Analysis of damage localization for ductile metal in process of shear band propagation

Available online at www. sciencedirect.comBCIENCETransactions ofNonferrous MetalsSociety of ChinaScienceTrans. Nonferrous Met. Soc. China 16(2006) 153 -158Presswww.csu.edu.cn/ysxb/Analysis of damage localization for ductile metalin process of shear band propagationWANG Xue-bin (王学滨)Department of Mechanics and Engineering Sciences, Liaoning Technical University, Fuxin 123000, ChinaReceived 10 March 2005; accepted 19 April 2005Abstract: Distribution of localized damage in shear band can' t be predicted thereically based on classical elastoplastic theory. Theaverage damage variable in shear band was considered to be a non-local variable. Based on non-local theory, an analytical expression forthe localized damage in strin-oftening region of shear band in the process of shear band propagation was presented using boundarycondition and symmetry of local damage variable, etc. The results show that dynamic shear softening modulus, dynamic shear strengthand shear elastic modulus influence the distribution of the localized damage in shear band. Internal length of ductile metal only govemnsthe thickness of shear band. In the strain softening region of shear band, the local damage variable along shear band's tangential andnormal directions is non-linear and highly non-uniform. The non-uniformities in the normal and tangential directions of shear band stemfrom the interactions and interplaying among microstructures and the non-uniform distibution of shear stress, respectively. At the tail ofthe strain-softening region, the maximum value of local damage variable reaches 1. This means that material at this postion fracturescompletely. At the tip of shear band and upper as well as lower boundaries, no damage occurs. Local damage variable increases asdynamic shear softening modulus decreases or shear elastic modulus increases, leading to difficulty in identification or detection ofdamage for less ductile metal matenial at higher strain rates.Keywords: shear band; ductile metal; damage localization; non-local theory; strain rate; shear stress; strain-softeningalloy and have been studied extensively[7, 11, 12].1 IntroductionWANG et al[13, 14] adopted gradient-dependentplasticity to investigate shear strain localization ofShear localization is an important and oftenductile metals, such as Ti and Ti alloy, in static[13] anddominating deformation and failure mechanism for Tidynamic loadings[14]. WANG[15] proposed a methodand Ti alloy in dynamic loadings[1- -10]. The eventualfor calculation of temperature distribution in adiabaticoutcome of localized deformation is ductile rupture andshear band in terms of the same theory. Beside shearmaterial separation. Shear localization occurs and playsstrainlocalization of ductile metal materials,an important role in engineering applications.gradient-dependent plasticity has been applied intoTo predict the distribution of plastic shear strain ininvestigation of tensile strain localization for Ti and Tishear band and the thickness of the band, somealloy[16]. .modifications and generalization from the standardSome experimental observations[1- 3, 10] showedcontinum description must be caried out. One of thethat for ductile metal materials, at the initial loadingmost promising approaches is the second order gradientstage， microcracks and induced damage appearcontinuum that incorporates the second order spatialrandomly and the distributions of deformation angradients of plastic strain in the yield function. Indamage variable are relatively uniform. With thgradient- dependent plasticity, the characteristic lengthincrease of strain at loading direction, damage and straindescribes the interactions and interplaying amongwithin the specimen progressively accumulate ormicrostructures. For Ti and Ti alloy the texture isconcentrate into a certain narrow zones and localizationheterogeneous to some extent and a certainofdaI中国煤化工. arrow zone is usuallymicrostructure will be influenced significantly by itsreferband. Afterwards, inneighborhoods. Interactions and interplaying amongthe prHC N M H Gf material, the lengthmicrostructures are of great importance for Ti and Tiof the band is increased and considerable damages andFoundatlon tem; Project(2004F052) supported by the Educational Department of Liaoning Province, Chinarresponding author: WANG Xue bin; Tel: +86 418-3351351; E-mail: wxbbb@ 263.net154WANG Xxue-bin/Trans. Nonferrous Met. Soc. China 16(2006) 153-158deformations are absorbed continuously by the bandband. At the tip of shear band, shear stress is maximumuntil ductile rupture and material separation take place.and attains the shear strength T。。The tip movesDistributions of damage and strain have hitherto beentowards the right in the process of shear bandmodeled numerically based on many kinds of modifiedpropagation. When a certain length of shear band iand generalized elastoplastic theories[17- -23]. However,reached, the shear stress at point 0 decreases to residualanalytical solution of localized damage in shear band inshear strength Tf. At the moment, strain-softeningthe process of shear band propagation has not beenzone is well formed. Afterwards, residual zone appearspresented yet so far.and its length increases. Strain-softening zone movesIn the paper, the average damage variable in sheartowards the right continually. For the sake of calculation,band was considered to be a non-local variable. Basedit is assumed that the shear deformation only occurs inon the non-local theory, an analytical expression forthe horizontal direction. The total length of shear bandlocalized damage in strain-softening region of sbearis L1+L2. The thickness of shear band is d. L is theband in the process of shear band propagation waslength of the residual zone. Lr is the length ofpresented using boundary condition and symmetry ofstrain-softening zone. In front of the tip of shear band,local damage variable, etc. Influences of relatedshear stress is lower tan the shear strength and materialparameters on the distribution of the local damagestill remains elastic.variable were investigated through a few examples.Some experimental results show that the post-peakstress-strain curve of Ti or Ti alloy under dynamic2 Analysisloadings exhibits approximately linear strain-softening2.1 Mechanical model for shear band propagationbehavior[6, 24]. Dynamic and static post-peak constitu-tive relations are shown in Fig.2.and basic assumptionsA mechanical model for shear band propagation[ 14](x)4is shown in Fig.1. A Ti block with a certain height andlength is loaded in horizontal shear stress t(x) and invertical compressive stress σ。y-axis and x-axis arecDynamievertical and horizontal, respectively.Strain-softeningStaticElastic zonezoneResidual%(x)Fig.2 Dynamic and static post-peak constiutive relations[14][(2.2 Analysis of elastic, plastic and total strains inTail ofstrain-softening zone of shear bandshear bandAccording to shear Hooke's law, the elastic shearstrain Y (x) in strain-softening zone can be expressedo叶L:asShear band |Tip of shear bandre(x)=T(x)(1)where G is the shear elastic modulus; T(x) is theFig.1 Shear stress acting on shear band and mechanical model forshear stress, and x∈[LI, L+L2].shear band propagation[14]According to WANG et al[13, 14], the plastic shearAn important outcome of shear localization is thestrain 7p(x, y) is non-uniform in the normal direction ofdecrease of the stress- carrying capability of the block.shear band and it is written as中国煤化工Therefore, when shear stress at point 0 reaches the(2shear strength, shear localization is initiated at the pointMHCNMHG.and a horizontal shear band is formed. Then， itpropagates towards the right and the shear stress at pointwhere y∈[- d/2; d/2]; τ。and τ。 are the dynamic0 begins to decrease. Point 0 is called the tail of shearnd static shear strengths, respectively, τc= frc; fWANG Xue bin/Trans. Nonferrous Met. Soc. China 16(2006) 153-158155is a coefficient considering strain rate effect anHerein, the average damage variable D(x) isf =1+Cln(γ/r)，where C is a material constant,considered to be a non-local variable. On the basis ofYo and Y are the strain rates in static and dynamicthe non-local elasticity model[25], the relation amongloading conditions, respectively; c' and c are thethe non-local damage variable D(x) , the local damagedynamic and static shear softening moduli, respectively,variable D(x,y) and its second spatial derivativeand c'=fc; and l is the intermal length parameter ofd2D(x, y)/dy2 can be derived as follows:ductile metal material describing the extent 0d2D(x,y)heterogeneity. According to gradient-dependentD(x)= D(x, y)+!2dy2(9)plasticity[13，14]， the relation between l and thethickness of shear bandis d = 2rl.It is noted that the derivation of Eqn.(9) is similarf Eqn.(2) is integrated with respect to thecoordinate y and then divided by d, then the averageto that of the local plastic shear strain yp() in Ref. [13].The following equation can be obtained by usingplastic shear strain 平p(x) in shear band can beEqn.(9)obtained:pd2Yp(xy)dy_ r。-t(x)d2D(x,y), D(x,)_ D(x)(10)后(x)=-切d(3)Obviously, this is a second order homogeneousThe total shear strain j(x) in shear band is theordinary differential equation. As mentioned above, thesum of elastic re(x) and plastic平(x) parts, namelythickness of shear band is determined by the intemallength parameter, i.e, d=2πl. Consequently, the fol-t(x)、tc-t(x)lowing two conditions are needed to solve the(x)=%。(x)+%(x)=(4)differential equation above:GD(x,y=+d12)=0(11)D(x, y)= D(x,-y)(12)2.3 Average damage variable in shear bandIn fact, Eqn.(11) is a boundary condition. ItAccording to classical damage mechanics, therequires that no any damage occur at upper and lowerrelation among the shear stress, the total shear strain r,boundaries of shear band. Eqn.(12) requires that thethe shear elastic modulus G and the damage variable Dlocal damage variable is symmetrical with respect to theiscoordinate y and it is a even function due to theτ= G(1- D)y(5)assumption of isotropic metal materials.The solution of Eqn.(10) can be obtained usingHerein, to establish the expression for the averagedamage variable D(x) in shear band, we generalizedEqn.(11) and Eqn.(12):and modified Eqn.(5) asD(x, y)= D(x) 1+ cos(13)r(x)= G[-2D(x)]v(x)(6)Substitution of Eqn.(8) into Eqn.(13) results in theThe differences between Eqn.(6) and Eqn.(5) arefollowing expression:obvious: D(x) is concemed with coordinate x; thecoefficient in front of D(x) is 2, not 1. Advantages ofthe present special definition will be discussed below.Using Eqn.(6)，D(x) is expressed as| 1+ cos(14)D(x)=1- (x)(7)G(x) )See Fig.I, if we use the following coordinatetransformationSubstitution of Eqn.(4) into Eqn.(7) leads tox=x'+L(15)D(x)=;(8)中国煤化工[岳一"(16)MHCNMHG2.4 Non-local theory and local damage variable inwhere x' ∈[0, L2] and y'∈[0, d], then Eqn.(14) can beshear bandwritten as156WANG Xue-bin/Trans. Nonferrous Met. Soc. China 16(2006) 153-158The length L2 of strain softening zone is assumed to beD(x',y)=20 times the thickness of shear band. For simplicity, weletτ, =0.Due to the non-uniform deformation of specimen(1+2y'-d~(17)beyond the onset of shear band or strain localization, themeasured stress-strain curve is not a purely mechanicalSee Fig.1， T(x') in the coordinate system ofproperty or constitutive relation. The measuredstress-strain curve also includes the contribution ofx'O'y' can be expressed asgeometrical size of specimen unless the size of theT(x')=-Tc-tf.x'+τj(18)specimen is small enough. Consequently， usually,2dynamic and static shear softening moduli cannot bedetermined through experimental tests. The pheno-3 Examplesmenon is similar to “size effect” in rock and soilmechanics. As a result, firstly, the influence of staticThe thicknesses of Ti and Ti-6A1-4V are about 10softening modulus c is studied and the distributions of- 55 μm[1, 24]. Herein, we let d=35 pμm. We can obtainthe local damage variable for different static softeningthe intemal length parameter describing themoduli are shown in Figs.3 and 4 with f =1, respec-heterogeneity is about l=5.57 μm using d=2 π l. Experi-tively.mental measurements show that the shear elastic moduliSecondly, the influence of strain rate on thefor Ti and many kinds of Ti alloy are about 45 GPa.distribution of the local damage variable is shown inAcordingly, we let G= =45 GPa. Static shear strength ofFig.5 with f=2..5 and r' = ft.=700 MPa. Finally, the .Ti (T<=σJ2, where σc is the yield stress in uniaxialinfluence of shear elastic modulus on the distribution oftension) is about 280 MPa. Herein, we letrc=280 MPa.the local damage variable is shown in Fig.6 withf=1 and.0.80.6! 10).4多20.2热心2040400x/1.75X 10-'m(2)(bFig.3 Three dimensional curved surface and contour map with e =30 GPa1.0， 100.43 20中国煤化工1015YHCNMHG(a(b)Fig4 Three-dimensional curved surface and contour map with c=3 GPaWANG Xue-bin/Trans. Nonferrous Met. Soc. China 16(2006) 153-1581570.850.42028 10 26m4x/1.75X 10~5m400+a)b)Fig.5 Three-dimensional curved surface and contour map withf=2.521.0个0.6S。至0.0.220100010s。0103040“》x/1.75X 10-5 ma)(bFig.6 Three dimensional curved surface and contour map with G=80 GPaG=80 GPa.present analytical prediction for the local damageAt the tip of shear band (x'=7X 104 m) and at twovariable in localized band is consistent with the relatedboundaries (y'=0 and y'=3.5X 10~5 m), the localnumerical results[17- 23]. .damage variable is always zero and no any damageThree- dimensional curved surface near the tip ofexists. At the tail of shear band (x =0), the maximumshear band becomes more steepdynamic shearlocal damage variable is I, which suggests that metalsoftening modulus decreases or shear elastic modulusmaterial at this site has fractured completely, as is inincreases, while it exhibits less steep at the tail of shearagreement with our common knowledge, reflecting theband, see Figs.3(a), 4(a), 5(a), 6(a). That is to say, theadvantage of the special expression Eqn.(6). If thelocal damage variable in shear band is increased. Areacofficient in front of D(x) in Eqn.(6) is 1, then thewith higher local damage variable is enlarged ilmaximum value of the calculated local damage variabletwo-dimensional contour maps as the local damagewill be 2, not 1, as is difficult to understand and notvariable increases, as can be seen from Figs.3(b), 4(b),consistent with usual viewpoints.5(b), 6(b).In shear band, the local damage variable in x' andThe result that increasing dynamic softeningy' directions is highly non-uniform. In y' direction,modulus leads to a decrease of the local damagethe reason for the non-uniformity is due to thevariable in shear band means that less ductile metalinteractions and interplaying among microstructures.material at higher loading rates possesses a lower localHowever, in x' direction, the non-uniformity is causeddan中国煤化工a certain difficulty inby the non-uniform distribution of shear stress r(x').ider iYHCNMHGIn the strain-softening zone of shear band, it is assumedthat I (x') is linear distribution in calculation. However,4 Conclusionsthe obtained distribution of the local damage variableexhibits non-linear characteristic. Qualitatively, the1) The average damage variable in shear band is.15WANG Xue-bin/Trans. Nonferrous Met. Soc. China 16(2006) 153-158considered to be a non-local variable. Based on themechanism in adiabatic shear band in TA2[]. Trans Nonferrous MetSoc China, 2004, 14(4); 670 - -674.non-local theory, an analytical expression for th[8] YANG Yang, WANG Zhao-ming. ZHANG Shao-rui. Somelocalized damage in strain-softening region of shearmetallurgical behaviours of adiabatic shear band on Ti side in theband in the process of shear band propagation isTV/mild steel explosive cladding interace[J]. Rare Metal and Materialspresented using boundary condition and symmetry ofand Engineering, 1997, 26(4); 13- 17. (in Chinesc)local damage variable, etc.19] TAMIZIFAR M, OMIDVAR H, MOHAMMAD TAGHI SALEHI s.2) The resulting theoretical expression for localizedEffet of processing parameters on shear bands in non-isothermal hotforging of Ti 6AI-4V[]. Materials Science and Technology. 2002, 18(1):damage in shear band shows that dynamic shear21-29.softening modulus, dynamic shear strength and shear[10] BAI y, BODD B. Adiabatic Shear Localization[M]. Oxford, UK:elastic modulus influence the distribution of thePergamon Press, 1992.localized damage in shear band. However, internal[11] WANG Jian-bua, YI Dan-qing, WANG Bin. Microstructure andlength parameter of ductile metal only goverms theproperties of 2618-Ti heal resistat alumioum aloyl]. TransNonferous Met Soc China, 2003, 13(3): 590- 594.thickness of shear band.3) In the strain-softening region of shear band, the[12] SHANG Jun-ing, uI Bang-sheng, GUO Jingjie. Microstucturecvolution during preparation of in-situ TiB reinforced titanium marixlocal damage variable along shear band's tangential andcomposite[J. Tans Nonterrous Met Soc China, 2003. 13(2): 315-normal directions is non-linear and highly non-unifom.Non-uniformities of the local damage variable in the[13] WANG Xue-bin, DAI Shu-hong, HAI Long, Analysis of localized shcarnormal and tangential directions of shear band stemdeformation of ductile metal based on gnadient dependcnt plastity[].Trans Noferrous Met Soc Chine. 2003, 13(6): 1348- 1353.from the interactions or interplaying armong[14] WANG Xuc-bin, YANG Mei, YU Hai-jun. Localized shear deformationmicrostructures and the non-uniform distribution ofduring shear band propegation in Titanium considering interactionsshear stress acting on the band, respectively.among micosrucures[]. Trans Nonferous Met Soc China, 2004,4) At the tail of the strain- softening region, the14(2): 335- -339.maximum value of the local damage variable reaches 1.[15] WANG Xue bin. Calculation of temperature distribution in adiabaticThis means that material at this position fails completely.sbear band based on gradient-dependeat plasticity{J]. Trans NonferrousMet Soc China, 2004. 14(6): 1062- 1067.At the tip of shear band and upper as well as lower[16] WANG Xue bin, YANG Mci, ZHAO Yang-feng. Analytical solutioo forboundaries, the local damage variable is always zerotrue sresstrue strain curve for titanium and T1-6AI-4V in uniaxialand no any damage occurs.tension considering intercions among microstructures[J]. Rare Metal5) Except for the tip and the tail of shear band andMalerials and Eginering. 2005. 34(3): 346- 349.its two boundaries, the local damage variable within[17] DE BORST R, PAMIN J, GEERS M G D. On coupledgradient-dependeat plasticity and damage theories with a view toshear band is increased as dynamic shear softeningmodulus decrease or shear elastic modulus increases,[18] COMI C. Computational modeling of gradient enhanced damage inleading to difficulty in identifcation or detection 0gquasi-brittle materials{J]. Mecbanics of Cobesive Frictional Materials,damage for less ductile metal material at higher strain1999,4(]): 17- -36.rates.{19} PIAUDIER-CABOT G, HAIDAR K, DUBE J F. Non-Jocal damagemodel with evolving intemnal lengtb[]. Int J Num Anal MethodsGcomech, 2004, 28(7- 8): 633- 652.References[20] TANG c Y, FAN J P, LEE T C. Simulation of necking using a danagecoupled finite element mecthod [0]. Journal of Materials Processing[{1] NESTRRENKO V F, MEYERS M A. WRIGHT T w. Self. organizationTechnology. 2003, 139(1- 3): 510- -513.in the iniation of adiabatic shear bands[J]. Acta Mater, 1998, 46(1):[21] 1.I C, ELLYIN F Mesomechanical prpach to inhomogeacous327- 340.particulate composite undergoing localized damage(Part 1l) - theory[2! XU Y B, ZHONG W I, CHEN Y J. Sheur localization andand pplicatio[J]. International Journal of Solids and Strutures, 2000,rersallization in dynarnic deformation of 8089 Al-Li alloy[小].37(10): 1389- 1401.Malerials Science and Engineering A, 2001, 299(1-2);: 28[22] CHUNG s W, CHOI Y」KIM s J. Computational simulation of|31 XUE Q MEYERS M A. NESTERENKO v F. Self organization ofloealized damage by finite element remeshing based on bubble packingshear bands in titanium and Ti 6Al-4V alloy[J]. Acta Materialia 2002,method[J]. Computer Modeling in Engineering and Sciences. 2003, 4(6):50(3): 575- 596.707 - -718.[4] YU Jin-qiang, ZHOU Hui-bua, SHEN Lc tian. Thermo plastic sheur[23] PATZAK B, JIRASEK M. Adaptive resolution of localized damage inbands induced during dynamic loading in T-55 alloys[J]. Actaqusibrite materials[J]. Journal of Engineering Mechanics. ASCE,Metalurgica Sinica, 1999, 35(4): 379- 383. (in Chincsc)2004. 130(6): 720- -732.[5] 1.U Wea sheng, HUANG Bai -yun, HE Yuc-hui. The room temnperalure[24] LIAO s C, DUFFY J. Adiabatic shear bands in a T-6AI 4V Titaniumfracture mechanics of a fuly-laelara TiAI aloy[J]I J Cent South UnivTchnol, 1997, 28(2): 152- 155.(in Chinese)[2s中国煤化工0301 23On nonlocal catitDt. In }[6] LI Qiang, XU Yong bo, SHEN Le-tian. Dynamic mechanical propertiesand damage churcteristics of ticanium alloy (Ti-17)UI. ActuMHCNMHG(Edited by LI Xiang-qun)Meallurgica Sinica, 1999, 35(5): 491- 494. (in Chinese)[7] YANG Yang, XIONG Jun, YANG Xu-yue. Microstructure evolution