigue behav-iour of welded joints depending on loading conditions andweldgeometryrdquo International Journal of Fatigue vol 31 no 1 pp 88ndash101 2009

[4] J Li Z-P Zhang Q Sun and C-W Li ldquoMultiaxial fatigue lifeprediction for various metallic materials based on the criticalplane approachrdquo International Journal of Fatigue vol 33 no 2pp 90ndash101 2011

[5] B C Li C Jiang X Han and Y Li ldquoA new path-dependentmultiaxial fatigue model for metals under different pathsrdquoFatigue amp Fracture of Engineering Materials amp Structures vol37 no 2 pp 206ndash218 2014

[6] X Chen D Jin and K S Kim ldquoFatigue life prediction of type304 stainless steel under sequential biaxial loadingrdquo Interna-tional Journal of Fatigue vol 28 no 3 pp 289ndash299 2006

[7] Z Gao T Zhao X Wang and Y Jiang ldquoMultiaxial fatigue of16MnR steelrdquo Journal of Pressure Vessel Technology vol 131 no2 pp 73ndash80 2009

[8] T Zhao and Y Jiang ldquoFatigue of 7075-T651 aluminum alloyrdquoInternational Journal of Fatigue vol 30 no 5 pp 834ndash849 2008

[9] D G Shang andD JWangMultiaxial Fatigue Strength SciencePress Beijing China 2007 (Chinese)

[10] N Shamsaei Multiaxial Fatigue and Deformation IncludingNon-Proportional Hardening and Variable Amplitude LoadingEffects The University of Toledo Dissertations 2010

[11] M Gladskyi and S Shukaev ldquoA newmodel for low cycle fatigueof metal alloys under non-proportional loadingrdquo InternationalJournal of Fatigue vol 32 no 10 pp 1568ndash1572 2010


[12] D Radaj C M Sonsino and W Fricke Fatigue Assessment ofWelded Joints by Local Approaches Wood Publishing LimitedCambridge UK 2006

[13] V Caccese P A Blomquist K A Berube S R Webber andN J Orozco ldquoEffect of weld geometric profile on fatigue lifeof cruciform welds made by laserGMAW processesrdquo MarineStructures vol 19 no 1 pp 1ndash22 2006

[14] S J Kim R T Dewa W G Kim and M H Kim ldquoCyclic stressresponse and fracture behaviors of Alloy 617 base metal andweld joints under LCF loadingrdquo Advances in Materials Scienceand Engineering vol 2015 Article ID 207497 11 pages 2015

[15] A Hobbacher Recommendations for Fatigue Design of WeldedJoints and Components International Institute ofWelding XIII-1539-96XV-845-96 2007

[16] Eurocode 3 Design of Steel Structures Part 1-9 Fatigue Euro-pean Standard EN 1993-1-9 2005

[17] BS5400 Steel Concrete and Composite Bridges Part 10 Code ofPractice for Fatigue BS5400 1980

[18] X Chen K An and K S Kim ldquoLow-cycle fatigue of 1Cr-18Ni-9Ti stainless steel and related weld metal under axial torsionaland 90∘ out-of-phase loadingrdquoFatigueampFracture of EngineeringMaterials amp Structures vol 27 no 6 pp 439ndash448 2004

[19] M Backstrom and G Marquis ldquoA review of multiaxial fatigueof weldments experimental results design code and criticalplane approachesrdquo Fatigue amp Fracture of Engineering Materialsamp Structures vol 24 no 5 pp 279ndash291 2001

[20] J Szusta and A Seweryn ldquoLow-cycle fatigue model of dam-age accumulationmdashthe strain approachrdquo Engineering FractureMechanics vol 77 no 10 pp 1604ndash1616 2010

[21] A Karolczuk and E Macha ldquoA review of critical plane orienta-tions in multiaxial fatigue failure criteria of metallic materialsrdquoInternational Journal of Fracture vol 134 no 3-4 pp 267ndash3042005

[22] D F Socie and G B Marquis Multiaxial Fatigue Society ofAutomotive Engineers Warrendale Pa USA 2000

[23] N Shamsaei and A Fatemi ldquoEffect of microstructure and hard-ness on non-proportional cyclic hardening coefficient and pre-dictionsrdquo Materials Science and Engineering A vol 527 no 12pp 3015ndash3024 2010

[24] Y Wang X Liu G Dai and Y Shi ldquoExperimental study onconstitutive relation of steel SN490B under cyclic loadingrdquoJournal of Building Structures vol 35 no 4 pp 142ndash148 2014

[25] A Fatemi and D F Socie ldquoA critical plane approach to multiax-ial fatigue damage including out-of-phase loadingrdquo Fatigue ampFracture of Engineering Materials amp Structures vol 11 no 3 pp149ndash165 1988

[26] Y Jiang ldquoFatigue criterion for general multiaxial loadingrdquoFatigue and Fracture of Engineering Materials and Structuresvol 23 no 1 pp 19ndash32 2000

[27] DMcClaflin andA Fatemi ldquoTorsional deformation and fatigueof hardened steel including mean stress and stress gradienteffectsrdquo International Journal of Fatigue vol 26 no 7 pp 773ndash784 2004

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


175 17540

55 55

185

18

R60

14120601120601 24120601

Figure 1 Geometry of the base-metal specimen

55 55

175 17540185

18

18 14120601120601

24120601

R60

Figure 2 Geometry of the welded-metal specimen

Table 1 Chemical composition of Q235B steel (wt)

C Mn Si P S V Alt Nb016 045 026 0021 0025 0006 0004 0003

loading with uniaxial one [20] Then the fatigue analysismethods developed for the uniaxial case can be employedto solve the multiaxial fatigue problems The well-knownManson-Coffin criterion which is widely used in the uniaxiallow-cycle fatigue analysis is modified for the multiaxialloading condition In the recent decades the Manson-Coffincriterion in terms of critical plane parameters such as KBMand FS parameters plays an important role in the multiaxialfatigue damage evaluation (more details are given in [21 22])

To study the multiaxial cycle deformation and low-cyclefatigue behaviors of mild carbon Q235B welded joints whichare more and more widely used in the steel constructions inChina the fatigue experiments are conducted on Q235B steeland its weldment by using thin-walled tubular specimensunder fully reversed strain-controlled loading conditionswith uniaxial in-phase and 90∘ out-of-phase loading TheKBM FS andMKBM critical plane parameters are evaluatedfor the experimental data gathered in this study

2 Experimental Procedure

21Materials and Specimens The investigatedmaterial in thepresent study is mild carbon Q235B structural steel which iswidely used in Chinarsquos steel constructions The investigatedQ235B steel has Youngrsquos modulus of 204GPa yield strengthof 270MPa ultimate strength of 390MPa Poissonrsquos ratio of03 and elongation of 369 The chemical composition ofQ235B steel is presented in Table 1

The base-metal specimen has a tubular geometry withthe outside and inside diameters of 18mm and 14mmrespectively The wall thickness in the gage section is 2mm

Table 2 Chemical composition of welding wire (wt)

C Mn Si P S V Ni Cr Mo Cu0077 154 092 0011 0012 0002 0006 0023 0004 0126

The geometry of the base-metal specimen is displayed inFigure 1

The welded specimen is made by the manual CO2 gas-shielded welding processThe welding wire of MG70S-6 witha diameter of 2mm is used The chemical composition ofwelding wire is presented in Table 2

The manufacture of the welded metal specimen is fol-lowed by [18] A well-designed notch is first machined at thecenter of the base metal bar and the notch is then filled withweld metal Finally the welded-metal specimen is machinedto the shape in accordance with the base-metal specimenThemanufacturedwelded-metal specimenhas an 18mm longwelded zone at the center of the gauge length The geometryof the welded thin-walled tubular specimen tested is identicalto the base one (see Figure 2)

The monotonic mechanical properties for the base-metaland welded-metal specimens are listed in Table 3

22 Fatigue Tests Fatigue tests under uniaxial in-phase and90∘ out-of-phase loading conditions are conducted under thefully reversed strain-controlled loading at constant ampli-tudes The applied waveforms for both base-metal andwelded-metal specimens are sinusoidal The three test strainpaths are displayed in Figure 3 The horizontal axis is theterm of axial strain 120576 and the vertical axis is the term ofshear strain 120574radic3 The correlation of the horizontal axis andthe vertical axis is derived from the von Mises criterion of120576 = radic1205762 + (13)1205742 in which 120576 is the equivalent von Misesstrain





120574radic3

120576

(a)

120574radic3

120576

(b)

120574radic3

120576

(c)



100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

120576 = 0006120576 = 0008

120576 = 0010

minus0012 0012minus0006 00060

(a)

120576 = 0006120576 = 0008

120576 = 0010


100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

minus0012 0012minus0006 00060

(b)







minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300

minus0008 0008minus0004 00040

radic3120591

(MPa

)

120574radic3

(a)

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300


(MPa

)

120574radic3

(b)





Δ1205762 = Δ1205761198902 + Δ1205761199012 = Δ1205902119864 + ( Δ12059021198701015840)11198991015840 (1)






Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400


100200300400

minus400


120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3120591

(MPa

)


(a)

minus100


Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400

minus400

minus0008 0008minus0004 00040

minus300minus200

100200300400


minus0008 0008minus0004 00040

120574

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3

radic3

120591(M

Pa)

(b)


(a) (b)





(a) (b)


0 0002 0004 0006 0008 001 00120

100

200

300

400

500




Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)






Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)













Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







120574radic3

120576

(a)

120574radic3

120576

(b)

120574radic3

120576

(c)



100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

120576 = 0006120576 = 0008

120576 = 0010

minus0012 0012minus0006 00060

(a)

120576 = 0006120576 = 0008

120576 = 0010


100200300400

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

minus0012 0012minus0006 00060

(b)







minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300

minus0008 0008minus0004 00040

radic3120591

(MPa

)

120574radic3

(a)

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300


(MPa

)

120574radic3

(b)





Δ1205762 = Δ1205761198902 + Δ1205761199012 = Δ1205902119864 + ( Δ12059021198701015840)11198991015840 (1)






Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400


100200300400

minus400


120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3120591

(MPa

)


(a)

minus100


Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400

minus400

minus0008 0008minus0004 00040

minus300minus200

100200300400


minus0008 0008minus0004 00040

120574

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3

radic3

120591(M

Pa)

(b)


(a) (b)





(a) (b)


0 0002 0004 0006 0008 001 00120

100

200

300

400

500




Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)






Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)













Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300

minus0008 0008minus0004 00040

radic3120591

(MPa

)

120574radic3

(a)

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

120576 = 0006

120576 = 0004

120576 = 0008

120576 = 0010

minus300

minus200

minus100

Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100

200

300

Shear strain

Shea

r stre

ss

minus0008 0008minus0004 00040

minus300

minus200

minus100

100

200

300


(MPa

)

120574radic3

(b)





Δ1205762 = Δ1205761198902 + Δ1205761199012 = Δ1205902119864 + ( Δ12059021198701015840)11198991015840 (1)






Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400


100200300400

minus400


120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3120591

(MPa

)


(a)

minus100


Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400

minus400

minus0008 0008minus0004 00040

minus300minus200

100200300400


minus0008 0008minus0004 00040

120574

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3

radic3

120591(M

Pa)

(b)


(a) (b)





(a) (b)


0 0002 0004 0006 0008 001 00120

100

200

300

400

500




Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)






Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)













Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400


100200300400

minus400


120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3120591

(MPa

)


(a)

minus100


Axial strain 120576

Axial

stre

ss 120590

(MPa

)

100200300400

minus400

minus0008 0008minus0004 00040

minus300minus200

100200300400


minus0008 0008minus0004 00040

120574

120576 = 0004

120576 = 0005

120576 = 0006

120576 = 0004

120576 = 0005

120576 = 0006

Shea

r stre

ss radic

3

radic3

120591(M

Pa)

(b)


(a) (b)





(a) (b)


0 0002 0004 0006 0008 001 00120

100

200

300

400

500




Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)






Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)













Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) (b)


0 0002 0004 0006 0008 001 00120

100

200

300

400

500




Equi

valen

t stre

ss a

mpl

itude

Δ1205902

(MPa

)






Δ1205762 = 1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888 (3)













Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









Δ120574max2 + 119878Δ120576119899 = [1 + ]119890 + (1 minus ]119890) 119878] 1205901015840119891119864 (2119873119891)119887

+ [1 + ]119901 + (1 minus ]119901) 119878] 1205761015840119891 (2119873119891)119888 (4)







0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








0001

0002

0004

000600080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 041E + 041E + 031E + 02


(a)

0001

0002

0004

0006

00080010




Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


1E + 021E + 01

(b)



Δ120574max2 (1 + 119896120590119899max120590119910 )

= [(1 + ]119890) 1205901015840119891119864 (2119873119891)119887 + (1 + ]119901) 1205761015840119891 (2119873119891)119888]

sdot (1 + 119896 12059010158401198912120590119910 (2119873119891)119887)

(5)




Δ120574max2 + (1 + 120590119899max120590119910 ) Δ1205761198992= [1205901015840119891119864 (2119873119891)119887 + 1205761015840119891 (2119873119891)119888](1 + 1205901015840119891120590119910 (2119873119891)

119887) (6)



0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0001

0002

0004

000600080010

0020


Uniaxial

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 031E + 02


(a)


0001

0002

0004

000600080010

0020

In-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

1E + 041E + 03


5E + 041E + 02

(b)


0001

0002

0004

000600080010

0020

Out-of-phase

Equi

vale

nt st

rain

ampl

itude

Δ1205762

5E + 031E + 03


1E + 02

(c)






M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




M p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)



KBM

par

amet

er

10minus3

10minus2

10minus1

104103102101


(b)


FS p

aram

eter

Base-metal specimen



10minus3

10minus2

10minus1

105104103102

(a)

FS p

aram

eter


10minus3

10minus2

10minus1


104103102101


(b)



6 Conclusions






KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




KBM

par

amet

er

Base-metal specimen


10minus3

10minus2

10minus1

105104103102


(a)M

KBM

par

amet

er


10minus3

10minus2

10minus1

104103102101



(b)





Competing Interests


Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



multiaxial cycle deformation and low-cycle fatigue behavior of...

Documents