Finite deformation analysis of mechanism-based strain gradient ...

International Journal of Plasticity 19 (2003) 235–251www.elsevier.com/locate/ijplasFinite deformation analysis ofmechanism-based strain gradient plasticity:torsion and crack tip fieldK.C. Hwang a , H. Jiang a , Y. Huang b, *, H. Gao ca Failure Mechanics Laboratory, Department of Engineering Mechanics, Tsinghua University,Beijing 100084, People’s Republic of Chinab Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, USAc Division of Mechanics and Computation, Stanford University, Palo Alto, CA 94305, USAReceived in final revised form 27 June 2001AbstractA finite deformation theory of mechanism-based strain gradient (MSG) plasticity is developedin this paper based on the Taylor dislocation model. The theory ensures the properdecomposition of deformation in order to exclude the volumetric deformation from the straingradient tensor since the latter represents the density of geometrically necessary dislocations.The solution for a thin cylinder under large torsion is obtained. The numerical method is usedto investigate the finite deformation crack tip field in MSG plasticity. It is established that thestress level around a crack tip in MSG plasticity is significantly higher than its counterpart(i.e. HRR field) in classical plasticity. # 2002 Elsevier Science Ltd. All rights reserved.Keywords: Finite deformation; Strain gradient plasticity; Torsion; Fracture1. IntroductionThe size dependence observed in recent experiments at the micron and submicronscales (e.g. Fleck et al., 1994; Lloyd, 1994; McElhaney et al., 1998; Stolken andEvans, 1988) and in direct dislocation simulations (Cleveringa et al., 1997, 1998,1999a,b; 2000; Shizawa and Zbib, 1999; Needleman, 2000; Zbib and de la Rubia,2001) have motivated the development of strain gradient plasticity theories based onthe concept of geometrically necessary dislocation (Nye, 1953; Cottrell, 1964;Ashby, 1970; Arsenlis and Parks, 1999; Gurtin, 2000). The strain gradient serves* Corresponding author. Tel.: +1-217-265-5072; fax: +1-217-244-6534.E-mail address: huang9@uiuc.edu (Y. Huang).0749-6419/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.PII: S0749-6419(01)00039-0

236 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251either as an independent measure of deformation besides the strain (e.g. Fleck andHutchinson, 1993, 1997; Shu, 1998; Gao et al., 1999; Shu and Fleck, 1999; Huang etal., 2000a,b; Shuand Barlow, 2000; Gurtin, 2001), or as an internal variable toincrease the plastic work hardening modulus (e.g. Nilsson, 1998; Acharya and Bassani,2000; Acharya and Beaudoin, 2000; Chen and Wang, 2000; Dai and Parks,2001; Meissonnier et al., 2001). There are also early works on strain gradient plasticitythat were proposed in order to avoid a spurious solution for the localized zoneand an excessive mesh dependence in classical plasticity (e.g. Aifantis, 1984, 1992;Lasry and Belytschko, 1988; Zbib and Aifantis, 1988; Muhlhaus and Aifantis, 1991;de Borst and Muhlhaus, 1991, 1992; Sluys et al., 1993).Since the aforementioned micron and submicron scale experiments involve largedeformation, Hwang et al. (2001) generalized Fleck and Hutchinson’s (1997) andGao et al.’s (1999) strain gradient plasticity theories from infinitesimal to finitedeformation. The equilibrium equations and boundary conditions are established inthe current configuration, while the constitutive relations are obtained by nominalgeneralization from the corresponding infinitesimal deformation theories. The infinitesimalstrain " ij and its gradient " ij,k are replaced by the Green–Lagrange strainE IJ and the corresponding gradient in the reference configuration, E IJ,K , respectively.Such a nominal generalization, however, does not always ensure the appropriatedecomposition of deformation into the volumetric and deviatoric parts. Forexample, " kk is the volumetric strain in infinitesimal deformation, but its nominalgeneralization E KK does not represent the volumetric strain in finite deformation.Instead, only the determinant J=det(F) of the deformation gradient F= @x@X representsthe volume change in finite deformation, where x and X denote the coordinatesof a material point in the current and reference configurations, respectively.The purpose of this paper is to develop a finite deformation strain gradient plasticitytheory that ensures the proper decomposition of volumetric and deviatoricdeformation. It is a generalization of a classical finite deformation plasticity theoryin Section 2 that excludes the volumetric deformation. We focus on the finitedeformation theory of mechanism-based strain gradient plasticity (Gao et al., 1999;Huang et al., 2000a,b), though the approach can also be applied to Fleck andHutchinson’s (1997) strain gradient plasticity theory. We then present the analyticalsolution for a thin cylinder under torsion, and numerical study of the crack tip field.2. A finite deformation theory of classical plasticityWe first present a finite deformation theory of classical plasticity that excludes thevolumetric deformation. Let X and x denote the coordinates of a material point inthe reference (undeformed) and current (deformed) configurations, respectively. Thedeformation gradient F= @x@X can be expressed in terms of the principal stretches l 1,l 2 and l 3 asF ¼ l 1 n 1 N 1 þ l 2 n 2 N II þ l 3 n 3 N III ;ð1Þ

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 237where n 1 ; n 2 ; n 3 and N I ; N II ; N III are the principal stretch directions in the currentand reference configurations, respectively. The volumetric deformation is characterizedby the determinant of F;J ¼ detðFÞ ¼ l 1 l 2 l 3 :ð2ÞThe right and left Cauchy–Green strain tensors C and B and the Green straintensor E are defined byC ¼ F T F; B ¼ FF T ; E ¼ 1 ð2 C 1 Þ; ð3Þwhere 1 is the second-order identity tensor.In order to exclude the volumetric deformation from F, we define a modifieddeformation gradientF ¼ J 1 3 F;ð4Þ which has no volumetric deformation since F =1. The isochoric right and leftCauchy-Green strain tensors, C and B , and the isochoric Green strain tensor E ,which all exclude the volumetric deformation, can be defined similar to (3) via F byC ¼ F T F; B ¼ F F T ; E ¼ 1 2 C 1 : ð5ÞIt can be shown that E degenerates to deviatoric strain tensor for infinitesimaldeformation. The invariants of C and B areI 1 ¼ I C ¼ I B ¼ ðl 1 l 2 l 3 Þ 2 3 l21 þ l2 2 þ 3 l2 ; ð6ÞI 2 ¼ II C ¼ II B ¼ ðl 1 l 2 l 3 Þ 2 3 l21 þ l 22 þ l 2 3 ;and III C ¼ III B =1. The effective strain is defined by" 2 ¼ 2 3 E : E ¼ 1 6 I 2 1 2I 1 2I 2 þ 1 2 ;ð7Þð8Þwhich degenerates to the von Mises effective strain for infinitesimal deformation.For a deformation theory of plasticity, the strain energy density function U (perunit volume in the reference configuration) of an isotropic solid generally takes theform U ¼ U J; I 1 ; I 2¼ U V ðÞþU JD ðÞ "ð9Þ

238 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251where U V and U D are the strain energy densities for the volumetric and deviatoricdeformation, respectively. The second Piola–Kirchhoff stress T is the work conjugateof the Green strain E, and is obtained from (9) as" ! #T ¼ @U@E ¼ J dUVdJ C1 þ 23" J 2 3 E " 2 þ E KKC 1; ð10Þ3where C 1 is the reciprocal of the right Cauchy–Green strain tensor C in (3), E KK isthe first invariant of E , and = dUDd" . The simplest form for UV and U D isU V ¼ K 2 ðJ 1Þ2; U D ¼ refN þ 1 "Nþ1 ;ð11Þwhich gives = dUDd" = ref" N , where K is the elastic bulk modulus, N(

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 239 l ¼ 18 2 2b; ð14Þ refwhich is indeed on the order of microns for typical metallic materials; via three dislocationmodels the effective strain gradient has been determined in terms ofdeviatoric strain gradient (Gao et al., 1999), and it is generalized for finite deformationby ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IJK IJK ð15Þ2Here IJK is the strain gradient tensor excluding the volumetric deformation and isgiven in terms of the isochoric Green strain E in (5) by IJK ¼ E IK;J þ E JK;I E IJ;K : ð16Þ4. A finite deformation theory of mechanism-based strain gradient plasticityGao et al. (1999) adopted a multiscale, hierarchical framework to develop amesoscale theory of mechanism-based strain gradient from the Taylor dislocationmodel on the microscale. The same framework, as shown in Fig. 1, is adopted in thepresent study in order to establish the corresponding finite deformation theory.Stress and strain are defined in the classical sense on the microscale, and aredenoted by T~ (second Piola–Kirchhoff stress) and E~ (Green strain), respectively,where tilde ½f... Š denotes the microscale measure. Concepts associated with strainFig 1. A schematic diagram of the multiscale framework to connect the mesoscale theory of strain gradientplasticity to the Taylor dislocation model on the microscale.

240 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251gradient plasticity are introduced on the mesoscale, such as the strain gradient tensor and higher-order stress , where is defined in terms of the mesoscale Greenstrain E by IJK ¼ E IK;J þ E JK;I E IJ;K ; ð17Þand is the work conjugate of . The microscale strain within the mesoscale cell(Fig. 1) is related to the mesoscale strain measures by the Taylor expansion,E~ IJ ¼ E IJ þ E IJ;K X~ K þ 0 X~ 2¼ E IJ þ 1 ð2 KIJ þ KJI ÞX~ K þ 0 X~ 2; ð18Þwhere X~ K is the local coordinate origined at the center of the mesoscale cell.4.1. Microscale analysisThe stress T~ and strain E~ on the microscale satisfy the constitutive relation (10)except the stress ~, which is governed by the Taylor dislocation model in (13),pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~ ¼ ref "~ 2N þ l;ð19Þwhere the microscale effective strain "~ is related to the microscale Green strain E~ inthe same way as in (8), and is the mesoscale effective strain gradient given in (15).The microscale constitutive relation now becomes 2 0T~ ¼ J~ dUV J~CdJ~~1 þ 2~ J~ 2 ~3 E "~ 2 þ E ~ 1 34 @KKAC 3"~3~1 5 ð20Þwhere the microscale variables can be expressed in terms of the mesoscale ones viaTaylor expansion up to the first order, such asJ~ ¼ J þ dJ1X~ K ¼ J þ JC MNKMNX~ K ; ð21ÞdX KC~ 1 IJ ¼ C 1 IJ C 1 IM C 1 JNð KMN þ KNM ÞX~ K ; ð22ÞE ~ IJ þ E IJ þ 1 ð2 KIJ þ KJI ÞX~ K : ð23Þ4.2. Mesoscale analysisThe mesoscale constitutive relations are derived from the work equality betweenthe micro- and mesoscales,

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 241ðT~ IJ E~ IJ dV ¼ ðT IJ E IJ þ IJK IJK ÞV cell ; ð24ÞVcellwhere the integration is over the mesoscale cell V cell in the reference configuration(Fig. 1), and stands for the virtual variation. Using the kinematics relation (18)between the strain measures on theses two scales, we obtain the mesoscale stress T IJand higher-order stress IJK in terms of microscale stress T~ IJ ,T IJ ¼ 1 T~ IJ dV; ð25ÞV cellðVcell IJK ¼ 1 T~ KI X~ J þ T~ KJ X~ I dV2V cellðVcellSubstituting the microscale constitutive relation (20) into (25) and (26), we obtainthe following finite deformation mesoscale constitutive relations for MSG plasticity," ! #T IJ ¼ KJðJ 1ÞC 1 IJ þ 23" J 2 3 E IJ " 2 þ E KKC 1 IJ ; ð27Þ3ð26Þ IJK þ l 2 "K6 V IJK þ " IJK þ 2 ref f ðÞf " 0 ðÞ "" IJK; ð28Þwhere is given in (13), l " =10 Yb and is less than 100 nm for typical metallicmaterials (Gao et al., 1999; Huang et al., 2000a,b), Y is the initial yield stress,V IJK ¼ 1 4 J ð2J 1ÞC 1 MN C 1 JK IMN þ C 1 IK JMNðJ 1Þ C 1 JM C 1 KNð IMN þ INM Þþ C 1 IM C 1 KNð JMN þ JNM Þ ; IJK ¼ 1 7243 J 4 13 C MNð IMN C JK þ IMN C IK ÞþJ 4 3ð2IJK þ IKJ þ JKI Þð Þ J 8 3 E MN IMN C 1 JK þ JMN C 1 IKþ 2 3 J 2 3 C1MN IMN JK þ IMN IK!23 IPP C 1 2JK3 JPP C 1 IK þ 2 " 2 þ E PP3!þ 2 " 2 þ E PPC 1 IM C 1 KNð JMN þ JNM Þ ;3C 1 JM C 1 KNð IMN þ INM Þð29Þð30Þ

242 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 IJK ¼ E " ! #MN54" 2 IMN J 2 3 E JK " 2 þ E PPC 1 JK3" ! #þ JMN J 2 3 E IK " 2 þ E PPC 1 IK :3ð31ÞThe above constitutive relations degenerate to Gao et al. (1999) and Huang et al.(2000b) for infinitesimal deformation.Based on the principle of virtual work, Hwang et al. (2001) derived the equilibriumequations and traction boundary conditions in terms of the second Piola–Kirchhoff stress and the higher-order stress. These equilibrium equations and tractionboundary conditions hold for finite deformation strain gradient theories,regardless of the constitutive law.5. Analytical and numerical studies5.1. Torsion of a thin cylinderWe investigate the torsion of a thin cylinder in this section. The cylinder has acircular cross section with an initial thickness h and mean radius R 0 , where R 0 is onthe order of microns or larger. For each material point, the cylindrical coordinatesin the reference (undeformed) and current (deformed) configurations are (R,Y,Z)and (r,,z), respectively, and they are related by (Fig. 2)r ¼ rðRÞ; ¼ þ K Z; z ¼ ð1 þ ÞZ; ð32Þwhere is the twist (angle of rotation per unit length in the axial direction), is thenominal axial strain to be determined due to finite deformation effect, and thefunction r(R) is also to be determined. The displacement isu ¼ re r þ ze zðRe R þ Ze z Þ; ð33Þwhere ðe r ; e ; e z Þ and ðe R ; e ; e z Þ are the unit vectors along the radial, circumferentialand axial directions in the current and reference configurations, respectively, andthey are related bye r ¼ cos K Ze R þ sin k Ze ; e ¼ sin K Ze R þ cos K Ze ; e z ¼ e Z : ð34ÞThe deformation gradient F is obtained from (33) asF ¼ drdR e re R þ r R e e R þ r R e e þ kre e z þ ð1 þ Þe z e Z ; ð35Þwhich has the determinant

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 243Fig 2. The cross section of a thin tube under torsion, where (r, , z) and (R, , Z) denote the cylindricalcoordinates in the current and reference configurations, respectively; is the twist (angle of rotation perunit length in the axial direction).J ¼ detðFÞ ¼ dr rðdR R 1 þ Þ: ð36ÞThe right Cauchy–Green strain tensor C in the reference configuration is obtainedfrom (3) 1 as0 dr 2 0 0dRrC ¼2 kr 20BR 2 R@ kr 20Rk2 r 2 þ 1 þ 21: ð37ÞCA

244 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251The Green strain tensor E and isochoric Green strain tensor E~ can be obtainedsimilarly from (3) 3 and (5) 3 .For a thin cylinder with the thickness h much less than the mean radius R 0 , theTaylor expansion of radial coordinate r in the current configuration givesr ¼ r 0 þdr ðR R 0 ÞþO ðR R 0 Þ 2 ; ð38ÞdR0where the subscript 0 denotes the variables at the center of cylinder wall. The rightCauchy–Green strain tensor C in (37) can also be evaluated at the center of cylinderwall to give the non-vanishing components asC RR ¼dr 2 ; C ¼ r 20; C Z ¼ C Z ¼ Kr2 0;dR0R 0R 0 ð39ÞC ZZ ¼ k 2 r 2 0 þ ð1 þ BÞ2:Similarly, the determinant of F in (36) becomes J= dr r 0dR 0R 0(1+), and the nonvanishingcomponents of the isochoric Green strain tensor E~ in (5) 3 are"E RR ¼ 1 # "2 J dr 2 23 1 ; E ¼ 1 #dR02 J r 2 2 03 1 ;R 0krE Z ¼ E 2 Z ¼ J 2 3 0; E ZZ ¼ 1 2R 0 2 J 2 3 k 2 r 2 0 þ 1 þ 2 ð40Þno1 :The isochoric strain gradient tensor in (16) can be written as ¼ 1; ð41Þ3 r0 J 2 3where the non-vanishing components are dr 2 RRR ¼ ; R ¼ R ¼ 2 r2 0dR0R 2 ;0 RZ ¼ RZ ¼ 2 kr2 0; R ¼ r 04 r 0þ 3 ;R 0 R 0 R 0 RZ ¼ ZR ¼ ZR ¼ ZR ¼ kro 23 ro þ 2R o R o RZZ ¼ ZRZ ¼ ZZR ¼ k 2 r 2 0 þ 1 þ 2 þ 3k 2 r 2 0drdR; ð42Þand = dr r 0dR 0 R 0. The effective strain and effective strain gradient are then obtainedby substituting (40)–(42) into (8) and (15), respectively. For a given twist and0

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 245cylinder radius R 0 , the above strains and strain gradients involve three parameters todrbe determined, namely r 0 ,dRand .0The constitutive law (27) gives the non-vanishing components of the second Piola–Kirchhoff stress as" ! #T RR ¼ 3" J 4 3 CRR J 2 3 2 " 2 þ E KKC 1 RR þ KJðJ 1ÞC 1 RR;3" ! #T ¼ 3" J 4 3 C J 2 3 2 " 2 þ E KKC 1 þ KJðJ 1ÞC 1 ;3" ! #ð43ÞT Z ¼ T Z ¼ 3" J 4 3 CZ 2 " 2 þ E KKC 1 Z þ KJðJ 1ÞC 1 Z;3" ! #T ZZ ¼ 3" J 4 3 CZZ J 2 3 2 " 2 þ E KK3C 1 ZZþ KJðJ 1ÞC 1 ZZ:The higher-order stress can be obtained similarly from (28). The substitution ofstress in (43) and higher-order stress into the equilibrium equations (Hwang et al.,2001) leadsto three ordinary differential equations involving a nondimensionall 2,parameter "R 0which is less 1% since R0 is larger than microns 2and l " is less thanl100 nm. Accordingly, we neglect the terms on the order of "R 0as compared tounity. This is consistent with the prior studies that suggest l " has essentially no effecton macroscopic quantities (Huang et al., 2000a,b). In conjunction with the tractionboundary conditions, the equilibrium equations can be integrated to give threealgebraic equationsT RR ¼ 0;T þ 2kR 0 T Z ¼ 0;ð44Þð45ÞT ZZ ¼ 0;drwhich are solved numerically to determine three unknown parameters, r 0 ,dR 0 and for each given twist .The Cauchy (true) stress can be obtained from the second Piola–Kirchhoffstress by ¼ 1 J FTFT ¼dr 1T Z ðe e z þ e z e Þ; ð47ÞdR0where the deformation gradient in (35) has been used. The Cauchy stress has onlythe shear component along the circumferential direction in the current configuration.In fact, it can be shown from kk =0 that three is no volumetric deformation inthe finite twist of a thin cylinder, i.e. J=1. The applied torque can be found fromð46Þ

246 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251(47) asT ¼ 2r 2 0 hT Z;ð48Þwhere h is the initial thickness of the cylinder and r 0 is the mean cylinder radius inthe current configuration.Fig 3 shows the normalized torque, T/(2R 2 0 h Y), versus the normalized twist,R 0 , for l/R 0 =1, 0.5, 0.1 and 0, where R 0 and h are the initial mean cylinder radiusand thickness in the reference configuration, l is the intrinsic material length in (14),the initial yield stress Y is 0.2% times the Young’s modulus E, andl/R 0 =0 correspondsto classical plasticity theory (without strain gradient effect). Other materialproperties include the Possion’s ratio =0.3,plastic work hardening exponentN.EN=0.2, and the reference stress ref = Y Y The empirical material constant inthe Taylor dislocation model and the Burgers vector b only appear through theintrinsic material length l in (14), and it is therefore not necessary to specify thevalues of and b for a given ratio l/R 0 . For a small twist, R 0

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 247effect becomes significant. For example, the classical plasticity theory (l/R 0 =0) predictsa maximum torque that occurs approximately at R 0 =0.5, but there is nomaximum torque for MSG plasticity (l/R 0 50). The curves for cylinder radius beingone or two times the intrinsic material length l (i.e. l/R 0 =1,0.5) are much higherthan that predicted by classical plasticity, which is clearly due to the strain gradientleffect. Even the curve for cylinder radius being ten times l (i.e.R 0=0.1) shows significantsize effect.Fig 4 shows the torque-twist relation for both infinitesimal and finite deformationtheories of classical plasticity (l/R 0 =0) and MSG plasticity (l/R 0 =1). The materialproperties and the normalizations are identical to those in Fig. 3. The curvesaccounting for finite deformation are significantly lower than those for infinitesimaldeformation, indicating the finite deformation effect is significant for R 0 >0.1.5.2. Mode-I fracture analysisJiang et al. (2001) used the infinitesimal deformation MSG plasticity theory (Gaoet al., 1999; Huang et al., 2000a,b) to investigate fracture around a stationary mode-I crack tip field. Due to the strain gradient effect, stress level around the crack tip inMSG plasticity is significantly higher than that in the calssical plasticity, i.e. theFig. 4. The normalized torque, T/(2R 0 2 h Y ), versus the normalized twist, R 0 , for both finite and infinitesimaldeformation theories of MSG plasticity (l/R 0 =1) and classical plasticity (l/R 0 =0), where h and R 0are the thickness and mean radius of the cylinder in the reference configuration, respectively; Y is theinitial yield stress, l is the intrinsic material length for MSG plasticity. Plasticity work hardening exponentN=0.2, Young’s modulus E=500 Y , and Poisson’s ratio =0.3.

248 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251HRR field (Hutchinson, 1968; Rice and Rosengren, 1968). The numerical resultsalso showed that the stress around the crack tip in MSG plasticity is not only moresingular than the HRR field, but also more singular than the classical elastic K field(square-root singularity). This provides a means to explain the cleavage fractureobserved in ductile materials (e.g. Elssner et al., 1994).We use the same loading, material properties, and finite element mesh as Jiang etal. (2001) but the finite deformation effect is accounted for. We take a circular domainof radius 10 3 l centered at the crack tip in our plane-strain finite element analysis,where l is the intrinsic material length in (14). The origin of the coordinate systemcoincides with the crack tip, while the traction-free crack faces coincide with thenegative x 1 axis. The classical mode-I elastic K field is imposed on the outer boundaryof the domain (10 3 l) with the elastic stress intensity K I increasing monotonically.The stress intensity factor K I is normalized by Y l 1/2 in the numerical analysis,where Y =0.2% E is the initial yield stress, E is the Young’s modulus, and l is theintrinsic material length in (14). The stress is normalized by Y , while the position(coordinate) is normalized by l. Other material properties include the plastic workhardening exponent N=0.2, Poisson’s ratio =0.3, and the reference stress in (13).The empirical material constant in the Taylor dislocation model and the Burgersvector b only appear through the intrinsic material length l in (14), and it is thereforenot necessary to specify the values of and b for the above normalizations.Fig 5 shows the distribution of the normalized stress ahead of the mode-I cracktip, i.e. / Y , versus the normalized distance r/l to the crack tip, where is theCauchy (true) stress along =1.014 ( =180 being the crack faces), and r is thedistance to the crack tip in the reference (undeformed) configuration. The remoteapplied stress intensity factor is K I =10 Y l 1/2 . The distribution of for infinitesimaldeformation MSG plasticity theory (Jiang et al., 2001) and the correspondingcurves for classical plasticity (i.e. without the strain gradient effect) are also shownfor comparison. For r>10l, all curves become the same straight line with the slopeof 1/2, corresponding to the elastic K I field (since the slope represents the order ofsingularity in the log–log plot). For r between 0.3l and 10l, all curves still coincidebut they deviate significantly from the straight line (of the slope 1/2). This indicatesthat the material undergoes significant plastic deformation, though neither thestrain gradient effect nor the finite deformation effect is important in this range.Once the distance r to the crack tip is less than 0.3l, the normal stresses predictedby MSG plasticity (for both finite and infinitesimal deformation) increase muchquicker than their counterparts in classical plasticity. A representative value of theintrinsic material length is l=5 mm [Fleck et al., 1994; Stolken and Evans, 1998; Eq.(14) with =0.5 and b=0.3nm]. At a distance r=0.02l=0.1 mm to the crack tip,which is still within the intended range of applications for MSG plasticity (Gao etal., 1999; Huang et al., 2000b), the normal stress predicted by MSG plasticity isapproximately twice of that for classical plasticity. For infinitesimal deformationtheories, the curve for classical plasticity has the slope of N/(N+1) and correspondsto the HRR field (Hutchinson, 1968; Rice and Rosengren, 1968), while thecurve for MSG plasticity has the slope higher than 1/2 (in absolute value), indicatingthe crack tip field in MSG plasticity is more singular than not only the HRR field,

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 249Fig. 5. The Cauchy (true) stress normalized by the initial yield stress Y versus the normalized distanceto the crack tip, r/l, ahead of the crack tip (polar angle =1.014 ), where l is the intrinsic materiallength for MSG plasticity, the plastic work hardening exponent N=0.2, Poisson’s ratio =0.3, the ratioof yield stress to elastic modulus Y /E=0.2%, and the remotely applied elastic stress intensity factor K I / Y l 1/2 =10. The results are presented for both finite and infinitesimal deformation of MSG plasticity andclassical plasticity.but also the classical elastic K field. The effect of finite deformation does not seem tobe significant in Fig. 5, as seen from the comparison between curves for infinitesimaland finite deformation. This is because we have taken a relatively small appliedstress intensity factor (K I =10 Y l 1/2 ). In fact, the strain is only a few percent (3%)at a distance of 0.1 mm (r=0.02l) to the crack tip. It can be established from Fig. 4that the strain level needs to be significantly higher (>10%) in order to display thedifference between the finite and infinitesimal deformation. We have indeed imposeda much larger applied stress intensity factor, but the finite deformation analysis hassome numerical difficulties in convergence.6. SummaryWe have developed a finite deformation theory of mechanism-based strain gradient(MSG) plasticity based on the Taylor dislocation model. The theory ensuresthe proper decomposition of deformation into volumetric and deviatoric parts suchthat the volumetric deformation does not contribute to the deviatoric strain gradient,which represents the density of geometrically necessary dislocations. We haveconducted the analytic study for a thin cylinder under large torsion, and establishedthat the effect of finite deformation is significant when R 0 >0.1, where is the twist

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